Paper Title: Non-Noetherian Conformal Cheshire Effect.






eloy.ayon-beato-at-cinvestav.mx 
Departamento de Física, CINVESTAV-IPN, A.P. 14740, C.P. 07360, Ciudad de México, México


hassaine-at-inst-mat.utalca.cl 
Instituto de Matemática, Universidad de Talca, Casilla 747, Talca, Chile


pedro.sanchez.s-at-cinvestav.mx 
Departamento de Física, CINVESTAV-IPN, A.P. 14740, C.P. 07360, Ciudad de México, México

§ ABSTRACT 
The gravitational Cheshire effect refers to the possibility of turning off the gravitational field while still leaving an imprint of the nonminimal coupling of matter to gravity. This allows nontrivial solutions in flat spacetime for which no backreaction is possible. The effect was originally shown to manifest itself for standard nonminimal couplings, such as those allowing conventional conformally invariant scalar fields. Recently, the most general scalar field action yielding a conformally invariant second-order equation was constructed, and entails a more involved nonminimal coupling explicitly breaking the conformal invariance of the action without spoiling it in the equation. We have succeeded in fully describing the spherically symmetric stealth solutions on flat spacetime supporting the Cheshire effect within this general non-Noetherian conformal theory. The allowed configurations are divided into two branches: The first one essentially corresponds to an extension of the solutions already known for the standard Noetherian conformal theory. The second branch is only possible due to the non-Noetherian conformal contribution of the action. The complete characterization of this branch is expressed by a nonlinear first-order partial differential equation. We have found the general solution of this equation using both seemingly new and well-established mathematical tools.


Non-Noetherian conformal Cheshire effect
    Pedro A. Sánchez 0009-0008-7437-0561
    August 5, 2024
========================================






§ INTRODUCTION 
 


It is evident that nonminimal couplings to gravity, arising due to the explicit presence of curvature in matter Lagrangians, vanish in the flat spacetime limit. However, it is less expected that after taking such a limit some nonminimal contributions to the variation persist, leaving an imprint on the energy-momentum tensor. This phenomenon has been dubbed the gravitational Cheshire effect <cit.>, making a reference to the Cheshire cat in Lewis Carroll's Alice's Adventures in Wonderland, which disappears leaving behind only its grin. The effect is manifested through the existence of nontrivial solutions where the nonminimal and minimal contributions cancel each other, leading to a vanishing energy-momentum tensor in flat spacetime. In fact, these are examples of gravitational stealth configurations whose definition extends beyond flat spacetime, being genuine nontrivial solutions to Einstein's equations by simultaneously canceling both their gravitational and matter contributions <cit.>. 

The Cheshire effect has been exhibited in particular for standard conformal scalar fields <cit.> whose action is 
 
    S_s[Φ,g] = - 1/2∫d^4 x √(-g)(∇_μΦ∇^μΦ + 1/6 R Φ^2 + 2λΦ^4 ),

yielding the scalar field equation 

    Φ-1/6RΦ-4λΦ^3=0,

and energy-momentum tensor 

    T_μν^s =   ∇_μΦ∇_νΦ - ( 1/2∇_αΦ∇^αΦ + λΦ^4 ) g_μν
        + 1/6( g_μν-∇_μ∇_ν+G_μν )Φ^2.


Here, λ is the coupling constant of the conformal self-interaction and the symbols , R, and G_μν stand for the d'Alembertian operator, the scalar curvature, and the Einstein tensor, respectively. The peculiarity of this system is that action (<ref>) is invariant under the conformal transformations 

    g_μν   ↦Ω^2(x) g_μν,    Φ   ↦Ω^-1(x)Φ.

In general, satisfying such property in the action will imply that the energy-momentum tensor is traceless modulo the equation of motion. To connect with our previous discussion, the last terms related to Φ^2 in (<ref>) are due to the nonminimal coupling in the action, and it is clear that in the flat limit g_μν→η_μν these terms do not vanish. In fact, they compensate with the previous minimal contributions in order to allow the existence of flat spacetime stealths that are the direct manifestation of the conformal Cheshire effect, as shown in <cit.>.

To make our terminology clear, the stealths are defined as the nontrivial extrema of any matter action not only with respect to the matter content but also with respect to the gravitational background. The first such example was shown to arise on the static BTZ black hole in three dimensions for a scalar field with the simplest nonminimal coupling to gravity, which contains the conformal coupling (<ref>) as a particular case <cit.>.  It was later on generalized for the same theory on (A)dS spacetimes of arbitrary dimension <cit.> or even on more exotic spacetimes, but likewise holographically relevant at non-relativistic limits <cit.>. Applications of stealth configurations in the context of cosmology were also reported in Refs. <cit.>.

Scalar-tensor theories, particularly those within the Horndeski class <cit.>, have proven to be an excellent laboratory for exhibiting stealth configurations. These examples are not limited to zero mass gravitational states but also include configurations defined on black holes, such as the Schwarzschild metric <cit.>, relying on having a constant kinetic term, see also Refs. <cit.>. More recently, it has been shown that in so-called beyond Horndeski theories <cit.>, there exist stealth solutions with a non-constant kinetic term <cit.>; for a detailed discussion on stealth solutions in the context of such theories, see <cit.>. The existence of such black hole stealths, which violate certain assumptions of the standard “no-hair” theorems valid in General Relativity, could prove to be of great use to distinguish such theory from scalar-tensor ones by means of future observations. In this perspective, a stealth on the top of the Kerr black hole was constructed in <cit.>, and it turns out that its disformal extension leads to another rotating black hole solution with different physical properties from those characterizing the unique prediction of General Relativity <cit.>.

In this article, we extend the known conformal realizations of the gravitational Cheshire effect provided in <cit.> by considering the most general action principle for a scalar field whose equation of motion is of second order and conformally invariant. This theory was identified in <cit.>, inspired by seminal results of <cit.>. The most innovative contribution to the conformally invariant equation is a nonminimal coupling, initially identified in <cit.>, whose inclusion breaks the conformal symmetry of the action but not of the equation; such symmetry was dubbed non-Noetherian conformal symmetry in <cit.>. Furthermore, it stands out that this new contribution makes the scalar field equation of motion fully non-linear, in contrast with the quasilinear standard conformal equation (<ref>).
 
To begin our exploration, it is natural to ask whether the standard conformal stealth configurations on flat spacetime <cit.>, retain its stealth behavior after introducing non-Noetherian conformal contributions. This task is straightforward even without imposing any symmetry on the scalar field because these Noetherian conformal configurations obey specific separability properties, as shown in <cit.>, which are also effective for the full theory. Conversely, examining the broader stealth spectrum of the full conformal theory beyond this separability presents a major challenge. The new stealth constraints are entirely nonlinear, making the quasilinearity of the standard case exploited in <cit.> inapplicable. To advance in this new study of the conformal gravitational Cheshire effect, we are focusing on spherically symmetric configurations that are still time-dependent. Unexpectedly, this simple assumption leads to the mutual cancellation of all second-order nonlinear contributions in the relevant independent constraints. This opens the door to a new branch of the gravitational Cheshire effect beyond separability, encoded in a single nonlinear first-order PDE. Remarkably, this branch exists only for a family of nonvanishing values of the non-Noetherian conformal coupling constant.

The full characterization of the new branch demands the use of sophisticated PDE tools. Indeed, finding the general solution of a fully nonlinear first-order PDE is not as straightforward as it is for its linear and quasilinear counterparts, where the method of characteristics is enough. The standard approach in the fully nonlinear case, originally formulated by Lagrange (see <cit.> and references therein), involves finding the envelope of a so-called complete integral, subject to the condition that one of its parameters is a general function of the others. This results in an implicit representation of the general solution even in the simplest nonlinear examples.

In contrast to the sophistication of Lagrange's methodology, we introduce a seemingly novel elementary strategy here, leveraging the fact that the involved nonlinear first-order PDE can be cast into a standard Hamilton-Jacobi equation. Despite being a well-known equation in mechanics, finding its general solution is not common since a complete solution suffices to obtain the general integral of the equations of motion of the mechanical system <cit.>. In this work, we address this by exploiting the quadratic dependence on the derivatives in the obtained Hamilton-Jacobi equation, allowing its identification with the definition of an hyperbola. Imposing integrability conditions on the hyperbola parameterization gives rise to a quasilinear first-order PDE for the local parameter. Naturally, this equation can be integrated using the more conventional method of characteristics.

The paper is organized as follows. In Sec. <ref>, we briefly present the most general conformally invariant second-order equation arising from an action principle and the related theory. In Sec. <ref>, we continue by reviewing the previously known separable stealth solutions defined on flat spacetime for the standard conformal case (<ref>). Then, we analyze how these solutions prevail when the most general non-Noetherian conformal theory is considered. In Sec. <ref>, we will show that turning on the non-Noetherian conformal coupling opens the possibility of obtaining radically new spherically symmetric conformal stealths beyond separability. We manage to prove that this new branch only exists for a precise family of nonvanishing values of the related coupling constant, and is exclusively determined by a single nonlinear first-order PDE. The general solution of the latter will be found using both the seemingly new and also the well-established PDE methods outlined above. Hence, we will provide several representations that can be useful for different applications and limits. Concretely, using the aforementioned quasilinearization we found three different representations for the general solution in Subsec. <ref>, two of them are implicit as expected. Notably, coordinate freedom also provides an explicit representation in a special system adapted to the problem, which is unfortunately awkward for expressing the Minkowski metric. We also apply the standard PDE approach relying on the Lagrange technique. When used on the complete integral derived from the so-called Lagrange-Charpit method, one of the representations obtained by quasilinearization is recovered in Subsec. <ref>. Since separable solutions are also a source of complete integrals we address them in Subsec. <ref>. There, we also get the unexpected result that the previously mentioned extension of the separable Noetherian conformal stealths of <cit.> can be also used as seed in the Lagrange approach to derive the most general non-Noetherian conformal stealths. Supplementary to this, in Subsec. <ref>,  we also present the so-called singular solution not contained in the general one, and show how several particular cases emerge from the general solution. The last section is devoted to a discussion concerning our results. To be as instructive as possible, we briefly review in App. <ref> all the PDE definitions and methods in which the main results are founded. Explicit expressions for relevant conformally invariant scalars are provided in App. <ref>.




§ NON-NOETHERIAN CONFORMAL SCALAR FIELDS

 


The most general second-order scalar field equation invariant upon a conformal transformation (<ref>) and arising from an action principle in four dimensions can be compactly written as <cit.>

    Φ-1/6RΦ-Φ^3(4λ + α/2𝒢̃
    -g̃^αβ∂
    f/∂g̃^αβ +2f)=0.

Here, we denote by

    g̃_μν=Φ^2g_μν,

an auxiliary metric which is conformally invariant by construction (<ref>). Furthermore, α is a coupling constant mediating a new nonminimal coupling to gravity expressed through 𝒢̃=R̃^2-4R̃_αβR̃^αβ +R̃_αβμνR̃^αβμν, which is nothing but the Gauss-Bonnet scalar built from the auxiliary metric (<ref>), and whose explicit expression is given in Eq. (<ref>). Additionally,  f=f(g̃_μν,C̃^α_ βμν) is an arbitrary function of the auxiliary Weyl tensor, only subject to the condition

    f(g̃,0) = 0.

Since the new ingredients in Eq. (<ref>) are all constructed from the auxiliary metric g̃_μν, it is clear that this equation is again conformally invariant. We remark that (<ref>) is a fully nonlinear equation since 𝒢̃ is quadratic on the second-order highest derivatives of Φ, see (<ref>), and only for α=0 this equation becomes quasilinear by involving exclusively linear contributions of the derivatives of Φ.

As was also shown in <cit.>, the
action yielding to  (<ref>) reads
 
    S[Φ,g] =    S_s[Φ,g] + S_nN[Φ,g] 
       - 1/2∫ d^4 x √(-g) Φ^4f(Φ^2g_μν,C^α_ βμν),

where S_s [Φ,g] was given in (<ref>) and <cit.> 
    S_nN[Φ,g]=   -α/2∫d^4 x √(-g)(ln(Φ)𝒢 
    -4/Φ^2G^μν∇_μΦ∇_νΦ
       -4/Φ^3(∇_μΦ∇^μΦ)Φ
    + 2/Φ^4(∇_μΦ∇^μΦ)^2).


Here, C^α_ βμν and 𝒢 refer to the components of the Weyl tensor and the Gauss-Bonnet scalar of the background metric g_μν, respectively. Notice that the full action is parity invariant in the scalar field, Φ↦-Φ. For α = 0 and f = 0, the action reduces to the standard conformal action (<ref>). Interestingly, not only this term is conformally invariant but also the last one defining the Weyl coupling in the full action. In fact, both can be rewritten only in terms of the auxiliary metric up to boundary terms

    - 1/2∫d^4 x √(-g̃)(1/6R̃ + 2λ + f(g̃_μν,C̃^α_ βμν) )+b.t.,

where R̃ is the scalar curvature of the auxiliary metric g̃_μν, see Eq. (<ref>) for its explicit expression. This is compatible with the result proved in <cit.> (see their Eq. (9) and the related discussion) that any conformally invariant scalar density built from the scalar field and the metric must exclusively be expressed in terms of the auxiliary metric. However, this is no longer the case for the α-extension described by action (<ref>), which when expressed in terms of the auxiliary metric keeps a dependence on the scalar field 

    S_nN[Φ,g]=-S_nN[Φ^-1,g̃] +b.t.,

and hence, explicitly breaks the conformal invariance of the action. This was first carefully stressed by Fernandes <cit.> using an infinitesimal argument. In summary, the conformal symmetry of Eq. (<ref>) does not have a Noetherian origin, hence, such equation describes non-Noetherian conformal scalar fields <cit.>.[See Refs. <cit.> for the two-dimensional non-Noetherian analog.] 

The variation of action (<ref>) with respect to the metric gives the energy-momentum tensor whose full expression is reported in Refs. <cit.>. It is more compactly expressed if the scalar field is redefined according to its conformal weight

    σ = 1/Φ,

and in this work we only need its expression evaluated on flat spacetime that reads 
    T_μν =    T_μν^s,flat
    -α{ 4σ_μασ_ν^ α/σ^2
    -2[2σ_μν/σ
    -η_μν(σ/σ - σ_ασ^α/σ^2)
    ]
       ×(σ/σ - σ_βσ^β/σ^2)
    + η_μν((σ_ασ^α)^2/σ^4 - 2σ_αβσ^αβ/σ^2)},

where the indexes of σ denote partial differentiation, and T_μν^s,flat corresponds to (<ref>) with g_μν=η_μν. Importantly, we remark that there is no contribution from the Weyl coupling, a conclusion that can be extrapolated to any conformally flat spacetime.

Finally, it is relevant to stress that the full action (<ref>) retains a global scaling symmetry when Ω=const. in (<ref>). Although the class of second-order theories allowing such property is much broader, see <cit.> and more recently <cit.>. An additional effective conformal symmetry appears for spacetimes with vanishing Gauss-Bonnet scalar, since the energy-momentum tensor again becomes traceless modulo the scalar equation of motion <cit.>. 





§ NOETHERIAN CONFORMAL STEALTHS

 


The Cheshire effect for the standard conformal scalar field (<ref>) was first shown to exist in Ref. <cit.>. The related nonminimal conformal grin manifests through a flat spacetime stealth  parameterized as

    Φ(x^μ) = 1/ a x_μ x^μ + b_μ x^μ - c ,

where x^μ = (t, x^i) are the Cartesian coordinates of Minkowski spacetime and the integration constants a, b_μ, and c are tied in terms of the coupling constant of the conformal potential as

    λ = 2ac + 1/2b_μ b^μ.

As proved in <cit.>, this is the general solution to the standard stealth constraints T_μν^s,flat=0, defining a Noetherian conformal stealth. The above integration constants are determined up to isometries so they can be restricted further. Indeed, for non-vanishing a one can fix b_μ = 0 without any loss of generality by choosing an appropriate translation. In the same way, one can set c = 0 via a translation for vanishing a, while simultaneously aligning the vector b_μ through a rotation so that it has a single independent component.

Remarkably, the parameterization (<ref>) is also effective  for the full non-Noetherian conformal action (<ref>). This can be readily seen using the redefinition (<ref>). The fact that the scalar field (<ref>) prevails as flat spacetime stealth even when α0, relies in that the specific sum-separable form implied for σ is characterized by having a Hessian proportional to the metric, σ_μν = (σ/4) η_μν. This in turn forces the energy-momentum tensor (<ref>) to be also proportional to the metric, T_μν = (tr (T)/4) η_μν, while the condition tr(T) = 0 amounts to the constraint 

    d_∓≡1 ∓√(1-48αλ)/24α=2ac + 1/2 b_μ b^μ.

For vanishing α only the upper-sign root is well defined giving the finite limit d_- = λ, recovering the known standard constraint (<ref>). The last configurations solve then the
non-Noetherian conformally invariant equation (<ref>). Since they observe the same local dependence than those of the case with conformally invariant action, and only the constraint obeyed by the integration constants is generalized, we shall refer to them as Noetherian-like conformal stealths.

As we will show in the next section, the Noetherian-like conformal configurations above do not exhaust all the stealths allowed by the general action (<ref>). Unfortunately, the goal of finding the general solution, as done in <cit.> for the α=0 case, turns out to be a formidable task given that the stealth constraints that follow from (<ref>) are no longer quasilinear. Hence, in what follows we will restrict ourselves to the spherically symmetric configurations, Φ = Φ(t,r),  defined in flat spacetime 

    ds^2 = -dt^2+dr^2+r^2dΩ^2.

For the Noetherian-like conformal configurations (<ref>) this entails that b_i = 0, allowing only two different cases up to isometries, depending on whether a is vanishing or not. For the former, one has the spatially homogeneous solutions 

    Φ(t,r) = 1/√( -2 d_∓)  t.

While the latter gives rise to the Lorentz-invariant family

    Φ(t,r) = 1/ a ( -t^2 + r^2 ) - d_∓/(2a) ,

depending on the single integration constant a, which is a consequence of the scaling invariance of action (<ref>) discussed at the end of Sec. <ref>. For the special value of the non-Noetherian conformal coupling constant α = 1/(48λ), the two roots (<ref>) degenerate to d_∓ = 2 λ and one recovers the particular solutions first found in <cit.>, cf. their Eqs. (A8) and (A9). In fact, it was through these particular examples that they were the first to recognize that Noetherian conformal configurations can remain stealth in the presence of non-Noetherian conformal contributions. It is noteworthy that the non-Noetherian conformally supported flat universe reported in <cit.> is just a conformal transformation of the stealth (<ref>). However, the transformed configuration is no longer free of backreaction like the original, due to the lack of conformal symmetry in the action.

The Noetherian and Noetherian-like conformal solutions studied in this section rely in the sum-separability of the inverse of the scalar field σ (<ref>). Fortunately,  when such separability is not attainable it is not the end of the story, at least for spherically symmetric configurations. In the next section, we show the existence of a new possibility, only valid in the non-Noetherian conformal regime α0.




§ NON-NOETHERIAN CONFORMAL STEALTHS




In this section, we will show that spherically symmetric generalized conformal stealths are necessarily divided into two branches. One corresponds to the separable sector already characterized in the previous section. Hence, we will concentrate in this section on the study of the new branch, which is radically different and its existence is only ensured for α≠0. Concretely, assuming spherical symmetry on conformal stealths gives a single off-diagonal component for the energy-momentum tensor (<ref>)

    T_t^r=Sσ_tr/3σ^3=0,

where

    S ≡ 1+12α(σ_t^2-σ_r^2
    +2/r σσ_r).

It is then evident that Noetherian conformal stealths, given by α = 0, must be necessarily sum separable since the unique option is σ_tr=0. Interestingly, for α0 the vanishing of the quantity S opens a new possibility that we will now explore. The remaining nontrivial stealth constraints can be conveniently rewritten from the energy-momentum tensor (<ref>) as


    T_t^t    = 1-48αλ
    +16ασ/r(σ_r-rσ_rr)S-S^2 /48ασ^4=0,
    
    T_r^r    = 1-48αλ
    +16ασ/r(σ_r+rσ_tt)S-S^2 /48ασ^4=0,
    
    T_θ^θ   =T_ϕ^ϕ= (r^2σ T_r^r)_r+r^2[σ_r
    T_t^t-σ(T_t^r)_t]/2r( σ-rσ_r )=0.


The last expression is a consequence of the conservation of the energy-momentum tensor (<ref>) and the fact that it is traceless in flat spacetime, since both conditions are satisfied via the scalar equation (<ref>). In other words, it is a by-product of two symmetries of action (<ref>): general covariance and the already discussed effective conformal symmetry on flat spacetime.

One can then conclude from the previous equations that there are in fact new possibilities beyond the sum separability of the Noetherian-like conformal stealths determined by the conditions 

    S   =0,    αλ   = 1/48.

This is clearly a net departure from the standard stealths behavior, and an obvious manifestation of a Cheshire effect due to the non-Noetherian conformal contribution, since necessarily α≠0. We shall denote the resulting configurations as non-Noetherian conformal stealths. From definition (<ref>) it follows that conditions (<ref>) give rise to a nonlinear first-order PDE. In what follows we will show how to obtain its general solution using different but equivalent approaches. We start with an elementary and straightforward method to quasilinearize Hamilton-Jacobi equations that does not seem to have been considered in the PDE literature. 




 §.§ Non-Noetherian from Hamilton-Jacobi 
quasilinearization 



If one rewrites the inverse scalar field (<ref>) as

    σ = 2√(|λ|)  r u,

equations (<ref>) with definition (<ref>) yield

    ϵ S/r^2=u_t^2-u_r^2+u^2+ϵ/r^2=0,

where ϵ=±1 is the sign of the non-Noetherian conformal coupling constant α=ϵ|α|. Redefining again the involved function u as

    u=sinhΘ,    α>0, 
    coshΘ,    α<0,

we obtain in both cases the following Hamilton-Jacobi equation

    Θ_r^2-Θ_t^2=1/r^2.

This equation defines a hyperbola that can be locally parameterized by a function H=H(t,r) as

    Θ_r   =cosh H/r,    Θ_t   =sinh H/r.

Such parameterization must be subject to the integrability condition ∂_tΘ_r=∂_rΘ_t, that implies the function H obeys the following inhomogeneous quasilinear first-order PDE

    X(H)   =sinh H/r,     
    X   ≡-sinh H∂_t+cosh H∂_r.

In other words, we have managed to transform the original fully nonlinear problem defined by Eq. (<ref>) to a quasilinear first-order PDE. The general solution to the latter can be found by integrating its characteristic system, see Eq. (<ref>), which is

    -dt/sinh H=dr/cosh H=r dH/sinh H.

The equation corresponding to the second equality is separable 

    dr/r=dsinh H/sinh H,

which implies that the following function is a first integral of the system, namely

    Ω   = sinh H/r,     X(Ω)    = 0.

As is custom in the treatment of quasilinear PDE, it is advantageous to use this integral as new coordinate

    (t,r) ↦( t,Ω=sinh H(t,r)/r).

Using these coordinates, the vector field becomes

    X=X(t)∂_t+X(Ω)∂_Ω=-sinh H∂_t,

and the PDE given in (<ref>) takes the simple form

    ∂_t(cosh H)=-Ω.

This allows its straightforward integration

    cosh H(t,Ω)=-Ω t +Ω h'(Ω),

depending on an arbitrary function of the invariant Ω, appropriately chosen for later convenience. In terms of the original coordinates, this means that the most general function H(t,r) satisfying the inhomogeneous quasilinear first-order PDE (<ref>) is implicitly given in terms of an arbitrary function by

    t+r H=h'(sinh H/r).

Equivalently, the most general function Ω(t,r) satisfying the homogeneous quasilinear first-order PDE involved in (<ref>) is given by the arbitrary implicit dependence

    t+1/Ω√(1+Ω^2r^2)=h'(Ω).


Summarizing, the starting system (<ref>) is only integrable for the found dependence of the local hyperbolic parameter H, expressed either in terms of the original or the adapted coordinates. This can be corroborated by writing the differential of the function Θ in terms of the new coordinates (<ref>)

    dΘ   =sinh H/r dt + cosh H/r dr
       =Ω dt + Ωcosh H/sinh H
    d(sinh H/Ω),

or 

    dΘ - cosh^2 H/sinh HdH = Ω dt - cosh H/ΩdΩ.

Using now the found dependence (<ref>) in the right-hand side yields

    d(Θ - cosh H - lnsinh H/cosh H + 1 - Ω t + h )=0.


We are now in a position to enunciate the main result of this subsection: the general solution to the Hamilton-Jacobi equation (<ref>) is given by


    Θ(t,r)=   t/rsinh H + cosh H 
    - arcsinh( 1/sinh H) 
        - h(sinh H/r),
    
    0=   t+r H-h'(sinh H/r),


where h is an arbitrary single-argument function in terms of which the hyperbolic parameter determining the solution is implicitly defined as a local function H(t,r) from the condition (<ref>). Notice that the integration constant appearing in (<ref>) can be absorbed into the function h, which is defined modulo a constant in (<ref>). An equivalent second statement is: the general solution to the Hamilton-Jacobi equation (<ref>) can be also expressed as
 

    Θ(t,r) =   Ω t + √(1+Ω^2r^2)
    - arcsinh( 1/Ω r) 
        - h(Ω),
    
    0 =    t+√(1+Ω^2r^2)/Ω-h'(Ω),


where the solution is now determined by the function Ω(t,r), which is defined by the arbitrary implicit dependence implied from condition (<ref>). This last expression for the general solution turns out to be just the one resulting from applying the procedure devised by Lagrange to the complete integral obtained from the Lagrange-Charpit method, as we will show in the next section.

Moreover, there exists a third possibility allowing an explicit expression for the general solution by using the integral Ω as a coordinate adapted to the problem. Concretely, using (<ref>), (<ref>) and (<ref>), the general solution to the Hamilton-Jacobi equation (<ref>) also allows the explicit expression


    Θ(t,Ω) =arcsinh(1/Ω [ h'(Ω) - t ]) 
    + Ω h'(Ω) - h(Ω),

after changing the radial coordinate according to

    r = √(Ω^2[h'(Ω)-t]^2-1)/Ω.


The drawback of this explicit expression for the general solution is that in such coordinates the flat spacetime metric (<ref>) acquires a very unsuitable form.

Finally, the previous results allow to conclude that a non-Noetherian conformal Cheshire effect is possible thanks to the existence of a new α-grin whose most general time-dependent spherically symmetric stealth is determined by


    Φ   =12√(|λ|)rsinhΘ,      λ>0, 
    
    12√(|λ|)rcoshΘ,      λ<0,
          
    α   =1/48λ,


where the function Θ allows either of the two implicit representations (<ref>) and (<ref>), or even the explicit representation (<ref>).




 §.§ Non-Noetherian from Lagrange-Charpit 




It is also possible to tackle the full characterization of the spherically symmetric non-Noetherian conformal Cheshire effect by directly dealing with the PDE (<ref>). However, since this equation is no longer quasilinear as the obtained for H in the previous subsection, the integration process leading to the general solution is less straightforward. It is then needed to appeal to the Lagrange approach reviewed in App. <ref>, where the general solution is constructed starting from a complete integral. Here we will apply such procedure to the complete integral for u obtained using the Lagrange-Charpit method summarized in App. <ref>.

We concretely use the version (<ref>-<ref>) of the nonlinear first-order PDE that can be written as

    F(t,r,u,p,q) ≡ p^2-q^2+u^2+ϵ/r^2=0,

where we have defined p=u_t and q=u_r. From Eq. (<ref>), the
associated Lagrange-Charpit characteristic system reads 
    dt/2p=-dr/2q=du/2(p^2-q^2)
    =-r^2dp/2up=-r^3dq/2(ruq-u^2-ϵ).

Using the original Eq. (<ref>) in the third equality, we
obtain the separable equation

    udu/u^2+ϵ =dp/p,

whose straightforward integration yields

    H(t,r,u,p,q)≡p/√(u^2+ϵ)=c_1=const.

Isolating the first derivatives from (<ref>) and
(<ref>), we obtain

    p=c_1√(u^2+ϵ),      q=√((u^2+ϵ)
    (1+c_1^2r^2))/r,

which allows to write du=pdt+qdr as

    du/√(u^2+ϵ)=c_1dt+√(1+c_1^2r^2)dr/r.

The above Pfaff equation is integrable for the following potential 
    d[ln(√(1+c_1^2r^2)+1/c_1r(u+√(u^2+ϵ)))
    -c_1t-√(1+c_1^2r^2)]=0,

from where the complete integral can be finally isolated using the redefinitions (<ref>) with 
    Θ(t,r;c_1,c_2)=c_1t + √(1+c_1^2r^2) 
    - arcsinh(1/c_1r) - c_2.


We now apply the Lagrange procedure to derive the general solution, starting from the above complete integral obtained via the Lagrange-Charpit method. 
Following App. <ref>, the first step is to impose in the complete integral (<ref>) with (<ref>), that is c_2 = h(c_1). Differentiating the result with respect to the remaining parameter, the next envelope condition is obtained 

    ũ_1=√(u^2+ϵ)(t+√(1+c_1^2r^2)/c_1-h'(c_1))=0.

The above condition can be explicitly solved for the parameter only for precise elections of the function h, defining its coordinate dependence c_1=c_1(t,r). On the contrary, for an arbitrary function such dependence can only be defined implicitly. Consequently, according to the Lagrange procedure the general solution to the nonlinear first-order PDE (<ref>), characterizing the spherically symmetric non-Noetherian conformal Cheshire effect, allows the following parametric representation in (<ref>)  

    Θ(t,r)    = c_1t + √(1+c_1^2r^2)
    - arcsinh(1/c_1r) 
    - h(c_1),
    
    0    = t+√(1+c_1^2r^2)/c_1-h'(c_1),


where c_1(t,r) is a function implicitly determined by the second condition. This is precisely the representation (<ref>) obtained in the previous subsection following the quasilinearization procedure and parameterized by the function Ω, which is just the local dependence of c_1 in the present context. Consistently, both approaches give rise to the same representation of the general spherically symmetric non-Noetherian stealth configuration (<ref>).




 §.§ Non-Noetherian from Noetherian 




As shown at the beginning of the section, conformal stealths with spherical symmetry in flat spacetime are divided into two branches: the Noetherian-like and the non-Noetherian ones. Surprisingly enough, it happens that the two classes of solutions intersect when αλ=1/48. Indeed, it is known that other source of complete integrals for nonlinear first-order PDE are the separable solutions; in the case of (<ref>) its most general sum-separable solution is

    σ(t,r;a,τ) = a [ -( t-τ )^2 + r^2 ] - λ/a,

which is nothing other than the Noetherian-like conformal stealth (<ref>) at α=1/(48λ), before performing the time translation to make b_0 vanish in (<ref>). In other words, the Noetherian-like conformal stealths also provide a complete integral for the nonlinear first-order PDE (<ref>) describing non-Noetherian conformal stealths!

With the different complete integral (<ref>), we can apply the Lagrange procedure again to obtain another parameterization of the general solution to the PDE (<ref>). Making that the original parameter arbitrarily depends on the reinstalled time-translation one, a = a(τ), and following App. <ref> again, it is obtained that the general solution can be also represented as


    σ(t,r)    = a(τ) [ -( t-τ )^2 + r^2 ] - λ/a(τ),
    
    a'(τ)    = -2 (t-τ) a^3(τ)/λ + [ -( t-τ )^2 + r^2 ] a^2(τ),


where the function τ(t,r) is implicitly defined by the last condition.

The above representation of the general solution is linked to the one given in terms of Θ in (<ref>) in the following way. First, consider a Legendre transform of the arbitrary function of the representation (<ref>)

    Z(z)   =Ω z - h(Ω),    
    z   =h'(Ω),

where, as usual, the last relation is used to solve for Ω=Ω(z). Then the arbitrary function of the representation (<ref>) and its argument are defined by the following map 
    (z,Z) ↦(
    τ = z - tanh(Z)/Ω(z) ,
     a = √(λ) Ω(z) cosh(Z) ),

when λ > 0. For λ < 0, the same map holds if λ is replaced by its absolute value as well as sinh and cosh are interchanged in every occurrence.





 §.§ Singular and particular solutions 




For λ < 0, there exists a solution, not contained in the general one, and called singular solution, cf. App. <ref>. Solving conditions (<ref>) for the equation (<ref>), using either its version (<ref>) or (<ref>), one obtains that the singular solution is given by

    σ(r) = ± 2 √(-λ)  r.

The singular solution is not only spherically symmetric but also static, properties that are incompatible for the Noetherian-like stealths considered in Sec. <ref>.[It may seem like a linear radial dependence as (<ref>) could cause a bad behavior on the angular stealth constraints (<ref>). But this is not the case; it is merely an artifact of how those components are expressed in terms of the others.]

Let us now turn our attention to those solutions which do arise as particular cases of the general solution for spherically symmetric non-Noetherian conformal stealths. The first question to answer is if there are more static solutions. In fact, choosing a(τ)=A=const. in (<ref>) gives τ = t and the solution becomes 

    σ(r) = A r^2 - λ/A.

This is another one-parameter class of static solutions, which in contrast to the singular one is defined for any sign of λ, as long as it is related to α through (<ref>). Remarkably, the previous static stealths can be also obtained as the flat spacetime limit of the second branch of static and spherically symmetric non-Noetherian conformally dressed black holes found by Fernandes in <cit.> precisely for αλ=1/48. These configurations were originally derived in <cit.> following such approach, cf. their Eq. (A4). In contrast, the first branch of black holes given in <cit.>, which exists for the different couplings relation αλ=1/144, does not have a well-defined flat spacetime limit. That sheds some light on the relevance of the couplings special tuning (<ref>) for the manifestation of a non-Noetherian conformal Cheshire effect. The previous two static solutions turn out to be the only ones within the spherically symmetric stealths giving rise to the conformal Cheshire effect. In fact, if one considers σ = σ(r), the PDE (<ref>) becomes a quadratic first-order ordinary equation whose two roots straightforwardly yield (<ref>) and (<ref>). 

Another interesting particular solution is obtained if one chooses a(τ) = B/τ, for which (<ref>) yields

    σ(t,r) = 2 ( B t ±√( (B^2+λ) (t^2-r^2) )).

Spatially homogeneous stealths can be obtained from the previous one by setting B = ±√(-λ) for λ < 0. They coincide with the spatially-homogeneous Noetherian-like conformal stealths (<ref>) when αλ=1/48,  first obtained in <cit.>. It is straightforward to check from PDE  (<ref>) that these are the only allowed spatially homogeneous configurations. On the other hand, taking B = 0 a new Lorentz-invariant solution different from the Noetherian-like conformal stealth (<ref>) is obtained 

    σ(t,r) = ± 2√(λ( t^2-r^2 )),

valid inside or outside the light cone for positive or negative λ, respectively. By similar arguments to those used in the static case, the above solutions and  (<ref>) are the only Lorentz-invariant stealths with spherical symmetry related to the conformal Cheshire effect.





§ CONCLUSIONS 



In this work, we have explicitly shown the existence of a non-Noetherian conformal Cheshire effect for scalar fields obeying the most general second-order conformally invariant equation derived from an action principle. 

We started by showing that the configurations supporting the original conformal Cheshire effect <cit.>, characterized by a sum separable inverse of the scalar field, are also  stealth solutions for the full theory. The original restriction of the separability constants in terms of the self-interaction coupling constant λ was generalized to involve also the non-Noetherian coupling constant α. Being the latter coupling the only one responsible for breaking the conformal invariance of the action, while preserving that of the scalar equation, we have dubbed such configurations Noetherian-like conformal stealths. 

Furthermore, we have shown that the presence of the α-coupling is responsible for the manifestation of a new conformal Cheshire effect, which is strictly non-Noetherian in the sense that it only arises for α≠ 0. We arrived to that conclusion by thoroughly studying the spherically symmetric stealths of the whole theory in flat spacetime. It results that the introduction of the α-terms  allow a new branch of stealth solutions beyond the separability of the Noetherian-like conformal configurations. We managed to prove that this new sector is ruled by a single nonlinear first-order PDE. Additionally, it only occurs for the tuning αλ=1/48 between the couplings, which in particular requires both to be non-vanishing. We have called the resulting configurations non-Noetherian conformal stealths. We fully characterized these latter using different but equivalent methods, with results related at most by a combination of Legendre and point transformations.

When a coordinate system conventionally adapted to the flat spacetime metric was used, implicit representations of the general solution were obtained as it is common in the study of nonlinear first-order PDE. Unexpectedly, an explicit representation was also obtained in a coordinate system adapted to the problem, but causing the flat metric to be represented in an unconventional manner

Another interesting result is that the generic Noetherian-like solutions are just the sum-separable sector of the nonlinear first-order PDE. An outstanding lesson is that, by avoiding a premature use of isometries, these solutions have enough integration constants to serve as a seed for generating the general solution describing the full non-Noetherian sector.

We explicitly show how to recover special solutions from the general picture, including the static ones, those that are spatially homogeneous, and finally the Lorentz-invariant examples. In particular, it results that there are only two static cases, and the remaining infinite family of new conformal stealths indeed overfly Minkowski spacetime without causing even the slightest backreaction.

It was pointed out that one of the particular non-Noetherian static stealths coincides with the one obtained in <cit.> also for αλ=1/48, as the flat spacetime limit of the static and spherically symmetric conformally dressed black hole found in <cit.> for Einstein gravity coupled to the theory (<ref>) without Weyl coupling, i.e. f=0. The exhaustive analysis of this work unveils that this was just the tip of the iceberg, and that an infinite class of configurations exists for the same couplings restriction. 

Hence, the emergence of the same couplings relation both for the non-Noetherian conformal Cheshire effect and for non-Noetherian conformally dressed black hole solutions should not be considered as a mere accident, but rather due to some deeper physical reasons that deserve future exploration. For the moment, we have provided further evidence that specific nontrivial non-Noetherian conformal couplings are favored by allowing new interesting physical configurations that are inconceivable in the Noetherian conformal case we are used to.


We would like to thank D. Flores-Alfonso, U. Hernandez-Vera, Julio A. Méndez, T. Simon, D. Uribe for useful discussions and reviewing this work. This work has been partially funded by Conahcyt grant A1-S-11548 and FONDECYT grant 1210889. P.A.S. work was supported by Conahcyt “Estancias Posdoctorales por México” contract I1200/320/2022.







§ NONLINEAR FIRST-ORDER PDE 




In order to make this work self-contained, in this appendix we will briefly summarize general methods to solve nonlinear first-order PDE heavily used in the main text. They can be found in the PDE literature <cit.>, see also <cit.> and references therein for some historical remarks. 

We consider a general nonlinear first-order PDE

    F(t, r, u, p, q)=0,

where p=u_t and q=u_r. There are different types of solutions, or integrals, allowed by the PDE. A general solution is the one given in terms of an arbitrary single-argument function. Additionally, a complete integral is a solution involving two arbitrary constants which cannot be combined into a single one.[The previous definitions are generalized to the case of n independent variables in the following way: a general solution is one given in terms of an arbitrary function with n-1 arguments and a complete integral is one involving n arbitrary constants and no less.] On the contrary, a solution that involves neither an arbitrary function nor arbitrary constants is called a particular solution if it is contained as a particular case of a general or complete solution, and a singular solution otherwise. The general and singular integrals can be obtained from a complete integral using envelopes; an approach to eliminate constants and obtain other solutions. The standard procedure to find the general solution of a nonlinear first-order PDE is due to Lagrange, see <cit.>, and consists in constructing the envelope resulting from assuming that one of the integration constants of a previously found complete integral is an arbitrary function of the other constants. This is why complete integrals are relevant in the context of nonlinear first-order PDE, and an established way to build them is the Lagrange-Charpit method. In the following sub-appendices we will review these methods to obtain complete, general, and singular solutions.




 §.§ The Lagrange-Charpit method 



The Lagrange-Charpit method is a systematic approach to determining complete integrals. The method to find the general solution of a quasilinear first-order PDE was originally given by Lagrange and rests in the integration of the associated characteristic ordinary system, see (<ref>) below. Charpit introduced a generalized characteristic system for fully nonlinear first-order PDE, providing a method to find a complete integral. Such characteristic system is defined by

    dt/F_p=dr/F_q=du/pF_p+qF_q
    =-dp/pF_u+F_t=-dq/qF_u+F_r.

From this system, one needs to find a first integral independent of (<ref>), let us say

    H(t,r,u,p,q)=c_1=const.

Then, isolating the first derivatives from both equations
(<ref>) and (<ref>) should give the explicit expressions

    p=p(t,r,u;c_1),      q=q(t,r,u;c_1).

The characteristic system (<ref>) warrants that the one-form

    du-p(t,r,u;c_1)dt-q(t,r,u;c_1)dr=0,

satisfies its integrability conditions, then it is possible to straightforwardly integrate the first-order quasilinear PDE system

    u_t=p(t,r,u;c_1),      u_r=q(t,r,u;c_1),

to obtain the desired complete integral

    u=u(t,r;c_1,c_2).


In the particular case in which the original PDE (<ref>) is quasilinear, namely given by

    a_1(t,r,u) p + a_2(t,r,u) q = a_0(t,r,u),

it is enough to consider only the first two equations of the characteristic system (<ref>), since they become the following subsystem for t, r, and u

    dt/a_1 = dr/a_2 = du/a_0.

Correspondingly, the previous two equations allow for two first integrals G(t,r,u) and H(t,r,u), but the two-dimensional nature of (<ref>) implies that they are not independent. Hence, the general solution u = u(t,r) is determined from

    G = h(H),

where h is an arbitrary single-argument function. This is precisely the method used in Subsec. <ref> to find the general solution for the spherically symmetric non-Noetherian conformal stealths after quasilinearizing the involved Hamilton-Jacobi equation.




 §.§ Envelopes and singular solutions 




Let us now consider a solution involving one or more parameters

    u = u(x^μ; c_i),

and construct the following system by differentiating with respect to the parameters

    u_i ≡∂_c_i u(x^μ; c_i) = 0.

Provided the above algebraic system can be solved for the parameters as functions of the independent variables, c_i = c_i(x^μ), the envelope of the multiparametric solution is defined as the function 

    v(x^μ)=u(x^μ; c_i(x^μ)),

and is also a solution by the conditions (<ref>). 

It turns out that the envelope of a complete integral precisely yields a singular solution, when it exists. Alternatively, the singular solution can be obtained without appealing to a complete integral by eliminating p and q from the following system

    F    = 0,     F_p    = 0,     F_q    = 0.

These procedures are followed in Subsec. <ref> to identify the singular solution involved in the problem. Envelopes are also essential in the Lagrange approach to find the general solution.




 §.§ Lagrange approach to general solutions 




Lagrange conceived the following approach in order to build the general solution of a nonlinear first-order PDE. Starting from a complete integral, u = u(t, r; c_1, c_2), consider that one of its parameters is an arbitrary function of the other, let say c_2=h(c_1). Then the general solution is the envelope, as defined in the previous sub-appendix, of the function

    û (t, r; c_1) ≡ u(t, r; c_1, h(c_1)).

In terms of the resulting local dependence of the parameter, c_1 = c_1(t,r), the general solution is formally expressed as

    u_gs = u(t,r;c_1(t,r),h(c_1(t,r))),

where is manifest the dependence on an arbitrary single-argument function. However, since the function h is arbitrary, such envelope cannot be explicitly found in general. Hence, the solution is parameterized in terms of the function c_1(t,r) which is determined implicitly by the envelope condition 


    u    = u(t,r;c_1,h(c_1)),
    
    0    = u_1(t,r;c_1,h(c_1)) + u_2(t,r;c_1,h(c_1)) h'.


This is the approach we follow in Subsecs. <ref> and <ref> to reach others parameterizations of the general spherically symmetric non-Noetherian conformal stealths.




§ SECOND-ORDER CONFORMALLY INVARIANT SCALARS 




For completeness, we report here conformally invariant scalars built from the auxiliary metric (<ref>), explicitly expressed in terms of the scalar field, Φ, and the background metric, g_μν. The first corresponds to the auxiliary scalar curvature 

    R̃=1/Φ^3(R-6)Φ,

and the second to the auxiliary Gauss-Bonnet scalar, which is compactly expressed as


    𝒢̃=1/Φ^4( 𝒢+∇_μ J^μ),

where

    1/8 J^μ=   1/ΦG^μν∇_νΦ
    +1/Φ^2(∇^μΦ)Φ
    -1/Φ^2(∇_νΦ)∇^μ∇^νΦ
       1/Φ^3(∇^μΦ)∇_νΦ∇^νΦ.


Here R and 𝒢 are the analogous quantities defined from the metric g_μν, being G^μν the Einstein tensor of the latter. Notice that compact expression (<ref>) is compatible with the topological nature of the Gauss-Bonnet terms. However, it has the inconvenience that seems to incorporate third-order contributions, but these actually cancel out when considering Bianchi and Ricci identities.








 Paper Title: Brane-vector dark matter and its connection to inflation and primordial gravitational waves.








nam.caohoang@phenikaa-uni.edu.vn (corresponding author) 
Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam

hung.tranngoc@phenikaa-uni.edu.vn  
Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam


§ ABSTRACT 
The scalar mode describing the fluctuation of the 3-brane (the observable universe) in a five-dimensional bulk spacetime compactified on a circle is absorbed by the Kaluza-Klein U(1) gauge field, leading to a massive brane-vector living on the 3-brane. The brane-vector can be responsible for dark matter because it is odd under a Z_2 symmetry, neutral under the Standard Model (SM) symmetries, and couples extremely weak to the SM particles due to its gravitational origin. Interestingly, the brane-vector dark matter could leave particular imprints on the cosmic microwave background (CMB) and the primordial gravitational waves. Hence, the precise measurements of the CMB and the observations of the primordial gravitational waves generated during the inflation can provide a potential way to probe the extra-dimensions and branes which are the main ingredients of string/M theory.



Brane-vector dark matter and its connection to inflation and primordial gravitational waves
    Tran N. Hung
    August 5, 2024
===========================================================================================



§ INTRODUCTION


Extra-dimensions and branes (hypersurfaces embedded in a higher-dimensional spacetime or the bulk) play a fundamental role in string/M theory which has been considered as a promising candidate for quantum gravity. String/M theory predicts that the spacetime has spatial extra-dimensions wrapped upon a compact internal space and open strings describing the non-gravitational sector with their endpoints are attached to branes. Whereas, closed strings describing the gravitational sector can propagate freely in the bulk. The low energy regime of string/M theory hence can imply a scenario that our observable universe is actually a 3-brane on which the Standard Model (SM) fields are confined, the directions transverse to the 3-brane are compact, and only the gravitational degrees of freedom and other exotic ones can move freely in the bulk <cit.>. In such a scenario, the constraints on the size of the extra-dimensions could be obtained from deviations from the usual Newton's law <cit.> and the experiments of gravitational waves (GW) <cit.>. Therefore, the experimental bounds of the size of the extra-dimensions can be relaxed, where the size of the extra-dimensions is as large as the micron scale. Interestingly, the compactified extra-dimensions can provide a possible solution to the hierarchy problem where the fundamental Planck scale can be much lower than the observed Planck scale <cit.>.

The brane tension is an important parameter that characterizes the typical size of the brane fluctuations along the extra-dimensions. If the brane tension is large enough, corresponding to the branes being too heavy, the branes can be considered as rigid objects. On the contrary, when the brane tension is much lower than the fundamental scale, the branes can fluctuate along the extra-dimensions. The fluctuations corresponding to the oscillating brane are realized as the Nambu-Goldstone bosons, also well-known as the branons, which arise as a result of the spontaneously broken isometry of the internal space due to the presence of the brane <cit.>. The branons are the physical scalar excitations in the low energy spectrum of the effective field theory generically only in the global case where the physical degrees of freedom describing the higher-dimensional gravitation are negligible. 

From the Kaluza-Klein (KK) theory, we know that the isometry of the internal space leads to the gauge transformations where the gauge fields are the off-diagonal components of the bulk metric <cit.> or the connection on the spacetime fiber bundle with the typical fiber being the internal manifold <cit.>. This implies when the isometry of the internal space is broken spontaneously by the presence of the brane, the KK gauge fields would absorb the branons as their longitudinal modes, corresponding to a gravitational Brout-Englert-Higgs mechanism. This leads to massive vector fields appearing on the brane, as well-known as brane-vector <cit.>. The mass of the brane-vector is determined by the brane tension, and hence it is a free parameter. Because the brane tension is much lower than the fundamental Planck scale for the case of the flexible branes, we expect that the mass of the brane-vector can range from the eV regime to the TeV regime. The brane-vector interacts with the SM fields via their energy-momentum tensor with the coupling determined by the observed Planck scale as a result of the fact that the brane-vector has a gravitational origin as new dynamic degrees of freedom describing the behavior of the spacetime at the short-distances <cit.>.

In this paper, we explore the brane-vector as a candidate for dark matter (DM) whose stability is guaranteed by
a Z_2 parity as a result of the 4D general coordinate invariance on the brane. The coupling of the brane-vector to the SM particles is strongly suppressed by the observed Planck scale, hence the relic abundance of the brane-vector DM could not generated by the thermal processes that appeared after the inflation. On the other hand, the brane-vector DM which couples extremely weak to the SM particles could be produced by the gravitational mechanism due to the quantum fluctuations in the inflationary stage <cit.>. The transverse polarizations of a massive vector field that couples minimally to 4D gravity are conformally invariant in analogy to the massless vector field, since its production is suppressed and could be negligible during the inflation <cit.>. In this sense, the observed relic abundance of the brane-vector DM should result from the production of its longitudinal polarization.

The Friedmann equation describing the evolution of the brane universe consists of linear and quadratic terms in terms of the energy density. The quadratic term is inversely proportional to the mass of the brane-vector or the brane tension since it plays an important role in the early universe. (Whereas, the linear term is dominant when the energy density is much smaller than the brane tension corresponding to the late time of the universe evolution.) This novel feature of the modified Friedmann equation combined with the production of the brane-vector DM during the inflation leads to a prediction of the tensor-to-scalar ratio and the reheating temperature in terms of the brane-vector mass. Note that, the reheating temperature can be estimated by the primordial GW spectrum <cit.>. In this way, the brane-vector DM could leave specific imprints on the cosmic microwave background (CMB) and the primordial GWs. Therefore, the precise measurements of the CMB as well as the future detections of the primordial GWs and the B-mode polarization in the CMB can allow us to probe the extra-dimensions and branes.

This paper is organized as follows. In Sec. <ref>, we study the four-dimensional effective field theory derived from a flexible 3-brane which propagates in a bulk spacetime compactified on a circle S^1 and on which the SM and inflaton fields are confined. We show that the low-energy spectrum of the effective theory includes a brane-vector that possesses a discrete Z_2 parity and hence could be responsible for the DM. Then, we compute the relic abundance of the brane-vector DM based on the production of its longitudinal mode during the inflation. In Sec. <ref>, we study the imprints of the brane-vector on the spectrum of the CMB and the primordial GWs carrying the information of the inflation era for two typical potentials of the inflation: i) α-attractor potential and ii) natural potential. Finally, we present a brief conclusion in Sec. <ref>.





§  BRANE-VECTOR DM


For simplicity without loss of generality, we consider the dynamics of a 3-brane propagating in a five-dimensional bulk spacetime compactified on a circle S^1. The action of the system is given by a sum of the bulk action and the brane action as follows

    S   =   S_bulk+S_brane,
    
    S_bulk   =   M^3_5/2∫ d^5X√(-G)(ℛ^(5)-2Λ_5),
    
    S_brane   =   ∫ d^4x√(-g)[-f^4+ℒ_SM+1/2g^μν∂_μϕ∂_νϕ-V(ϕ)].

In the bulk action (<ref>), M_5 is the five-dimensional Planck scale, ℛ^(5) is the Ricci scalar of the bulk spacetime, G is the determinant of the bulk metric, X^M with M=0,1,2,3,4 are the coordinates of the points on the bulk spacetime, and Λ_5 refers to the bulk cosmological constant which is necessarily negative to restore the conventionally Hubble expansion law in the low-energy regime <cit.>. In the brane action (<ref>), f is the tension of the 3-brane, the coordinates of the points on the 3-brane are parameterized by x^μ with μ=0,1,2,3, and g is the determinant of the induced metric g_μν on the 3-brane. The SM fields that are described by the Lagrangian ℒ_SM and the inflaton field ϕ are confined to the 3-brane. 

The presence of the 3-brane could be created spontaneously by a proper physical process, such as the tachyon condensation when turning on the magnetic fluxes <cit.>. This configuration should be realized as the ground state. As a result, the formation of the 3-brane would break spontaneously the local isometry of the bulk spacetime in the compactified extra dimension. As we see below, the Nambu-Goldstone (NG) mode describing the excitation of the 3-brane would be eaten by the KK gauge field, corresponding to the local U(1) isometry of the circle S^1, hence it becomes massive. 



 §.§ Dimensional reduction of the bulk action


Let us study the compactification of the five-dimensional bulk spacetime on the circle S^1 that is parametrized by an angle θ. The bulk metric can be decomposed into the four-dimensional fields as follows

    G_MN=(
    [ g_μν+ρ^2 A_μ A_ν          ρ^2 A_μ;          ρ^2 A_ν              ρ^2;                  ]
    ),

where g_μν is the four-dimensional metric tensor, A_μ is the KK gauge field corresponding to the local isometry in the fifth compactified dimension, and ρ is a scalar field whose vacuum expectation value (vev) determines physically the radius of the circle S^1. With this decomposition, we can write the bulk action (<ref>) in terms of the four-dimensional fields as follows.

    S_bulk   =   M^2_P/2⟨ρ⟩∫ d^4x√(-g)ρ(ℛ^(4)-2Λ_5-ρ^2/4F_μνF^μν+2/3∂^μρ∂_μρ/ρ^2),

where g is the determinant of the four-dimensional metric g_μν, ℛ^(4) is the four-dimensional Ricci scalar, F_μν≡∂_μ A_ν-∂_ν A_μ is the field strength tensor of the KK gauge field, and M_P is the observed Planck scale determined as M^2_P=2π⟨ρ⟩ M^3_5 with ⟨ρ⟩ denoting the vev of the scalar field ρ or the radius of the circle S^1. We can check that the gravitational action (<ref>) is invariant under the following transformation 

    θ   ⟶   θ+α(x),
    
    A_μ   ⟶   A_μ-∂_μα(x).

A set of such transformations forms the local isometry in the compactified dimension associated with a particular class of general coordinate transformations. In particular, the off-diagonal element of the bulk metric transforms as a gauge field of the U(1) local symmetry.

Reducing the bulk action (<ref>) on the internal manifold S^1, we obtain an effective action that consists of the four-dimensional Einstein gravity plus a cosmological constant coupled to the massless U(1) gauge field and the scalar field as follows

    S_4D   ⊃   ∫ d^4x√(g)[M^2_P/2(ℛ^(4)-2Λ_5)-1/4κ^2F_μνF^μν+1/2∂^μ r∂_μ r+⋯],

where we have expanded the scalar field ρ around its vev as ρ=⟨ρ⟩(1+√(3)r/2) with r to be the radion field, κ is the KK gauge coupling given as 

    κ≡√(2)⟨ρ⟩^-1/M_P,

and the ellipse refers to the terms related to the self-couplings of the radion field and its couplings to the four-dimensional metric and the KK gauge field.



 §.§ Effective action on the 3-brane


The induced metric on the 3-brane can be determined by the embedding geometry as ds^2_brane=g_μνdx^μ dx^ν where g_μν=G_MN∂_μ Y^M∂_ν Y^N with Y^M(x) being the embedding function which describes the position of the 3-brane in the bulk spacetime. In the static gauge, the embedding function is given by Y^M(x)=(x^μ,Y^θ(x)). With the ansatz of the bulk metric as given by Eq. (<ref>), the induced metric g_μν is found as follows

    g_μν   =   g_μν-ρ^2(A_μ+∂_μ Y^θ)(A_ν+∂_ν Y^θ),
       =   g_μν-⟨ρ⟩^2(A_μ+∂_μ Y^θ)(A_ν+∂_ν Y^θ)+𝒪(r).


When the 3-brane is created at a certain point Y^θ_0 in the fifth compactified dimension which minimizes the brane action (<ref>) and corresponds to the ground state. In this situation, the presence of the 3-brane breaks spontaneously the U(1) local isometry in the fifth compactified dimension. The fluctuations of the 3-brane around the ground state in the fifth compactified dimension can be parametrized as Y^θ(x)=Y^θ_0+π(x) where π(x) is the NG mode associated with the excitation of the 3-brane along the direction of the U(1) local isometry breaking. As a result, the KK gauge field would absorb this NG mode as its longitudinal mode, leading to the brane-vector. This brane-vector is expressed by the definition X_μ≡ A_μ+∂_μπ. The field strength tensor of the brane-vector is given by 

    X_μν≡∂_μ X_ν-∂_ν X_μ=F_μν.


It should be noted that the NG mode describing the fluctuation of the 3-brane in the fifth compactified dimension would be the physical excitation in the decoupling limit of the KK gauge field corresponding to take the limit κ→0. This physical excitation is well known as the branon which is massive in the situation that the U(1) global isometry in the fifth compactified dimension is realized as an approximate symmetry. Interestingly, it was indicated that the branon can be responsible for the DM candidate <cit.>. However, as indicated in Ref. <cit.>, this limit leads to the violation of the Sublattice Weak Gravity Conjecture <cit.>, the Swampland Distance Conjecture <cit.>, and the Festina Lente bound <cit.> which are recognized as the constraints of quantum gravity on the low-energy effective theories. On the other hand, this limit would belong to the Swampland which consists of the effective field theories that cannot be ultraviolet completed in quantum gravity. 

In this way, we can rewrite the induced metric on the 3-brane in terms of the four-dimensional metric and the brane-vector as follows

    g_μν=g_μν-⟨ρ⟩^2X_μ X_ν+𝒪(r).

Using this result, we can expand the square root of the determinant of the induced metric on the 3-brane as follows

    √(-g)=√(-g)(1-⟨ρ⟩^2/2X^μ X_μ+⋯),

where g refers to the determinant of the four-dimensional metric g_μν. With Eqs. (<ref>) and (<ref>), we can find an effective action on the 3-brane as follows

    S_4D   ⊃   ∫ d^4x√(-g)(-f^4+m^2_X/2κ^2X_μ X^μ+λ/κ^2X^μ X^ν T^mt_μν+ℒ_SM+1/2g^μν∂_μϕ∂_νϕ-V(ϕ)+⋯),

where m_X=√(2)f^2/M_P is the mass of the brane-vector which is acquired due to the spontaneous breaking of the U(1) local isometry in the fifth compactified dimension, T^mt_μν is the energy-momentum tensor of the matte fields (the SM and inflation fields) which is given by

    T^mt_μν=2δℒ_mt/δ g^μν-g_μνℒ_mt,

with ℒ_mt denote the Lagrangian of the matter fields living on the 3-brane, and λ=1/M^2_P is the coupling constant of the brane-vector to the matter fields.

From the actions (<ref>) and (<ref>), we observe that first, the brane-vector is odd under a discrete symmetry Z_2. This means that the effective action (<ref>) is invariant under the transformation X_μ→-X_μ, which means that the brane-vector always appears in the pairs. Hence, this discrete symmetry would guarantee the stability of the brane vector. Second, in analogy to the graviton, the brane-vector couples to the energy-momentum tensor of the matter fields with the coupling strength suppressed strongly by the observed Planck scale. This is due to the fact that the brane vector has a geometric or gravitational origin associated with the off-diagonal component of the bulk metric. Because the brane-vector is stable, neutral, and couples very weakly to the matter fields, it behaves as the DM. 



 §.§ Production of brane-vector particles during the inflation


Because the brane-vector is a massive vector field, it has three physical polarizations corresponding to two transverse components and one longitudinal component which is the NB mode describing the fluctuation of the 3-brane in the fifth compactified dimension. However, the production of the transverse polarizations of the brane-vector by the quantum fluctuations is suppressed during the inflation because the transverse polarizations are conformally invariant. This implies that the contribution of the transverse polarizations to the production of the energy density of the brane-vector can be negligible. Whereas, unlike transverse polarizations, the production of the longitudinal polarization is unsuppressed because it is not conformally invariant. Therefore, the relic abundance of the brane-vector DM which is observed today is essentially due to the production of its longitudinal polarization during the inflation.

The relic abundance of the brane-vector observed today is

    Ω_X=ρ_X/ρ_c,

where ρ_c is the critical density and ρ_X is the present energy density of the brane-vector. As discussed above, the contribution to the present energy density of the brane-vector is approximately that of its longitudinal mode and it is calculated as follows <cit.>

    ρ_X≃ρ_X,L=H^2_i/8π^2a_*k^2_*/a^3_0∫ dln kk^2/k^2_*[a_0/a_*|X_L(k,t_0)/X_0(k)|^2+a^3_0/a_*(k^2+a^2_0m^2_X)|∂_tX_L(k,t_0)/X_0(k)|^2],

where H_i refers to the Hubble parameter during the inflation, a_0 and a_* are the values of the scale factor at the present time and the moment of H=m_X, respectively, the wavenumber k_* is defined as k_*=a_*m_X. X_L(k,t_0) is the solution of the equation of motion for the longitudinal polarization of the brane-vector at the present time, given by

    (∂^2_t+3k^2+a^2m^2_X/k^2+a^2m^2_XH∂_t+k^2/a^2+m^2_X)X_L(k,t)=0,

with an initial condition at the horizon exit given as

    X_L(k,t)|_exit=X_0(k)(1-ik/aH_i)exp{ik/aH_i}.

By solving numerically Eq. (<ref>), we can find X_L(k,t) and then substitute it into the integral given in Eq. (<ref>) to obtain an approximate result which is equal to 1.2 <cit.>. As a result, we derive the energy density of the brane-vector observed at the present time as follows

    ρ_X≃3m^2_XH^2_i/20π^2(a_*/a_0)^3.

From the expansion law along the 3-brane as <cit.>

    H^2+k^2/a^2=(ρ_m+M^2_Pm^2_X/2/6M^3_5)^2+Λ_5/6,

and the relation a_*≃ a_mre√(H_mre/m_X) with a_mre and H_mre denoting the scale factor and the Hubble parameter at the time of the matter-radiation equality, we can determine the ratio a_*/a_0 in Eq. (<ref>) as

    a_*/a_0=(ρ_r,0/(√(7)-1)M^2_P)^1/41/√(m_X),

where ρ_r,0 is the energy density of the radiation at the present time. In addition, using the tensor power spectrum amplitude in the brane-world scenario as A_t≃12x_0[H_i/(2π M_P)]^2 with x_0≡2√(3H^2_i M^2_P/(2f^4)) <cit.> and the relation between the tensor power spectrum amplitude A_t and the scalar power spectrum amplitude A_s given by the tensor-to-scalar ratio r as r=A_t/A_s, we can determine the Hubble parameter H_i in terms of the inflationary quantities that can be observed in the CMB spectrum as

    H^3_i=π^2A_sr/6√(3)m_XM^2_P.

Finally, we find an expression for the relic abundance of the brane-vector DM in terms of its mass, the tensor-to-scalar ratio and the scalar power spectrum amplitude as

    Ω_Xh^2   ≃   0.16×(rA_sM^2_P/10^26GeV^2)^2/3(m_X/10^3 GeV)^7/6.


Using the constraints from both the Planck data on the observed DM relic abundance Ω_CDMh^2≃0.12 <cit.> and the 95% CL upper limit on the tensor-to-scalar ratio as r<0.056 <cit.>, we depict the allowed parameter space for the tensor-to-scalar ratio r and the brane-vector mass m_X in Fig. <ref>.

We observe that these constraints impose a lower bound on the brane-vector mass as m_X≳262.45 GeV. This bound can be transferred to an upper bound on the radius of the compactified extra-dimension. First, we need to determine the brane-vector mass in terms of the radius of the compactified extra-dimension. We observe that the expansion law on the 3-brane as given by Eq. (<ref>) would lead to the standard expansion law in the low energy limit of ρ_m≪ M^2_Pm^2_X/2 and the bulk cosmological constant satisfying 

    Λ_5=-6M^6_5/M^4_P.

Using this relation, we find an expression for the brane-vector mass in terms of ⟨ρ⟩ as follows

    m_X=√(3)/π⟨ρ⟩.

Finally, we determine the constraint on the radius of the compactified extra-dimension as ⟨ρ⟩≲4.14×10^-19 m in order for the brane-vector responsible for the DM candidate. Obviously, this upper bound satisfies the constraint coming from the fifth-force experiments <cit.> which set an upper bound ⟨ρ⟩≲𝒪(40) μm. In addition, we find that the brane-vector mass is about a few hundred GeV for the region r∈[𝒪(10^-3),𝒪(10^-2)] which is predicted in some potential inflation models and is experimentally accessible in the near future/present experiments.




§  CONNECTION OF BRANE-VECTOR DM TO INFLATION AND PRIMORDIAL GRAVITATIONAL WAVE

As discussed above, the brane-vector DM has the geometric origin by which its coupling to the SM particles would be strongly suppressed by the observed Planck scale. Hence, it is extremely hard to detect the signatures of the brane-vector DM at the colliders. However, the inflation and the primordial GWs generated during the inflation are very sensitive to the presence of the brane-vector DM. This can be seen from the non-standard expansion law given by Eq. (<ref>) with the allowed mass region of m_X≳262.45 GeV. The very different behavior of the expansion rate compared to the usual one due to the presence of the brane-vector DM becomes significant in the energy density region around/above 5.32×10^39 GeV^4. Therefore, the production of the brane-vector DM due to the quantum fluctuations during the inflation would leave imprints in the spectrum of the CMB and the primordial GWs whose accurate measurements allow us to probe the brane-vector DM and hence the compactified extra dimension and the brane universe. 

In the following, we assume that the brane-vector particles constitute almost the observed relic abundance by which the tensor-to-scalar ratio can be expressed in terms of the brane-vector mass as follows

    r≃(50.55 GeV/m_X)^7/4.

From this relation, we observe that if the tensor-to-scalar ratio is not too small, the energy density during the inflation is much larger than M^2_Pm^2_X/2. As a result, the expansion law during the inflation is given by

    H_i≃V(ϕ)/√(3)M^2_Pm_X.

The equation of motion which plays the behavior of the inflaton during the inflation in the slow-roll approximation reads

    3H_iϕ̇≃-V'(ϕ).

Also, we can determine the slow-roll parameters in terms of the inflation potential, its derivatives, and the brane-vector mass as follows

    ϵ≃M^4_Pm^2_X/V(ϕ)( V'(ϕ)/V(ϕ))^2,   η≃M^4_Pm_X^2/V(ϕ)V”(ϕ)/V(ϕ).

These slow-roll parameters are related to the scalar spectral index n_s and the tensor-to-scalar ratio r which can be measured in the CMB spectrum as

    n_s-1=-6ϵ+2η,   r=16ϵ.

The number of e-folds between the moment that the comoving mode with the wave number k exits the horizon and the end of the inflation is computed as 

    N_k   =   ∫^ϕ_e_ϕ_kH_i/ϕ̇dϕ
       =   -1/M^4_Pm^2_X∫^ϕ_e_ϕ_kV^2(ϕ)/V'(ϕ)dϕ,

where ϕ_k and ϕ_e refer to the values of the inflaton at the time of the mode k exiting the horizon and the end of the inflation, respectively. We can determine ϕ_e from ϵ≈1 which occurs at the end of the inflation. 

With the equations related to the inflation and the brane-vector DM production during the inflation, we can predict the scalar spectral index n_s and the tensor-to-scalar ratio r in terms of the number of e-folds N_k. The relation between r and N_k is given by

    r^15/7/1078.47=M^4_PGeV^2/V(ϕ_k)( V'(ϕ_k)/V(ϕ_k))^2,

where ϕ_k=ϕ_k(N_k,r) which is a function of the number of e-folds and the tensor-to-scalar ratio and can be determined from Eqs. (<ref>) and (<ref>).
Whereas, the relation between n_s and N_k is determined by 

    n_s=1-3/8r(N_k)+2η(N_k).




 §.§ α-attractor potential

The inflation with the α-attractor potential is given by <cit.>

    V(ϕ)=Λ^4[ 1-exp( -ϕ/M̅) ] ^2n,

where ϕ is the inflaton field coupled minimally to gravity on the 3-brane, Λ is related to the inflationary energy scale, M̅≡√(3α/2)M_P with α to be a parameter determining the inverse curvature of the Kähler manifold, and n is a parameter. Note that, in the case of α=1 and n=1, the α-attractor potential leads to the Starobinsky model <cit.>.

By substituting the potential (<ref>) into Eq. (<ref>), we obtain the slow-roll parameters as follows

    ϵ   =   [2nβ e^-ϕ/M̅/(1-e^-ϕ/M̅)^n+1]^2,
    η   =   (1-e^ϕ/M̅/2n)ϵ,

where

    β≡M_P^2m_X/M̅Λ^2.

At the moment that the mode k crosses the comoving Hubble horizon corresponding to X_k=exp(-ϕ_k/M̅)≪ 1, we obtain the following approximation

    X_k   ≃   √(ϵ)/2nβ,
    η   ≃   ϵ-β√(ϵ).

As a result, we can find approximate expressions for the scalar spectral index n_s and the tensor-to-scalar ratio r as follows

    ϵ   ≃   1/16[√(4(1-n_s)+β^2)-β]^2,
    
    r   ≃   [√(4(1-n_s)+β^2)-β]^2.

Note that, from Eqs. (<ref>) and (<ref>), we can express the parameter β in terms of the scalar spectral index n_s and the brane-vector mass m_X as follows

    β≃ 0.02[4(1-n_s)-958.48(GeV/m_X)^7/4](m_X/GeV)^7/8.


The value of the inflaton at the end of the inflation corresponding to ϵ≃ 1 reads

    X_e≃1+n+2nβ-√(1-n^2+4n(1+n)β+4n^2β^2)/n(1+n),

where 
X_e≡exp(-ϕ_e/M̅). Then, the number of e-folds is calculated by

    N_k=1/2nβ^2[B(1-X_k,2+2n,-1)-B(1-X_e,2+2n,-1)],

where B(z,p,q)=∫^z_0t^p-1(1-t)^q-1dt is the incomplete beta function.


In order to find the relation between N_k and n_s, we solve numerically Eq. (<ref>) for various values of the number of e-folds. The result is shown in Fig. <ref> where n is taken the values from 1 to 10 and the mass of the brane-vector DM is considered as 350 GeV and 1000 GeV.

We observe that the brane-vector DM would be consistent with the inflation which requires 50 - 60 e-folds for its high mass region. The low mass region of the brane-vector DM may be favorable to the inflation if the parameter n in the inflation potential is large enough.

By solving numerically Eq. (<ref>) and using Eq. (<ref>), we obtain the relation between the number of e-folds N_k and the mass of the brane-vector DM, as depicted in Fig. <ref>, with n=1 and n=10. Our prediction is shown by the purple bands with the scalar spectral index taking the value from 0.961 to 0.969. In adition, we show the 68% and 95% CL experimental constraints for n_s and r <cit.> corresponding to the (dashed) blue and (dashed) red contours.  

We see that the number of e-folds required by the inflation is satisfied in the high mass region of the brane-vector DM. Whereas, in order for the low mass region to meet this criterion, the parameter n in the inflation potential needs to be large enough. These are completely consistent with what is observed in Fig. <ref>.

We arrive to investigate the reheating constraint for the brane-vector DM produced during the inflation. The connection between the parameters of the inflation and those of the reheating can be obtained by the following relation 

    ρ_re/ρ_e=e^-3N_re(1+ω_re),

where ρ_re and ρ_e are the energy density at the end of reheating (re) and the end of inflation, respectively, and N_re is the number of e-folds during the reheating corresponding to the equation-of-state parameter denoted by ω_re. The energy density at the end of reheating is determined in terms of the reheating temperature T_re and the effective number of relativistic degrees of freedom g_re at the reheating epoch as ρ_re=π^2g_reT^4_re/30. With the energy density at the end of inflation given by the inflation potential at the end of the inflation V_e as ρ_e≃3V_e/2, we express the reheating temperature as follows

    T_re   =   [(45/π^2g_re)^3π^2A_s/2]^1/12(M^2_Pm_X)^1/3(50.55 GeV/m_X)^7/48e^-3/4N_re(1+ω_re)
          ×[n(n-2β)-1+√(1-n^2+4n(1+n)β+4n^2β^2)/n(n+1)]^n/2.


In order to determine N_re, we use the relation k/(a_0H_0)=a_kH_k/(a_0H_0) with k=0.002 Mpc^-1 being the pivot scale and a_k (H_k) is the scale factor (the Hubble parameter) at the moment which the mode k crosses the horizon, which leads to <cit.>

    k/a_0H_0=e^-(N_k+N_re)a_re/a_mrea_mreH_mre/a_0H_0H_k/H_mre.

From the conservation of entropy, we can find the ratio of a_re to a_0 as follows

    a_re/a_0=(43/11g_s,re)^1/3T_0/T_re,

where g_s,re is the effective number of relativistic degrees of freedom that contribute to the entropy and T_0 is the present temperature of the universe. Then, the number of e-folds at the end of reheating is given by

    N_re   =   4/1-3ω_re[67.38-N_k-ln( k/a_0H_0)-1/12ln g_re +1/3lng_re/g_s,re+1/4ln(π^2A_s/3).
          .+7/16ln(50.55 GeV/m_X)+n/2ln(n(n+1)/n(n-2β)-1+√(1-n^2+4n(1+n)β+4n^2β^2))].


From Eqs. (<ref>) and (<ref>), we can find the relation between the reheating temperature T_re and the mass of the brane-vector DM. In Fig. <ref>, we depict this relation for n=1 (the left panel) and n=10 (the right panel). The region is bounded by the solid and dashed purple curve represents our prediction with the 68% CL value of the scalar spectral index as n_s=0.965±0.004.

We see that the present observation of n_s leads to the large uncertainty for the prediction of the reheating temperature in terms of the mass of the brane-vector DM. The CMB experiments in the future which will be able to decrease the uncertainty of the n_s measurement will provide a more accurate prediction for this relation.


The spectrum of inflationary GWs <cit.>, which is a function of the frequency f, is connected to the brane-vector DM as

    Ω_GWh^2(f)   ≃   A_s/3(πΩ_mhf/a_0H_0)^2(50.55 GeV/m_X)^7/4exp{-1/8(50.55 GeV/m_X)^7/4log(f/f_0)}
          ×g_in/3.36[3.9/g_s,in]^4/3[3j_1(z_k)/z_k]^2T^2_1(f/f_eq)T^2_2(f/f_re),

where “in" denotes the moment that the mode k crosses the horizon, j_1(z) is the spherical Bessel function of the first kind, z_k≡4π f/(a_0H_0), the functions T_1,2(x) describe the change of the primordial GW spectrum at the frequency f_re≃0.26Hz× T_re/(10^7GeV) <cit.> due to the presence of the reheating and are given as follows

    T_1(x)   =   (1+1.57x+3.42x^2)^1/2,
    
    T_2(x)   =   (1-0.32x+0.99x^2)^-1/2.

In Fig. <ref>, we show the prediction of the primordial GW spectrum for various values of the brane-vector mass and the parameter n. In addition, we include the projected sensitivities of the different GW observations, represented by the colored regions, which are the future SKA <cit.>, the space-based LISA <cit.>, BBO <cit.>, DECIGO <cit.>, and Ultimate-DECIGO <cit.>.

We observe that Ultimate-DECIGO can probe the reheating temperature and the brane-vector DM in its allowed mass region if the parameter n in the inflation potential is large enough. On the contrary, if n is low, Ultimate-DECIGO only allows us to probe the low mass region of the brane-vector DM. In this sense, the observation of the primordial GWs produced during the inflation provides a potential way to probe the extra-dimensions and branes: the reheating temperature and the mass of the brane-vector DM which are determined by the spectrum of the primordial GWs are compared to the prediction which is given in Eqs. (<ref>) and (<ref>).




 §.§ Natural potential

We study the connection of the brane-vector DM to the inflation with the natural potential or the axion potential <cit.> which appears naturally in the compactifications of M/string theory <cit.> and provides a plausible solution to the strong CP problem <cit.>. The inflation potential in the natural inflation is given as follows

    V(ϕ)=Λ^4[1-cos(ϕ/f)],

where f is a parameter. Using Eq. (<ref>) with the inflation potential (<ref>), we find the slow-roll parameters as follows

    ϵ = γ^21+cos(ϕ/f)/[1-cos(ϕ/f)]^2
    η = γ^2cos(ϕ/f)/[1-cos(ϕ/f)]^2

where

    γ≡M_P^2m_X/fΛ^2.

At the horizon crossing of the mode k, corresponding to ϵ≪ 1, we find the value of the inflaton field in terms of the spectral index n_s and the brane-vector mass as follows

    cos(ϕ_k/f)≃ -1 + 4/25γ^2[5(1-n_s)-γ(γ+√((5-5n_s)/2+γ^2))]≡ X_k,

where the parameter γ can be expressed as

    γ/√(2)≃1-n_s-300(GeV/m_X)^7/4/√(1-n_s-240(GeV/m_X)^7/4).


The number of e-folds is

    N_k=1/γ^2[X_k-X_f - 2log( 1+X_k/1+X_f)  ],

where X_f≡cos(ϕ_f/f) with ϕ_f being the value of the inflaton field at the end of inflation which is determined by the condition ϵ≃ 1 leading to

    X_f≃ 1+γ/2(γ-√(γ^2+8)).


In Fig. <ref>, we show the dependence of the scalar spectral index n_s on 
the number of e-folds N_k.

We can see that the brane-vector DM is only consistent with the natural inflation in the 2σ region for the value of n_s and its low mass region around 280 GeV. However, as we see above, the low mass region of the brane-vector DM is hard to detect in the spectrum of the primordial GWs in the near future.



§ CONCLUSION


In the present paper, we have explored dark matter and its connection to the inflationary observations and the spectrum of the primordial gravitational waves in a scenario in which our observable universe is realized as a flexible 3-brane that could oscillate in a five-dimensional bulk spacetime compactified on a circle. (Such a scenario can be considered as a low-energy limit of ten-dimensional string theory which is reduced on a five-dimensional internal manifold.) The off-diagonal components of the bulk metric that is the gauge field corresponding to the U(1) isometry of the internal space would absorb the scalar mode describing the fluctuation of the 3-brane in the bulk. As a result, this leads to a massive vector field living on the 3-brane, so-called the brane-vector. Because the coupling of the brane-vector is strongly suppressed by the observed Planck scale, the observed relic abundance of the brane-vector dark matter is essentially due to the production of its longitudinal mode by the quantum fluctuations in the inflationary stage. In this way, the tensor-to-scalar ratio is determined in terms of the brane-vector mass or the brane tension. In addition, the mass of the brane-vector enters into the Friedmann equation describing the evolution of the brane universe, which is very sensitive with respect to the region of the high energy density such as the inflation scale. Therefore, we have studied the imprints of the brane-vector dark matter in the inflationary observables and the spectrum of the primordial gravitational waves. This can open a promising observation window to detect the extra-dimensions and branes which play a fundamental role in string/M theory.


99

Horava1996 P. Horava and E. Witten, Nucl. Phys. B 475 (1996) 94.

Lukas1999 A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram, Phys. Rev. D 59, 086001 (1999).

Adelberger2003 E. G. Adelberger, B. R. Heckel, and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 53 (2003) 77.

Adelberger2009 E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl, and S. Schlamminger, Prog. Part. Nucl. Phys. 62 (2009) 102. 

Kapner2007 D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle,
and H. E. Swanson, Phys. Rev. Lett. 98, 021101 (2007).

Murata2015 J. Murata and S. Tanaka, Class. Quant. Grav. 32, 033001 (2015).


Trashorras2016 J. García-Bellido, S. Nesseris, and M. Trashorras, JCAP 07 (2016) 021.

Visinelli2018 L. Visinelli, N. Bolis, and S. Vagnozzi,	Phys. Rev. D 97, 064039 (2018).

SenGupta2018 S. Chakraborty, K. Chakravarti, S. Bose, and S. SenGupta, Phys. Rev. D 97, 104053 (2018).

Dvali1998 N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429, 263 (1998).

Nam2021 C. H. Nam, Eur. Phys. J. C 81, 1102 (2021).

Sundrum1999 R. Sundrum, Phys. Rev. D 59, 085009 (1999). 

Bailin1987 D. Bailin and A. Love, Rep. Prog. Phys. 50, 1087 (1987).

Overduin1997 J. M. Overduin and P.S. Wesson, Phys. Rep. 283, 303 (1997).

Coquereaux1998 R. Coquereaux and A. Jadczyk, World Sci. Lect. Notes Phys. 16, 1 (1988).

Nam2019 C. H. Nam, Eur. Phys. J. C 79, 384 (2019).

Dobado2001 A. Dobado and A. L. Maroto, Nucl. Phys. B 592, 203 (2001).

Kang2001 G. Kang and Y. S. Myung, Phys. Rev. D 63, 064036 (2001).

Clark2007 T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Phys. Rev. D 75, 065028 (2007).

Clark2008 T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Phys. Rev. D 78, 115004 (2008).

Clark2009 T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Phys. Lett. B 671, 383 (2009).

Nam2023 C. H. Nam, Phys. Lett. B 841, 137930 (2023).

Ford1987 L. Ford, Phys. Rev. D 35, 2955 (1987).

Lyth1998 D. H. Lyth and D. Roberts, Phys. Rev. D 57, 7120 (1998). 

Dimopoulos2006 K. Dimopoulos, Phys. Rev. D 74, 083502 (2006).

Arias2012 P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, JCAP 1206, 013 (2012). 

Graham2016 P. W. Graham, J. Mardon, and S. Rajendran, Phys. Rev. D 93, 103520 (2016).

Nakayama2008 K. Nakayama, S. Saito, Y. Suwa, and J. Yokoyama, JCAP 0806, 020 (2008).

Clifton2012 T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rept. 513 (2012) 1.

Majumdar2002 M. Majumdar and A.-C. Davis, JHEP 0203 (2002) 056.

Cembranos2003 J. A. R. Cembranos, A. Dobado, and A. L. Maroto, Phys. Rev. Lett. 90, 241301 (2003).

Reece2017 B. Heidenreich, M. Reece, and T. Rudelius, JHEP 08 (2017) 025.

Reece2018 B. Heidenreich, M. Reece, and T. Rudelius, Eur. Phys. J. C 78, 337 (2018).

Ooguri2007 H. Ooguri and C. Vafa, Nucl. Phys. B 766, 21 (2007).

Montero2020 M. Montero, T. Van Riet, and G. Venken, JHEP 01 (2020) 039.

Vafa2021 M. Montero, C. Vafa, T. Van Riet and G. Venken, JHEP 10 (2021) 009.

Binetruy2000 P. Binetruy, C. Deffayet, and D. Langlois, Nucl. Phys. B 565, 269 (2000).

Deffayet2000 P. Binetruy, C. Deffayet, U. Ellwanger, and D. Langlois, Phys. Lett. B 477, 285 (2000).

Langlois2000 D. Langlois, R. Maartens, and D. Wands, Phys. Lett. B489 (2000) 259.

Planck2020 Y. Akrami et al. (Planck collaboration), Astron. Astrophys. 641, A6 (2020).

Planck2020b Y. Akrami et al. (Planck collaboration), Astron. Astrophys. 641, A10 (2020).





Carrasco2015a J. J. M. Carrasco, R. Kallosh, and A. Linde, JHEP 10 (2015) 147.

Carrasco2015b J. J. M. Carrasco, R. Kallosh, and A. Linde, Phys. Rev. D 92, 063519 (2015).

Starobinsky1980 A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).

Liddle2003 A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003). 

Yokoyama2010  K. Nakayama and J. Yokoyama, JCAP 1001, 010 (2010).

Kuroyanagi2011 S. Kuroyanagi, K. Nakayama, and S. Saito, Phys. Rev. D 84, 123513 (2011).

SKA G. Janssen et al., PoS AASKA14, 037 (2015).

LISA LISA collaboration, Laser Interferometer Space Antenna, arXiv:1702.00786.

BBO J. Crowder and N. J. Cornish, Phys. Rev. D 72, 083005 (2005).

DEC N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett. 87, 221103 (2001).

UDEC H. Kudoh, A. Taruya, T. Hiramatsu, and Y. Himemoto, Phys. Rev. D 73, 064006 (2006).

Freese1990 K. Freese, J. A. Frieman, and A. V. Olinto, Phys. Rev. Lett. 65, 3233  (1990).

Svrcek2006 P. Svrcek and E. Witten, JHEP 06 (2006) 051.

Peccei1977 R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).






 Paper Title: Transmission Coefficient of Super-Janus Solution.



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        Transmission Coefficient of Super-Janus Solution
        
        
        
        Saba Baig,
        Andreas Karch,
        Mianqi Wang
        
         
        Theory Group, Weinberg Institute, Department of Physics 
         University of Texas, 2515 Speedway, Austin, TX 78712-1192, USA
         2em
        
        § ABSTRACT 
        We calculate the transmission coefficient of the super-Janus interface conformal field theory, both at weak and at strong coupling, where latter is described holographically as a domain-wall solution on AdS_2× S^2× M_4×Σ. Surprisingly we find perfect agreement between the free and strong coupling answer, mirroring a similar unexpected equivalence previously found for the entanglement entropy.
        
        
         
        
        § INTRODUCTION
        
        Interface conformal field theories (ICFTs) <cit.> share the same symmetry as the much more widely studied boundary conformal field theories (BCFTs), but display several novel features, which only recently have found wide appreciation both in the high energy and condensed matter communities.
        
        While ICFTs can always be mapped to BCFTs via the folding trick, they give rise to novel physical quantities that, while they of course also exist in the folded picture, only make natural sense in the ICFT: the entanglement entropy (EE) of generic regions including the interface <cit.> and the energy transmission across the interface <cit.>. In particular, in this paper the focus is on the transmission coefficient of a codimensional-1 interface dividing CFT_1 and CFT_2 with central charges c_1 and c_2, which is given by the correlation function of the stress tensor across the interface:
        
        
            𝒯=2c_LR/c_1+c_2=⟨ T^1T^2+T^1T^2⟩_1|2/⟨ (T^1+T^1)(T^2+T^2)⟩_1|2
        
        
        Together, c_eff and c_LR define two novel “central charges" governing the properties of the interface: c_LR appearing in the transmission, and c_eff as the coefficient of log divergent terms in the EE for regions ending on the interface. Some universal bounds on these two quantities have recently been derived in <cit.>. Another interesting quantity related to the EE is the usual boundary entropy or g-factor g=⟨ 0|B⟩ associated with symmetric intervals, which fold to the standard BCFT EE which is uniquely determined by g. For generic asymmetric intervals, g actually becomes a function of l_L/l_R where l_L/R stands for the length of the part of the interval on either side of the interface. Little is known about this function g(l_L/l_R). For the particular example of the bosonic Janus ICFT it has for example been worked out in <cit.> in the strong coupling limit.
        
        An interesting class of examples of ICFTs is provided by holographic ICFTs, that is ICFTs which in a suitable large N and strong coupling limit have an equivalent description in terms of classical gravity in a higher dimensional spacetime. Simple solutions to Einstein equations that have the right symmetries to correspond to the dual of an ICFT can be constructed in terms of branes <cit.>. Generically the field theory dual to these “bottom-up" constructions is not known and may not even exist. To realize a concrete dual pair, we rely on “top-down" constructions as they actually arise in string theory. The simplest such top-down ICFTs are the Janus solution and its supersymmetric generalization.
        
        The original Janus solution <cit.> describes a 4d ICFT. The CFT on both sides of the interface is given by N=4 SYM, but with different values of the coupling constant. The 2d Janus ICFT has been constructed in <cit.> and its supersymmetric generalization in <cit.>.
        
        The 2d Janus ICFT is based on a famous holographic CFT, the D1-D5 system. 
        In the free limit, the D1-D5 CFT is described by a symmetric orbifold. We start with a theory of 4N free compact scalars together with their free fermion superpartners. This gives a sigma model with a (T^4)^N/S_N target space, which we consider in the large N limit. For simplicity we are working at a special point in moduli space where T^4 is just a product of 4 circles with identical radii. This orbifold CFT is what we call the free limit. All properties of the orbifolded theory can be deduced from the free seed theory of free compact bosons and fermions. In order to have a holographic description, we need to drive the theory to strong coupling. Here the coupling corresponds to a marginal deformation that in terms of the sigma model is a blow up mode of the orbifold singularity. In the infinite coupling limit holography applies and the D1-D5 CFT is dual to type IIB supergravity on AdS_3 × S^3 × T^4<cit.>.
        
        The (bosonic) Janus ICFT now corresponds to a deformation of the D1-D5 system where the radius of the T^4 jumps across the interface taking the values r_1,2 on the right and left of the interface respectively. While this jump breaks all supersymmetries, some supersymmetries can be restored by imposing non-trivial boundary conditions across the interface for the fermions as well. This latter is the super-Janus ICFT of interest to us. 
        
        The g-factor for a symmetric interval in the bosonic Janus ICFT has been calculated at both weak and strong coupling in <cit.>. One should note that this is not a single number, but a function of r_1/r_2, the ratios of the T^4 radii on the two sides of the interface. While the two functions obtained in the two limits are definitely not identical, the difference is numerically remarkably small.
        
        In contrast, in the supersymmetric case it was shown that both g<cit.> as well as c_eff<cit.> as a function of r_1/r_2 are identical in the free theory and in the strong coupling limit, that is the CFT calculation of S=log g on the orbifold target space T^4N/S_N matches with the holographic calculation of the RT surface <cit.> in the bulk. While supersymmetry often gives rise to non-renormalization theorems, the remarkable insensitivity of these two entropic quantities to the coupling constant is, as of now, not understood.  
        
        In <cit.> a similar comparison was performed for the energy transmission coefficient, or equivalently c_LR, for the bosonic Janus ICFT. Once again, they studied the free orbifold CFTs and its holographically dual (non-susy) Janus solution of type IIB supergravity, which is valid at strong coupling. Similar to the EE calculation, they found that without supersymmetry for a generic ratio of r_1/r_2, the strongly coupled transmission coefficient 𝒯 by the holographic calculation is different, in fact strictly bigger, than that in the weakly coupling regime given by the free CFT calculation at the orbifold point. 
        
        Given that for both entropic quantities, c_eff and g, perfect agreement between weak and strong coupling answers had been found in the supersymmetric case, it is natural to ask whether this miracle repeats itself for the transmission coefficient. We set out to answer this question in this work by once again calculating this quantity in the free orbifold CFT, this time with a supersymmetric boundary condition, and its holographic dual valid at strong coupling. Amazingly enough we once again find perfect agreement between the weak and strong coupling answer for all values of r_1/r_2. All key quantities in this supersymmetric ICFT seem to be remarkably insensitive to the coupling constant of the bulk CFT, a fact which begs for a deeper understanding.
        
        This paper is organized as follows: in Section <ref> we calculate the transmission coefficient in the 2d 𝒩=(4,4) orbifold CFT, in presence of jumping radii and ℤ_2 orbifold twist operators for the free boson sector. In Section <ref> we calculate it in the bulk type IIB Janus model on AdS_2× S^2×Σ by KK reducing the 6D supergravity to a 3D toy model, and find a perfect match with the CFT result. In Section <ref> an alternative way of deriving the transmission coefficient of the super-Janus model is presented by solving the spin-2 fluctuations in AdS_2/CFT_1. Finally in Section <ref> we discuss a surprising conjecture on the relation between c_eff and c_LR for supersymmetric theories, and other future directions on the subject.
        
        
        
        § CFT CALCULATION
        
        
        To determine the transmission coefficient in the free orbifold CFT one simply calculates the transmission coefficient for 4N real scalars. As shown in <cit.>, the fact that we are orbifolding does not affect the transport of the conserved total energy. So in the bosonic theory the task is to calculate the transmission coefficient for a free boson with compactification radius r_1 and r_2 on the two sides of the interface respectively. In fact, this system allows many different conformal interface, after folding we are looking for a BCFT of two compact scalars, or equivalently all D-branes on a torus with side lengths r_1 and r_2. As argued already in <cit.>, following similar arguments for the original 4d Janus ICFT in <cit.>, the particular interface boundary conditions corresponding to Janus are uniquely obtained by varying the action of the compact free bosons  without adding any extra terms on the interface.
        
        As shown in <cit.> these particular boundary conditions fit into the general framework of bosons whose components to the left and right of the interface are related to each other by a matrix S that was analyzed already in the early work of <cit.>. To calculate the transmission coefficient in the free Janus CFT all that needed to be done is to extract the correct form of the matrix S from the boundary conditions in <cit.>.
        Here we want to redo this calculation for the supersymmetric case. Specifically, the CFT here is a 2d 𝒩=(4,4) NLSM with an interface. The free-field limit again corresponds to a (supersymmetric) sigma model where the target space is an orbifold (T^4)^N/S_N.
        
        One subtlety in the free theory is that the theory has more than one conserved stress tensor, so the analysis of <cit.> does not directly apply. In principle we could get a different transmission coefficient whether the energy comes in as fermions or as bosons: the stress tensor of free fermion and boson is separately conserved. Of course this ambiguity would disappear the moment we move away from the orbifold point which couples bosons and fermions. But we really want to calculate the transmission coefficient at the orbifold point itself. Fortunately we will see that both fermions and bosons have the same transmission coefficient in the free super-Janus ICFT, so this subtlety is immaterial. To see this note that the boundary condition for super-Janus worked out in  (6.8) of <cit.> once again has exactly the form of a generic free fermion interface also worked out in <cit.>, see eq. (3.12).[To see this one has to identify in the region to the left of the interface Ψ from the first reference with ψ_- in the second, and Ψ̃ in the first with ψ_+ in the second, whereas in the region to the right of the interface the roles of ψ_± are reversed. This flip can be attributed to the fact that the former reference works in the folded picture.]
        
        In the comparison one finds that the angle ϑ characterizing the interface in <cit.> is related to the radii of T^4 on the two sides by
        
            sin(2 ϑ) = 2 r_1 r_2/(r_1^2+r_2^2) .
        
        
        This is exactly what one finds in the bosonic case: 
        fermion and boson in the free super-Janus ICFT have the same transmission coefficient, which hence is equal to the transmission coefficient of total energy. That is, in free super-Janus we again get the same state-independent transmission coefficient as in the in free bosonic case:
        
            T_weak = 4/ ( r_1/r_2  +  r_2/r_1 )^2
        
        
        More generally, one can obtain a class of  ICFTs by adding in an exactly marginal deformation, which in turn gives the gluing condition at the boundary/interface <cit.>. In addition to the radii deformation above, let us consider a ℤ_2 orbifold twist deformation. It is obtained by perturbing the theory with a dimension (h,h̅)=(1,1) twist operator T_0, with an OPE of T_0(z,z̅)T_0(w,w̅)∼ 1/|z-w|^4. Under this twist, the orbifold CFTs on the two sides live on different points in their moduli space. Following <cit.> it was further argued that the orbifold twist deformation is one of the many possible Kähler moduli deformations. By counting distinct two forms on the orbifold point of a Kähler manifold (e.g. a T^4/ℤ_2 orbifold point in K_3), one can naturally extrapolate the presence of orbifold twist-fields to jumping world-sheet B fields across the interface. 
        
        Concretely, let us follow Appendix C of <cit.> and consider two free compact bosons on each side of the interface σ=0 coupling to anti-symmetric B field, with both jumping radii r_1,r_2 and B field b_1,b_2 on the left and right side of the interface. The action before folding is
        
            ∫_σ>0(r_1^2η^μν∂_μ X^i∂_ν X^i+2ϵ^μνb_1∂_μ X^1∂_ν X^2)   +∫_σ<0(r_2^2η^μν∂_μX̃^i∂_νX̃^i+2ϵ^μνb_2∂_μX̃^1∂_νX̃^2)
        
        
        Let us do a change of variables from r_1, r_2 and the jump in B fields Δ b = b_1-b_2 to parameters k, ψ and θ:
        
            Δ b=4ksinhθ,   r_1=√(2ke^ψ/coshψ),   r_2=√(2ke^-ψ/coshψ)
        
        and hence in particular
        
            r_1/r_2 = e^ψ.
        
        
        The new parameters give a simpler expression for the transmission coefficient 𝒯, and as we will see in Section <ref> they arise naturally in the bulk supergravity solution. An overall shift on the B fields will be a total derivative and leave all quantities of interest invariant.
        
        Let us do a transformation X^i→ r_1X^i,X̃^i→ r_2X̃^i as in <cit.> to normalize the energy-momentum tensor. The boundary conditions are
        
            r_1∂_σ X^i-r_2∂_σX̃^i+ϵ_ij(b_1/r_1∂_τ X^j-b_2/r_2∂_τX̃^j)=0,   i=1,2
            
                r_2∂_τ X^i=r_1∂_τX̃^i,   i=1,2
        
        
        One can then write down the scattering matrix of this 2-boson system under this boundary condition at the interface, 
        
            ([  ∂_- X^1; ∂_+ X̃^1;  ∂_- X^2; ∂_+ X̃^2;          ]) = S ([  ∂_+ X^1; ∂_- X̃^1;  ∂_+ X^2; ∂_- X̃^2;          ])
        
        
        where we only write out the 2× 2 transmission submatrix for ∂_- X^i and ∂_- X̃^i
            ([ ∂_- X^1; ∂_- X^2;         ]) = ([  2r_1r_2(r_1^2+r_2^2)/(r_1^2+r_2^2)^2+Δ b^2            2r_1r_2Δ b/(r_1^2+r_2^2)^2+Δ b^2;            2r_1r_2Δ b/(r_1^2+r_2^2)^2+Δ b^2 -2r_1r_2(r_1^2+r_2^2)/(r_1^2+r_2^2)^2+Δ b^2;                                             ])([ ∂_- X̃^1; ∂_- X̃^2;          ])
        
        
         Following <cit.>, we can fold the theory via σ→-σ (and adding a minus sign in front of the B field), and calculate the energy transmission coefficient using the above S matrix on the conformal boundary state |b⟩. Explicitly, if we set the boundary condition to be
        
            (L_m^i-S_ijL̅_-m^j)|b⟩=0,
        
        
        
        then the energy transmission coefficient is 
        
            𝒯_weak=2/c_1+c_2(⟨ 0|L_2^1L̅_2^2|b⟩/⟨ 0|b⟩+⟨ 0|L_2^2L̅_2^1|b⟩/⟨ 0|b⟩+⟨ 0|L_2^1L̅_2^4|b⟩/⟨ 0|b⟩+⟨ 0|L_2^4L̅_2^1|b⟩/⟨ 0|b⟩)
        
        
        Plugging in the S matrix leads to the result
        
            𝒯_weak=4/ ( r_1/r_2  +  r_2/r_1 )^2+Δ b^2/r_1^2r_2^2=1/cosh^2ψcosh^2θ
        
        
        § HOLOGRAPHY CALCULATION WITH KK REDUCTION
        
        In this section we consider the holographic dual of the 2D supersymmetric Janus ICFT. As mentioned above, the dual spacetime is a solution of type IIB supergravity of the form AdS_2× S^2× M_4×Σ dictated by symmetry <cit.>. The solution is supported by R-R and/or NS-NS charges. The manifold  M_4 can be T^4 or K_3, corresponding to the target space of the sigma model on the CFT side. While in the CFT calculation we focused on the special case of a T^4 made from 4 identical S^1 factors, the gravity side calculation is completely insensitive to the details of M_4. For calculational simplicity, we will only consider the case with a simple self-dual R-R three-form flux where the harmonic functions in <cit.> only have two singularities, and leave the cases of arbitrary charge deformations to future work (these would correspond to the case of F1-NS5 solutions or more general branes preserving the same amount of supersymmetry). 
        
        To calculate the transmission coefficient 𝒯, we need to study a linearized gravity wave propagating on this background, corresponding to a boundary wave incoming from (say) the left, leading to a transmitted and reflected wave due to the presence of the interface.
        In <cit.> this was done for the holographic ICFT dual to a thin-brane model. 𝒯 was related to the tension of the thin brane dividing the two parts of the spacetime. Soon after <cit.> showed that the generalization to two thin branes is almost trivial: the tensions simply add. This was generalized to the 'thick-brane' model in <cit.> where 𝒯 of the non-supersymmetric Janus solution was presented. Anytime the geometry only depends non-trivially on a single coordinate, as is in the case of an AdS_2 slicing of AdS_3, one can approximate the smooth stress tensor corresponding to the fields supporting the geometry by an array of delta functions, effectively treating the spacetime as one with multiple thin branes. In the limit that the comb of delta functions goes back to a smooth configuration one obtains the transmission coefficient of the original geometry[It would be reassuring to reproduce the formula for 𝒯 directly from studying fluctuation around the continuous geometry. This was left for future work in <cit.>. Here we assume that the answer presented there as one obtains it from taking the continuum limit of the discretized geometry is, in fact, correct.] 
        
        However a caveat with this method is that if there is non-trivial dependence on some extra compact dimensions (like in our super-Janus solution where the metric and matter fields depend non-trivial on both coordinates on Σ), the method needs to be modified. 
        To resolve this issue, in this section we will use Kaluza-Klein compactification to reduce the 10D supergravity above to a 6D supergravity in Einstein frame, and then further  to an effective 3D model, where we can compute the transmission coefficient via the thick-brane model. An alternative way of matching the higher dimensional model with the effective 3D model in the spirit of the above discussion is in Section <ref> by considering the spin-2 fluctuation wavefunction.
        
        
        
         §.§ 6D supergravity with R-R charge
        
        We consider a type IIB solution on AdS_2× S^2× M_4 over a strip Σ.  We will use the most general metric with two asymptotic AdS_3× S^3 regions, which contains five parameters θ, ψ, L, k, b. Following <cit.>, the relevant functions for our solution are two independent holomorphic functions A and B, and two independent harmonic functions H and ĥ. The complex coordinate on the strip Σ will be written as w=x+iy below, where x is non-compact and y∈ [0,π].
        
        The metric reads
        
            ds^2=f_1^2ds_AdS_2^2+f_2^2ds_S^2^2+f_3^2ds_M_4^2+ρ^2ds_Σ^2,
        
        where the dilaton and the various functions appearing in the metric are given by 
        
            e^-2ϕ   =1/4(A+A̅-(B+B̅)^2/ĥ)(A+A̅-(B-B̅)^2/ĥ)
            
                    f_3^4   =4e^-ϕĥ/A+A̅
            
                    f_1^2   =e^ϕ/2f_3^2|H|/ĥ((A+A̅)ĥ-(B-B̅)^2)
            
                    f_2^2   =e^ϕ/2f_3^2|H|/ĥ((A+A̅)ĥ-(B+B̅)^2)
            ρ^4   =e^-ϕĥ|∂_wH|^4/H^2A+A̅/|B|^4
        
        
        Let us consider the case with nonzero R-R charge and zero NS-NS charge. The generic solutions to the harmonic and holomorphic functions are
        
            H   =-iLsinh(w+ψ)+c.c.
            
                    A   =ik^2coshθ+sinhθcosh w/sinh w+ib
            
                    B   =ikcosh(w+ψ)/coshψsinh w
            ĥ   =icoshθ-sinhθcosh w/sinh w+c.c.
        
        
        
        Under (<ref>), the dilaton is 
        
            e^-2ϕ=k^4cosh^2(x+ψ) sech^2ψ+(cosh^2θ-sech^2ψ)sin^2y/(cosh x-cos ytanhθ)^2,
        
        and metric factors are
         
            ρ^4   =e^-ϕL^2/k^2cosh^2xcosh^2θ-cos^2ysinh^2θ/cosh^2(x+ψ)cosh^4ψ,
            
                     f_3^4   =e^-ϕ4/k^2cosh xcoshθ-cos ysinhθ/cosh xcoshθ+cos ysinhθ,
            
                     f_1^2   =ρ^2cosh^2(x+ψ)/cosh^2θcosh^2ψ,
            
                     f_2^2   =ρ^2(1/sin^2y+cosh^2θcosh^2ψ-1/cosh^2(x+ψ))^-1.
        
         
        The pure AdS_3× S^3 solution corresponds to θ=ψ=0. At the asymptotic regions of the strip x→±∞ near the ICFT boundary the six dimensional dilaton dual to the volume of T^4 is
        
            lim_x→±∞e^-ϕf_3^4=4k^2sech^2ψ e^∓ 2ψ
        
        
        Similarly, the asymptotic values for the combination of the axion χ and the RR four form C_4 near the boundary are
        
            lim_x→±∞e^ϕ/2f_3^2χ-4e^-ϕ/2/f_3^2  C_4=± 4ksinhθ
        
        and is dual to the ℤ_2 orbifold twist operator T_0 mentioned above. From the above two equations it is clear that the parameters k,ψ,θ are the same ones inside the variable change (<ref>).
        
        
        
        
        
        
        
        
        
         §.§ KK compactify to 3D
        
        
        Our goal is to reduce the system to an effective 3D geometry, so we can use the expression in <cit.> to derive the transmission coefficient for the system on the strip. 
        
        We can consider compactifying the above 10D metric down to 6D, keeping it in the Einstein frame. The 6D metric is AdS_2× S^2 fibred over Σ, which simplifies to <cit.>
            ds_6D^2=R^2 K(x,y)(cosh^2(x+ψ)/cosh^2ψcosh^2θds_AdS_2^2+dx^2+dy^2)+R^2sin^2y/K(x,y)ds_S^2^2
        
        where R^2=2Lcoshψcoshθ is the AdS_3 radius and
        
            K(x,y)=√(1+(cosh^2θcosh^2ψ-1)sin^2y/cosh^2(x+ψ)).
        
        For the AdS_2 metric, we will use the Poincare patch ds_AdS_2^2=(dz^2-dt^2)/z^2.
        
        The reduction so far was fairly standard as the metric was completely independent of the coordinates on T^4. The non-trivial dependence of the overall size of the T^4 on Σ as encoded in f_3 simply got absorbed into the 6D dilaton. To further reduce to 3D, we have to be very careful as, in addition to the S^2, we need to reduce on the compact part of Σ, on which all functions depend. 
        
        We start with the 6D Einstein-Hilbert action S=1/(16π G_6)∫√(-g_6)R_6, and integrate out the sphere S^2 trivially. This gives us an effective 4D Einstein-Hilbert action with x,y-dependent Newton constant S=1/16π∫ G_4(x,y)^-1√(-g_4)R_4.
        
        
        
        
        
        
        
        
        
        
        
        
        Now we further compactify along the compact y direction by integrating the above action over y∈ [0,π]. Effectively, we now have a 3D action S=1/16π∫ G_3(x)^-1√(-g_3)R_3 with a x-dependent Newton constant G_3(x), and a certain 3D metric that obeys the Einstein equation. After changing the frame for the 3D metric g_3→ g_3 G_3(x)^1/2, we got the standard Einstein-Hilbert action S=1/16π G_3 ∫√(-g_3)R_3^EH. Explicitly, after integrating out S^2 and y and gauging away G_3(x), the action becomes
        
            S=-R^4/48π G_6∫_AdS_2×ℝ_x3cosh 2(x+ψ)+cosh 2ψ+2/z^2cosh^2ψcosh^2θ.
        
        
        
        The ansatz for such a 3D solution is an AdS_2-slicing 3D effective model with warpfactor e^2A(x):
        
            ds_3D^2=R^2(e^2A(x)ds_AdS_2^2+dx^2).
        
        
        It has an Einstein-Hilbert action of
        
            S_3D=-R/16π G_3∫e^2A(2e^-2A+6A'^2+4A”)/z^2.
        
        
        Matching the above ansatz with the integrated 6D action from (<ref>) and using the KK relation of Newton constants G_6=R^3G_3/3, the solution is
        
            A(x)=log(cosh(x+ψ)/coshψcoshθ),
        
        
        and that gives us the effective 3D metric
        
            ds_3D^2=R^2(cosh^2(x+ψ)/cosh^2ψcosh^2θds_AdS_2^2+dx^2)
        
        
         §.§ 𝒯 from the 3D effective model
        
        In order to calculate the transmission coefficient from the thick brane method as in <cit.>, we do a change in coordinate x to a compact x̃, lining up with the ones used in the cited paper. Consider x̃∈(-πcoshψcoshθ/2,πcoshψcoshθ/2) where
        
            cosx̃/coshψcoshθ=1/cosh(x+ψ)
        
        
        
        
        
        
        
        
        The effective 3D model above then becomes
        
            ds_toy^2=R^2e^2A(x̃)(ds_AdS_2^2+dx̃^2)
        
        
        where
        
            e^2A(x̃)=(cosh^2ψcosh^2θ cos^2x̃/coshψcoshθ)^-1.
        
        
        
        From Eq. (2) and Eq. (17) in <cit.>, we finally obtain the strong coupling transmission coefficient of the super-Janus solution
        
            𝒯_strong=2c_LR/c_L+c_R=1/cosh^2ψcosh^2θ
        
        which, as advertised, is identical to the free field theory answer of (<ref>) with jumping radii and B field deformation.
        
        
        
        
        
        § ALTERNATIVE DERIVATION: GRAVITON REDUCTION FOR ADS_2/CFT_1
        
        Another way to derive the transmission coefficient is via graviton fluctuation as illustrated in Section 5 of <cit.>. Previously,  the reduction of the AdS graviton from a higher-dimensional solution has been done in various dimensions <cit.>. Below, we will derive the fluctuation wavefunction Ψ for the AdS_2 graviton from the 10D solution, and will indirectly read off 𝒯 from the asymptotic behavior of Ψ at the ends of the strip. 
        
        
        
         §.§ Fluctuation wavefunction in AdS_2/CFT_1
        
        We wish to obtain the spin-2 spectrum reduced from a top-down supergravity solution to AdS_2. Following <cit.>, we rewrite the perturbations of the 10D metric (<ref>) as
        
            ds^2=e^2A(w)(g̅_μν+h_μν)dX^μ dX^ν+ĝ_abdw^adw^b,
        
        
        where X^μ are the AdS_2 coordinates, w^a are for the internal manifold, and e^2A is the warpfactor. We want to find the AdS_2 graviton fluctuation of the form 
        
            h_μν(X,u)=h_μν^[tt](X|λ)Ψ(w|λ)
        
        
        where h_μν^[tt](X|λ) is the Pauli-Fierz transverse-traceless solution for a spin-2 particle in AdS with mass m^2=λ+2. For our 10D supergravity, the internal wavefunction Ψ(w|λ) is solved by <cit.>
            -1/√([ĝ])(∂_a√([ĝ])ĝ^abe^2A∂_b)Ψ=m^2Ψ,
        
        
        where [ĝ] is the determinant of ĝ_ab.
        
        
        
         We can see that the lowest mass solution has l=0 and the lowest energy Δ_4=0 for M_4. 
        
        In our metric the above ĝ is 
        
            ĝ_abdy^ady^b=f_2^2ds_S^2^2+f_3^2ds_M_4^2+ρ^2dzdz̅,
        
        
        and we will use the spherical harmonic function for Ψ as
        
            Ψ=Y_lmΨ_l(x,y).
        
        
        Plugging this ansatz into (<ref>), we have
        
            -f_1^2(δ^ij/ρ^2∂_i∂_j+2δ^ij/ρ^2(∂_ilog(f_1f_2f_3^2))∂_j-l(l+1)/f_2^2-Δ_4/f_3^2)Ψ_l(y^1,y^2)=m^2Ψ_l(y^1,y^2).
        
        
        Since we will only consider Δ_4=l=0, below we abuse the notation and use Ψ for the 0th component Ψ_0. We can substitute Ψ with a more convenient Ψ̃:
        
            Ψ̃=cosh(x+ψ)sin y/2coshψcoshθΨ,
        
        
        
        and write the equation for the l=0, Δ_4=0 component as
        
            -cosh^2(x+ψ)(∂_x^2+∂_y^2)Ψ̃=m^2cosh^2ψcosh^2θ Ψ̃.
        
        
        
        The variables are separable in this case, and imposing the Neumann boundary condition on y for Ψ, we have the general solution Ψ̃_0=∑_nξ̃_n(x)sin((2n+1)y), where ξ̃_n satisfies the Legendre equation, and has solution
        
            ξ̃_n=c_1 P_κ^2n+1(tanh (x+ψ))+c_2 Q_κ^2n+1(tanh (x+ψ)),  κ(κ+1)=m^2cosh^2ψcosh^2θ.
        
        
        For a nonzero and convergent ξ̃_n, κ has to be an integer greater than 0, and c_2=0. At first glance, the lowest massive excitation of AdS_2 is m^2=2/(cosh^2ψcosh^2θ).
        
        However, the m^2=0 excitation corresponds to the wavefunction satisfying
        
            (∂_x^2+∂_y^2)Ψ̃=0,
        
        
        which after seperating variables gives a nontrivial equation
        
            Ψ̃=∑_n(A_ncosh(2n+1)(x+ψ)+B_nsinh(2n+1)(x+ψ))sin(2n+1)y.
        
        
        The non-divergent terms for Ψ are then only the n=0 components:
        
            Ψ=A_0+B_0tanh (x+ψ).
        
        
        In principle we should be able to extract 𝒯 from this fluctuation.
        
        
        
         §.§ From Ψ to 𝒯
        
        
        The fluctuation wavefunction (<ref>) in principle encodes all the information about the fluctuation and one should be able to directly read off 𝒯 from it. A general recipe of how one would go about doing this has been outlined in <cit.>, even though the calculation hasn't been carried out explicitly as one has to deal with matching the various asymptotic forms of the metric which proves to be surprisingly cumbersome.
        
        As a workaround, we are going to pursue the following alternate strategy. As the transmission coefficient is encoded in the graviton fluctuation wavefunction (<ref>), we can simply construct a toy model in 3D that has the same fluctuation function. Explicitly, the PDE (<ref>) on x,y for the wavefunction Ψ(x,y) with mass zero is
        
            cosh^2(x+ψ)/cosh^2ψcosh^2θ(2tanh(x+ψ) ∂_x+∂_x^2+2 y ∂_y+∂_y^2)Ψ(x,y)=0.
        
        
        Its solution (<ref>) is actually the solution to an ODE eigenvalue equation on x. Namely, let us reduce the y direction of the above equation and consider the fluctuation of a 3d gravity
        
            (∂_x^2+2tanh(x+ψ) ∂_x+M^2)Ψ(x)=0.
        
        
        Recall the ansatz for 3D Einstein gravity with isometry SO(2,1) in the form of (<ref>). Analogous to the modeling in <cit.> for AdS_4-slicing of a 5d model, the fluctuation wavefunction of the transverse-traceless mode Ψ(x) of 3D gravity obeys the ODE
        
            (∂_x^2+2A'(x)∂_x-e^-2A)Ψ(x)=0
        
        
        Comparing (<ref>) with (<ref>), the warpfactor A(x) satisfies
        
            A'(x)=tanh(x+ψ).
        
        Integrating the above equation we get the warpfactor e^2A=kcosh^2(x+ψ). To determine the constant k we demand that the toy model has the same asymptotic behavior in the x→±∞ region as the original metric (<ref>). Low and behold we reach a 3D toy model with a metric that is identical to what we found by explicitly doing the KK reduction  in (<ref>). 
        
        Strictly speaking, this alternate method can be viewed as a check of the KK-reduction procedure in the previous chapter: we explicitly verified that the fluctuation equation in the full 6D metric is identical to the one of the effective 3D metric. 
        
        As before, we now can simply plug in the 3D toy model metric into the formulas of <cit.> and, of course, once more obtain (<ref>).
        
        More generally, this procedure allows us to derive the transmission coefficient 𝒯 from the graviton fluctuation wavefunction, in particular for top-down models with internal dimensions dependence where the spin-2 wavefunction happens to be separable.
        
        
        
        § DISCUSSION AND FUTURE DIRECTIONS
        
        
        In this work we demonstrated that the remarkable agreement between strongly coupled super-Janus that was previously seen in c_eff and g continues to hold for the transmission coefficient. The obvious next future direction is understanding why. Clearly novel non-renormalization theorems are at work in these supersymmetric ICFTs.
        
        Another interesting finding involves the two important quantities in an (1+1)D ICFT: entanglement entropy c_eff and transmission coefficient c_LR. They are interestingly bounded by each other <cit.>. According to the calculations in <cit.>, the entanglement entropy across the interface in the super Janus solution in Section <ref> is 
        
            c_eff=c/coshψcoshθ
        
        
        in the strong-coupling gravity calculation, and agrees with the weak-coupling limit of the dual 2d 𝒩=(4,4) orbifold ICFT[The weak-coupling CFT calculation for the entanglement entropy was given under no jumping RR modulus, i.e. θ=0, and was not yet performed in presence of the orbifold twist θ.], just like the transmission coefficient. Morever, we observe that in presence of supersymmetry, the relation between c_eff and c_LR becomes more stringent: in the case of super Janus solution where c_LR is given by (<ref>) and c_L=c_R=c, we observe a simple relation 
        
            𝒯=c_LR/c=(c_eff/c)^2.
        
        
        One can ask whether this relation holds true for most general type IIB supergravity BPS solutions in the form of (<ref>) with two poles in the harmonic functions corresponding to two asymptotic AdS_3 regions. Concretely, from the BPS equations for the metric factors f_1^2f_2^2f_3^4=H^2 we can write the general 6D metric in terms of the harmonic functions H,ĥ and holomorphic functions A,B as
        
            ds^2=ρ^2f_3^2   (H^2|B|^2/|∂_wH|^2((A+A̅)ĥ-(B+B̅)^2)ds_AdS_2^2+dx^2+dy^2)
               +|∂_wH|^2((A+A̅)ĥ-(B+B̅)^2)/|B|^2ρ^2f_3^2ds_S^2^2
        
        
        and the KK reduced 3D action after integrating the y direction is expected to have the 3D effective model in the form of
        
            ds_3D^2=R^2(k^2cosh^2(x+s) d_AdS_2^2+dx^2)
        
        
        which leads to the above relation between c_eff and c_LR. In fact, this relation has already been pointed out for supersymmetric free theories with c=3/2 across a conformal interface <cit.>. It is curious to see whether it is a general rule for supersymmetric ICFT with a conformal interface.
        
        
        Another interesting question is to further explore the role of the internal manifold. Our supergravity answer did not at all depend on the shape of the internal torus, or even whether we replace T^4 with K3 altogether. On the field theory side we limited ourselves to a very symmetric torus. The supergravity answer makes one suspect that in the field theory 𝒯 should also be independent of the shape of the internal manifold. It would be interesting to verify this explicitly.
        
        
        
        § ACKNOWLEDGEMENTS
        
        
        We'd like to thank Costas Bachas for collaboration at initial stages of this work, and him as well as Stefano Baiguerra, Shira Chapman, Giuseppe Policastro, and Tal Schwartzman for discussions of their work in <cit.>. Thanks to Ilka Brunner, Michael Gutperle, Yuya Kusuki, and Hiroshi Ooguri for comments on a draft of our manuscript.
        This work was supported in part by the U.S. Department of Energy under Grant No. DE-SC0022021 and a grant from the Simons Foundation (Grant 651678, AK). 
        
        JHEP


 Paper Title: Position space equations for generic Feynman graphs.

	


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    ∂ tr  Tr θNcoloring MITP-TH-18/24
    
    
     ITEP-TH-24/24
    
    
     IITP-TH-19/24
    
    
    
    
    
    	Position space equations for generic Feynman graphs
    
     V.Mishnyakov ^a,[victor.mishnyakov@su.se], 	A. Morozov^b,c,d,[morozov@itep.ru],	M.Reva^b,c,[reva.ma@phystech.edu]
    
    ^a Nordita, KTH Royal Institute of Technology and Stockholm University,
     
    			Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
    ^b MIPT, Dolgoprudny, 141701, Russia
    ^c NRC “Kurchatov Institute", 123182, Moscow, Russia
    ^d Institute for Information Transmission Problems, Moscow 127994, Russia
    ^e ITEP, Moscow, Russia
    
    
     Abstract
    
    We propose the extension of the position space approach to Feynman integrals from the banana family to generic Feynman diagrams. Our approach is based on getting rid of integration in position space and then writing differential equations for the products of propagators defined for any graph. We employ the so-called ”bananization” to start with simple Feynman graphs and further substituting each edge with a multiple one. We explain how the previously developed theory of banana diagrams
    can be used to describe what happens to the differential equations
    (Ward identities) on Feynman diagrams after this transformation. Our approach works for generic enough (large enough) dimension and masses. We expect that after Fourier transform our equations should be related to the Picard-Fuchs equations. Therefore, we describe the challenges of Fourier transform that arise in our approach.
    
    
    
    
    
    
    
    
    
    
    
    
    
    	
    
    
    § INTRODUCTION
    
    
    Feynman integrals of perturbative QFT are known to be notoriously hard to calculate. In recent years, however, great progress has been achieved both on the quantitative and the qualitative fronts <cit.>.  A lot of algorithmically effective tools have been developed for computing specific Feynman integrals, leading, in particular, to leaps in precision collider phenomenology. At the same time, various mathematical properties have been established both generic and for special (families) of graphs. Among these are number theoretic properties <cit.>, geometric <cit.>, Hopf-algebraic <cit.>, integrable <cit.> and others. Among others, an especially effective way to treat Feynman integrals is finding differential equations that they satisfy. These come in various shapes and forms, such as the systems coming from IPB identities <cit.>, the Picard-Fuchs equations <cit.>, GKZ-hypergeometric systems <cit.>, and the recently introduced position space ones <cit.>. Only in some cases relations between these various kinds are well understood.
    
    A common drawback of various differential equations approach is a lack of generic statements, that would be valid for any graph. In principle even the existence of these equations is not completely clear, and deriving them involves  quite a lot of non-generic choices. 
    Despite that there is progress on in that direction too. 
    
    We continue the development of the position space approach. In previous works <cit.> we were focused mostly on the so-called banana graphs, which are a certain family of multi-loop two-point functions. In the latest work we have hinted a possible generalization to other graphs. Here we would like to take a step further and propose that the position space approach allows demonstrating from very simple first principles that differential equations exist for generic Feynman integrals.
    
     Our proposal relies heavily on the idea of considering what we think is appropriate to be called unintegrated correlators, even though in practice we deal with single diagram contributions to correlators. As we explain below, this means that we consider position space expression which are products of propagators which do not have any integration over the would-be internal vertices. These functions have two key properties. They satisfy conceptually very simple differential equations (which might be quite involved technically) and in momentum space they correspond to a function, whose various evaluations correspond to different choices of integrated vertices.
     
    
    
    
    
    § THE SETTING
    
    
    Consider a graph Γ, and let it be a simple graph for now, meaning that it does not have multiple edges between any vertices. Denote V_Γ,E_Γ the set of vertices and edges. The corresponding ”Feynman integral” which may be called the ”Feynman function” is just a product of propagators over all edges:
    
        I_Γ( 𝐱 | 𝐦^2 ,D ) =∏_<ij> ∈ E_ΓG(x_i^μ-x_j^μ| m^2_ij,D)
    
    where 𝐱 is a collection of vertex coordinates x_i^μ∈ℝ^D are coordinates correspond to each vertex i ∈ V_Γ and, in principle each propagator carries its own mass m_ij^2, collectively denoted by 𝐦^2. Feynman diagrams can have multiple edges, which we could incorporate in the definition (<ref>) straight away. However, for the purposes of writing equations we would like to think of starting with a simple graph and then multiplying each edge - a procedure we call ”bananization” - meaning that we substitute each propagator with the corresponding banana function, which is just a product of propagators:
    
        B_n(x_i-x_j)= ∏_a=1^n G(x_i^μ-x_j^μ| m^2_ij,a,D)
      
    These functions I_Γ( 𝐱 | 𝐦^2 ,D ) are what we refer to as ”unintegrated correlators”. This means that any true position space Feynman integral can be obtained by declaring some edges to be internal and integrating over them.
    
    
    
    To set further set up our story consider the Fourier transform of these functions:
    
        I_Γ( 𝐱 | 𝐦^2 ,D )    = ∫∏_<ij> ∈ E_Γ d^Dp_ijexp(∑_<kj> ∈ E_Γ i p_kj· (x_k-x_j) ) ∏_<ij> ∈ E_ΓB_n(p_ij)  = 
        			
            =∫∏_<ij> ∈ E_Γ d^Dp_ijexp(∑_k ∈ V_Γ i  (x_1-x_k)·( ∑_j<k p_jk - ∑_j>k p_kj) ) ∏_<ij> ∈ E_ΓB_n(p_ij)
     
    where by p_ij we mean all momenta from i-th to j-th vertex. After a partial change of variables we separate the external from the loop momenta:
    
        k_i= ( ∑_j<k p_jk - ∑_j>k p_kj)    i =1,… |V_Γ| - 1  ,
           l_i= orthogonal complement to k_i
    
    and obtain 
    
        I_Γ( 𝐱 | 𝐦^2 ,D ) =∫∏_i=1^|V_Γ| d^D k_iexp(∑_j ∈ V_Γ i  (x_1-x_j)· k_j ) I_Γ(
        	𝐤 )
    
    where:
    
        I_Γ( 	𝐤)  = ∫∏_i=1^L_Γ d^D l_i ∏_j=1^|E_Γ| B_n_j({l,k})
    
    is a loop integral. The function I_Γ( 	𝐤) depends on the same number of independent variables as the coordinate representation, i.e. |V_Γ|(|V_Γ|-1)/2 squared momenta and their scalar products for sufficient big space-time dimension D. We will elaborate this statement in the next section. Any ”true” Feynman integral corresponds to declaring a subset of vertices to be integral. In momentum space this means that one has to set all the corresponding momenta to zero k_i =0 for i ∈ V^(int)_Γ. In this sense I_Γ( 	𝐤) it is a more general function than any of the particular loop integrals. 
    
    
    
    § ON THE CHOICE OF INDEPENDENT VARIABLES
    
    
    As usual, the Feynman functions do not depend on all the position variable x_i^μ, but on their Poincare invariants. The most obvious choice of invariants is just the lengths:
    
        X_ij=√((x_i^μ-x_j^μ)(x_i,μ-x_j,μ))
    
    not for just the edges, but for all pairs of coordinates. Note, that we use upper case for invariants and lower case for the vectors.  Hence the function depends on |V_Γ|(|V_Γ|-1)2 variables. This is true if all the variables are independent, which requires for the dimension D to be big enough. 
    It's easy to see that in D=1 for, say |V_Γ|=3 these variables are not independent. For D=2 they are, however, for |V_Γ|=4 one has to at least D=3 for the length of side of the tetrahedron to be independent variables. In general, the variables X_ij^2 are independent if the following condition is satisfied:
    
        D ≥ |V_Γ|-1   .
      
    
    The synopsis of the next section is to show that coordinate space equations are straightforward in terms of the invariants. However, this one could argue that these variables are not completely natural. A priory the Feynman function is formulated as the function of vertex position. In particular, the Fourier transform utilizes such presentation. Indeed, as we later discuss in  section <ref> writing the equation in terms of the invariants  causes the Fourier transform to be complicated, even though, in principle, possible. Moreover, when written in terms of invariants, x_ij^2 the topology of the graphs is somewhat lost.
    
    
    
    To demonstrate that, we consider two graphs:
    
    
     (A) at (-1,0) Γ_1  =; [Circle-Circle] (0,0) – (2,0);
        [-Circle] (2,0) – (4,0);
          (B) at (6,0) Γ_2  =; [Circle-Circle] (7,0.5) – (9,0.5);
        [Circle-Circle] (7,-0.5) – (9,-0.5);
    
    
    
    These two graphs corresponding to similar functions:
    
        I_Γ_1 = G(X^2_12)G(X^2_23)   ,    I_Γ_2= G(X_12^2)G(X_34^2)
    
    As functions of invariants these are the same functions: product of two propagators each depending on its own variable. Therefore if we want to actually distinguish between the two and keep track of the topology we should keep some part of the data from the initial variables x_i^μ. Mainly we should remember, that X_ij^2=(x_i-x_j)^μ (x_i-x_j)μ. This actually means, that we know that the function I_Γ_1 actually involves three points and therefore generally depends on the variable X_13^2 as well, just that the dependence is constant. Similarly we remember that I_Γ_2 is a function of four vertex positions and therefore depends (though constantly) on X_13^2,X_14^2, X_23^2 as well. We will see how to take this into account in the next section.
    
    
    
    
    
    § DIFFERENTIAL EQUATION
    
    
    In this section we demonstrate the position space equations that the unintegrated correlators satisfy. A review of the banana case is available in the appendix <ref>.
    
    
    
     §.§ Generic graphs and bananization
    
    
    Let us first start with the simple graph again (before bananization).
     Each propagator satisfies the (Klein-Gordon) equation of motion, which we write in Λ-form for convenience:
     
        (Λ_ij^2 +(D-2)Λ+X_ij^2m_ij^2 ) G(X_ij^2|m_ij,D ) =0
    
    where Λ_ij=X_ij∂∂ X_ij just as before 
    
        X_ij^2=(x_i-x_j)^μ (x_i-x_j)_μ
    
    We also denote the respective differential operator as:
    
        ℰ̂_1(X_ij) := (Λ_ij^2 +(D-2)Λ+X_ij^2m_ij^2 )
    
    as it is first in the series of banana graph operators, introduced in previous works <cit.>. Therefore the Feynman function I_Γ satisfies a system of equations
    
        _1(X_ij) I_Γ(𝐱) = 0   ,    <ij> ∈ E_Γ
    
    Promoting to bananized diagrams we substitute each edge by a banana graph and, hence, the resulting differential equation for each line is the respective banana operator. The corresponding operators is constructed by taking the symmetric product of the equations of motion for a single propagator (<ref>) via an explicit procedure nicknamed the Λ-formalism <cit.> and reviewed in <ref>. Therefore for the banananized graph, where the i ↔ j edge has been substituted by and n_ij-multiple edge one has the following system:
    
        _n_ij(X_ij) I_Γ(𝐱) = 0   ,    <ij> ∈ E_Γ
    
    
    To really specify the function we also need to tell which of the pairs <i,j> do not belong to the set of edges, meaning that the power of the propagator is vanishing. This can be actually also nicely incorporated in the whole scheme, just by declaring a constant function to be a 0-banana function, and writing the corresponding equation as:
    
        ℰ̂_0(X_ij) B_0:= Λ_ij B_0 = 0  ↔  B_0 = const
    
    Therefore the full system of equations is given by:
    [colback=champagne!30]
    	
        _n_ij(X_ij) I_Γ(x) = 0   , i,j ∈ V_Γ
    
    where n_ij is the number of edges between two vertices, including zero's if the vertices are not connected. 
    
    
    
    Finally, it is crucial that we remember that the invariants are indeed made out of the initial coordinates, which will further be important in Fourier transform. This means that we want to distinguish between say a triangle graph and three independent lines. Specifying the set of vertices and including the 0-banana  operators is important for that. Consider the two graphs from (<ref>):
    
        I_Γ_1 = G(X^2_12)G(X^2_23)   ,    I_Γ_2= G(X_12^2)G(X_34^2)
    
    However to truly differentiate between them we should treat the first as a function of three vertex coordinates and the second as a function of 4 vertex coordinates with the following equations:
    
        _1(X_12)	I_Γ_1(𝐱) =0 ,  _1(X_23)	I_Γ_1(𝐱) =0,   _0(X_13)	I_Γ_1(𝐱) =0
    
    while for the second graph:
    
        _1(X_12)	I_Γ_2(𝐱) =0   _1(X_34)	I_Γ_2(𝐱) =0
        			 
           _0(X_1i)	I_Γ_2(𝐱) =0,   i =2,… 4   _0(X_2i)	I_Γ_2(𝐱) =0,    i =3,4
    
    
     §.§ Examples in the massless case
    
    
    When the propagators are massless, the position space equations are the simplest and can be given explicitly. The derivation and notations are explained in appendix <ref>. For a single banana graph, the annihilating differential operator, denoted by DET_n is given by:
    
        DET_n =   Λ̂∏_j=1^n  (Λ̂+j(D-2))
    
    Therefore, say for the ice-cone, graph, which is a particular case of a bananized triangle (with only one multiple edge) one has:
    
        I_Triangle(x_1,x_2,x_3)=G(X_12)G(X_23)G(X_13)^n
    
    
     (0,0) – (1.25,2) – (2.5,0);
       (0,0) to[out=15,in=180-15]  (2.5,0);
       (0,0) to[out=30,in=180-30]  (2.5,0);
     (0,0) to[out=-15,in=180+15] (2.5,0);
         (0,0) to[out=-30,in=180+30]  (2.5,0);
         (x1) at  (0-0.25,0-0.25) x_1;
         (x2) at (1.25,2+0.25) x_2;
         (x3) at (2.5+0.25,0-0.25) x_3;
        (ldots) at (1.25,0) 900.65… ;
     
      
       
        
    
    
    
    The equations are:
    
        Λ̂_12(Λ̂_12+(D-2))  I_Triangle(x_1,x_2,x_3) =  Λ̂_23(Λ̂_23+(D-2))  I_Triangle(x_1,x_2,x_3) = 0
                 
        Λ̂_13∏_j=1^n  (Λ̂_13+j(D-2))I_Triangle(x_1,x_2,x_3) = 0
    
    
    It is also instructive to consider the four point case, where one could have more interesting topologies. Consider, for example, a bananized kite graph:
    
    
     (0,0) – (2.5,0) – (2.5,2.5) – (0,2.5) – (0,0);
       (0,0) to[out=10,in=-100] (2.5,2.5);
        (0,0) to[out=30,in=-120] (2.5,2.5);
         (0,0) to[out=90-10,in=-180+10] (2.5,2.5);
         (0,0) to[out=90-30,in=-180+30] (2.5,2.5);
         (x1) at (-0.25,-0.25) x_1;
         (x2) at (0-0.25,2.5+0.25) x_2;
         (x3) at (2.5+0.25,2.5+0.25) x_3;
         (x4) at (2.5+0.25,0-0.25) x_4;
         (ldots) at (1.25,1.25) -450.75… ;
        [color=black,fill=white] (0,0) circle (1pt);
         [color=black,fill=white] (2.5,2.5) circle (1pt);
          [color=black,fill=white] (2.5,0) circle (1pt);
           [color=black,fill=white] (0,2.5) circle (1pt);
    
    
    
        I_Γ(x_1,x_2,x_3,x_4) = G(X_12)G(X_23)G(X_34)G(X_14) G(X_13)^n
    
    This function satisfies a system of equation:
    
        Λ̂_12(Λ̂_12+(D-2)) I_Γ(x_1,x_2,x_3,x_4) = Λ̂_23(Λ̂_23+(D-2)) I_Γ(x_1,x_2,x_3,x_4) =0  ,
                
        Λ̂_34(Λ̂_34+(D-2)) I_Γ(x_1,x_2,x_3,x_4) = Λ̂_14(Λ̂_14+(D-2)) I_Γ(x_1,x_2,x_3,x_4) =0 ,
        Λ̂_13∏_j=1^n  (Λ̂_13+j(D-2))I_Γ(x_1,x_2,x_3,x_4) =0 ,
                
        Λ̂_24I_Γ(x_1,x_2,x_3,x_4) =0.
    
    The last equation is necessary to fully specify the topology of the diagram — it ensures that there is no edge between the 1 and 4 vertex.
    
    
    
    Massless case was chosen only for simplicity reason and all the techniques in this article are applicable to case with masses too.
    
    
    
    § FOURIER TRANSFORM
    
    One nice feature of the banana case is that by writing the very straightforward equation in position space and Fourier transforming it, one could obtain a momentum differential equation for the non-trivial loop integral. In the generic mass case the resulting equation seemed to be in a sense non-minimal (reducible), which, however, is a separate issue. Still it was demonstrated <cit.> that for the known examples  the rightmost factor of the differential operator is the irreducible Picard-Fuchs operator 
    
    
    
    Therefore it is natural to ask if it is possible to do so for the equations we presented for a generic graph. In this section, we demonstrate that this is indeed the case. Here we don't want to complicate things by discussing minimality. Moreover, we should concede that at the current stage, the resulting momentum differential operators are of completely intractable form. This phenomenon has already been visible for the triangle examples in <cit.>.  However, in this paper we demonstrate a kind of proof of concept - that the equations always exist. The problem of choosing the correct variables/bases should be solved in the future.	  
    
    
    
    The main technical issue, which also lead to the equations being complicated is the fact the momenta couple to the positions x^μ_i, while the equations are given in terms of the invariants X_ij^2. Moreover, due to symmetries, the momenta k_i^μ in (<ref>) couple to (x_1-x_i)^μ.  Hence we have to find a set of differential operators that would on one hand reproduce Λ_ij and X_ij^2, when acting on invariant function, and would be suitable for Fourier transform. 
    
    
    
    Naively one could attempt to write:
    
        Λ_ij= X_ij∂∂ X_ij      (x_i-x_j)^μ∂^μ∂ (x_i-x_j)^μ
    
    However, this is not correct, since the set of vectors  (x_i-x_j)^μ are not independent. Only their lengths are for suitably large D. Therefore an alternative solution should be found. One way to resolve this issue is to consider a set of operators:
    
        𝒟_ab = (x_a-x_b)^μ(∂^μ∂ x_a^μ+∂^μ∂ x_b^μ)
    
    The plus sign between the derivatives makes the Fourier transformed operators simpler. After Fourier transform we get:
    
        (x_a-x_b)^μ     ⟶  ∂∂ k_b^μ - ∂∂ k_a^μ
        ∂∂ x_a^μ       ⟶  -k_a^μ
        ∂∂ x_1^μ       ⟶  ∑_a=2^n k_a^μ
        
        			(x_1-x_a)^μ     ⟶  ∂^μ∂ k_a^μ
    
    
    Hence the 𝒟_ab operators can be Fourier transformed. On the other hand, when acting on invariant functions we have the following equality.
    
        D_ab   = ∑_i ≠ a, b(-X^2_ib+X^2_ab+X^2_ia) ∂∂ X_ia^2 -∑_j ≠ a, b (-X^2_aj+X^2_ab+X^2_jb) ∂∂ X_jb^2= 
        		
           =∑_i ≠ a, b (-X^2_ib+X^2_ab+X^2_ia)2 X^2_ia·Λ_ia -∑_j ≠ a, b (-X^2_aj+X^2_ab+X^2_jb)2X^2_jb·Λ_jb
    
    Recall that we want to express the set of operators Λ_ij in terms of D_ab and x_ij^2. The system of equations (<ref>) as a linear system on Λ_ij almost does the job. A complication,  that requires a minor modification is due to the fact that this system is degenerate. It's easy to check that while the size of the system is n(n-1)/2 it has corank one. Despite that, the problem is easily resolved by substituting one of the equations in (<ref>) by:
    
        Λ_total = ∑_i<jΛ_ij
    
    where
    
        Λ_total : = ∑_a=1^n x_a^μ∂∂ x_a^μ
    
    is the total dilatation/Euler operator. It has a Fourier transform:
    
        ∑_a=1^n x_a^μ∂∂ x_a^μ  ⟶   - ∑_a=2^n∂∂ k_a^μ k^μ_a
    
    The resulting system of equations, (<ref>) with one equation substituted with (<ref>) is now non-degenerate, hence we can express:
    
        Λ_ij = ∑_ab c_ij^ab( X^2 ) 𝒟_ab + c_ij^Λ(X^2) Λ_total
    
    
    This means that we can now express the differential operators _n_ij in terms of D_ab and X^2_ij, meaning that it can be Fourier transformed:
    
        _n_ij(X^2_ij,|Λ_ij)   ⟶  𝐄̂_ij(s_ab,  s_ab)
     
    where s_ab is some choice of momentum space invariants. For example, by choosing the following parametrization:
    
        s_ab=2k^μ_a k_μ,b
    
    we get:
    
        𝒟̃_ab F( { s_ab})   = (∂∂ k_b^μ - ∂∂ k_a^μ)(k_a^μ+k_b^μ)  F( { s_ab}) = 
           =∑_c (s_ab+s_bc) (∂∂ s_bc- ∂∂ s_ac) F( { s_ab})
    
    
     §.§ Example of triangle
    
    The position space equation for a three point triangle function are written in terms of three operators Λ_12,Λ_23,Λ_12. According to our general prescription we can express them through operators, which are written in terms of the vertex positions x_1^μ,x_2^μ,x_3^μ. To do this, we solve the following operator equations:
    
        Λ = x_1^μ∂∂ x_1^μ+x_2^μ∂∂ x_2^μ+x_3^μ∂∂ x_3^μ   = Λ_12+Λ_13+Λ_23
        𝒟_12=(x_1^μ-x_2^μ)( ∂∂ x_1^μ+ ∂∂ x_2^μ)    = X_12^2+X_13^2-X_23^22 X_13^2Λ_13 - X_12^2+X_23^2-X_13^22 X_23^2Λ_23
        𝒟_13=(x_1^μ-x_3^μ)( ∂∂ x_1^μ+ ∂∂ x_3^μ)    = X_12^2+X_13^2-X_23^22 X_12^2Λ_12 - X_13^2+X_23^2-X_12^22 X_23^2Λ_23
    
    Solving this system we will get the matrix M:
    
        [                                                                      X_12^2 (X_12^2 - X_13^2 - X_23^2)                                    -2 X_12^2 X_13^2 (X_12^2 - X_13^2 - X_23^2)X_12^2 + X_13^2 - X_23^2 - 2 X_12^2 (X_12^2 X_13^2 - X_13^4 + X_12^2 X_23^2 + 2 X_13^2 X_23^2 - X_23^4)X_12^2 + X_13^2 - X_23^2;                                                                                                                                                                                                                                                                                                                     ;                                                                     -X_13^2 (X_12^2 - X_13^2 + X_23^2)   2 X_13^2 (X_12^4 - X_12^2 X_13^2 - 2 X_12^2 X_23^2 - X_13^2 X_23^2 + X_23^4)X_12^2 + X_13^2 - X_23^2                                     2 X_12^2 X_13^2 (X_12^2 - X_13^2 + X_23^2)X_12^2 + X_13^2 - X_23^2;                                                                                                                                                                                                                                                                                                                     ;                                                                     -X_23^2 (X_12^2 + X_13^2 - X_23^2)                                                                                        2 X_13^2 X_23^2                                                                                        2 X_12^2 X_23^2;                                                                                                        ]
    
    To decrease the size of matrix we omitted the common multiplier:
    
        den=1X_12^4 - 2 X_12^2 X_13^2 + X_13^4 - 2 X_12^2 X_23^2 - 2 X_13^2 X_23^2 + X_23^4
    
    Thus the solution is:
    
        [ Λ_12; Λ_13; Λ_23 ]
            = den· M ·[    Λ; 𝒟_12; 𝒟_13 ]
    
    It can be seen from the size of matrix, that even in the simplest case the Fourier transformation becomes rather challenging, what makes its practical usage, i.e. connecting simple coordinate equations with difficult momentum ones, obstructed.
    
    
    
    § RELATION TO PF
    
    Comparing with the case of banana integrals studied in <cit.>, connecting coordinate and momentum equations here has additional complications. There are two new arising problems, namely:
    
      *  Instead of single equation we should “factorize” the system of equations.
        
      *  Previously we had single variable non-fuchsian equation, what already produced a lot of complications. However, in this case, every equation has derivatives in multiple variables.
    
    Summing up, because of these reasons Fourier transformed equations cannot be considered independently but as the system. Thus there is no factorization for single equations and it is replaced by checking that the differential ideal generated by Fourier transformed system lies in the ideal generated by Picard-Fuchs system. In other words, the Fourier system should be the consequence of Picard-Fuchs one. Unfortunately, the complexity of these systems does not allow to present “factorization” rules, even in massless case, as we have done for equal mass case for banana diagrams in <cit.>.
    
    
    
    
    § CONCLUSION
    
    
    In this short letter, we have demonstrated that differential equations exist for any Feynman integral and have a very transparent origin. The two main principles that we utilize is to work with unintegrated functions and work in position space initially. Let us summarize the main advantages of our approach.
    
      *   We have shown that existence and even shape of equations in position space can be determined from first principle. One can immediately trace its origin to just the equations of motion, i.e. the Gaussian part of the action.
    	
      *   The existence of equations in momentum space is an immediate corollary and  is in principle defined by explicit Fourier transform rules.
    	
      *  In fact the discussion did not really depend that much on the shape of the e.o.m.. In principle, one could substitute the Klein-Gordon equation by any differential operator (perhaps of finite order) (see <cit.> for a similar treatment in quite a different setting). Thus, the position space equations exist in any theory (defined perturbative through a quadratic action). 
    	
      *  The position space equations can by potentially defined in cured space, to deal with loop correction in curved space QFT <cit.>. 
    
    The last point also prompts us to discuss another feature of the proposal. A particular part of the discussion of the Fourier transform is the possibility to integrate over certain vertices by setting some momenta in Fourier space to zero. However, the operation of an integration over a vertex can perhaps be archived directly in position space on the level of equations. The idea here is in the spirit of D-module theory. There one can rigorously define the so-called integration ideal in the D-module, and its dual restriction ideal in Fourier space. Effectively it gives a way to define differential equations that are satisfied by an integral of a function over some subset of the variables. In this context one requires certain holonomic properties from the functions and D-modules in question, which have to be investigated in our context. An additional complication that we deal with here is that the integration is over multi-dimensional vertex coordinates, while the differential equations are initially given in terms of invariants. Here again, the discussion in sec. <ref> is of relevance. Despite that we can still speculate that after appropriate modification techniques of D-modules and holonomic functions theory can be applied in our context exactly in the desired way: obtaining differential equation for integrated diagrams from the unintegrated ones. Concluding this discussion we mention that that would be the supposed approach to curved space integrals, where, in general, only position space methods are available.
    
    
    
    § ACKNOWLEDGEMENTS
    
    Nordita is supported in part by NordForsk. The work 
    is supported by the Russian Science Foundation (Grant No.20-71-10073).
    
    
    utphys 
    
    § THE BANANA CASE AND REVIEW OF THE Λ-FORMALISM
    
    
    
    
    In <cit.> a method of computing coordinate–space differential equations was proposed, which are essentially tensor powers of the equations of motion. Here we review some parts this method, that produces the differential operators _n(X).   All the operators are written using Λ:=x^μ_μ.Then each propagator satisfies 
    
        (Λ^2 + (D-2)Λ + X^2 m^2)G_m (X) = 0
    .
    The banana function the product of propagators:
    
        B_n(𝐦|X)=∏_i=1^n G_m_i(X)
    .
    Firstly, we notice that one can reduce any power of the dilatation operator acting on the propagator to first derivatives and multiplication by X^2 using the equations of motion. Namely:
    
        Λ^k G_m =  ∑_i=0^k a_k,i X^2iΛ G_m + ∑_i=1^k b_k,i X^2i  G_m
    
    There does not seem to be any simple expression for these coefficients, however they can be defined recursively. Next, let us introduce a notation for a product of Λ-derivatives of the propagators:
    
        I_k⃗=I_(k_1,… k_n ):= ∏_i=1^nΛ^k_i G_m_i
    
    where k_i can be zero. In these notations the initial banana function is just:
    
        B_n(x)=I_0⃗:= I_(0,…)
    
    Thus, applying the dilatation operator to the product of propagators we get:
    
        Λ^n I_0⃗ = ∑_k⃗ : ∑ k_i=nnk_1,… k_n I_k⃗
    
    Using the expansion (<ref>) for each of the I_k⃗ functions we obtain:
    
        Λ^k I_0⃗=  ∑_ϵ⃗ : ϵ_i=0,1 C_k^ϵ⃗ I_ϵ⃗
    
    where the coefficients C_k^ϵ⃗ are made up of the a_k,i,b_k,i  of (<ref>) and the multinomial coefficients in (<ref>).
    
    
    
    The key idea is to notice that these relations for k=1,… 2^n  imply that the homogeneous linear system with matrix elements (C_k^ϵ⃗ | Λ^k I_0⃗) has a nontrivial solution given by the 2^n+1–tuple ( {I_ϵ⃗}_ϵ_i=0,1 , -1 )
    This means that the determinant of this matrix should vanish which implies a differential equation on I_0⃗=B_n:
    
        ℰ^(n)_x I_0⃗:=_k = 0,…, 2^n 
        ϵ⃗∈ℤ_2^n ( C_k^ϵ⃗ | Λ^k I_0⃗)  = 0
    
    
    The coefficients C_k^ϵ⃗ are in principle computed via the recursion for and a_k,i,b_k,i, however, there is no explicit formula for them in terms of the masses and dimensions. Clearly, the final determinant is a more complicated object. However, formula (<ref>) establishes the existence and the form of the position space equations for generic n-banana functions with distinct masses. 
    
    
    
    To illustrate the construction, consider the example of n=2 generic mass banana diagram. In this case, the basic function is I_0,0=B_2 = G_m_1 G_m_2 := G_1 G_2 and there are three ways to apply the dilatation operators:
    
        I_1,0 = (Λ G_1) G_2; I_0,1 = G_1 (Λ G_2); I_1,1 = (Λ G_1) (Λ G_2).
     To get an equation we need to apply derivatives to the banana function four times. By iterated application of Λ operator and using the relations (<ref>) we obtain:
    
        ^0Λ I_0,0 = I_1,0 + I_0,1
           Λ^2 I_0,0 = 2 I_1,1 -(D-2)I_1,0 - (D-2) I_0,1 - (m_1^2 + m_2^2 )x^2 I_0,0
           Λ^3 I_0,0 = -6 (D-2)I_1,1 - (x^2 m_1^2  + 3 X^2 m_2^2 - (D-2)^2)I_1,0 -
        		
           Λ^4 I_0,0= - (3 X^2 m_1^2 + m_2^2 X^2 -(D-2)^2)I_0,1 + (D-4) (m_1^2 + m_2^2) X^2 I_0,0
           Λ^4 I_0,0 = (-4 (m_1^2+m_2^2) X^2+14 (D-2)^2) I_1,1 +(2 X^2 ((D-4)m_1^2 + (5D-14)m_2^2) -(D-2)^3)I_1,0 +
           Λ^4 I_0,0=+(2 X^2 ((5D-14) m_1^2+(D-4) m_2^2)-(D-2)^3) I_0,1 +
           Λ^4 I_0,0= (-(m_1^2+m_2^2)(D^2 -6 D +12)+ 6
        		m_2^2 m_1^2 X^2+m_1^4 X^2+m_2^4 X^2) I_0,0
    
    The matrix (C_k^ϵ⃗ | Λ^k I_0⃗) is then explicitly given by:
    
        [                                    1                                    0                                    0                                    0                                I_0,0;                                    0                                    1                                    1                                    0                              Λ I_0,0;                - (m_1^2 + m_2^2 )x^2                               -(D-2)                               -(D-2)                                    2                            Λ^2 I_0,0;              (D-4)(m_1^2 + m_2^2)x^2 - (m_1^2 X^2 + 3 m_2^2 X^2 -(D-2)^2) -(3 m_1^2 X^2 + m_2^2 X^2 - (D-2)^2)                              -6(D-2)                            Λ^3 I_0,0;                               C_4^00                               C_4^10                               C_4^01                               C_4^11                            Λ^4 I_0,0;                                      ]
    
    where
    
        C_4^00   =  X^2 (-(m_1^2+m_2^2)(D^2 -6 D +12)+ 6
        		m_2^2 m_1^2 X^2+m_1^4 X^2+m_2^4 X^2),	 
        
        		C_4^10   = (2 X^2 ((D-4)m_1^2 + (5D-14)m_2^2) -(D-2)^3),  
        
        		C_4^01   =(2 X^2 ((5D-14) m_1^2+(D-4) m_2^2)-(D-2)^3),  
        
        		C_4^11   = -4 (m_1^2+m_2^2) X^2+14 (D-2)^2
    
    The equations (<ref>) imply that the matrix has a kernel of the form (I_0,0, I_1,0, I_0,1, I_1,1,-1).
    Therefore, its determinant vanishes, which produces an equation:
    
        ℰ^(2)_x I_0⃗ =    4 X^2 (m_1^2 - m_2^2) {Λ^4 + 2 (2 D - 5) Λ^3  +
        		(2 X^2 (m_1^2 + m_2^2)+(D-2)(5D-16)) Λ^2  +
        		
           + 2 ((m_1^2+m_2^2) X^2 (2D-3)+(D-4) (D-2)^2 ) Λ  + 
           + X^2 ((m_1^2 -m_2^2)^2 X^2 + 2 (D-1)(D-2) (m_1^2 +m_2^2)) } I_0,0 = 0
    
    
    As we see, the equation is of order 4 in x derivatives just as expected. 
    
    
     §.§ Vanishing masses
    		
    
    		Taking the limit of vanishing masses greatly simplifies the calculations. It is to be considered a ”solvable” example: we can completely derive the form of the general position space operator starting from the equations of motion.  In this case the cut propagator satisfies:
    		(Λ^2 + (D-2)Λ)G = 0
    		
    		which makes applying powers of Λ trivial:
    		
        Λ^nG = (2-D)^n-1Λ G   , n>1
    
    		Which makes the whole calculation much simpler. As a result we obtain the equation to be a determinant, analogous to the one in (<ref>): 
    		
        _n(X) = ( n!(n-i)!)  C_n^(red)
    
            where the reduced matrix is given by:
    		
        C_n^(red)={( S_2(i,k) (2-D)^i-k| Λ^i) }_i=1,…, n+1 
        
        					k=1,… ,  n
    
      where S_2(i,k) are the Stirling numbers of the second kind. Computing the determinant gives:
      	
        _n(X)  =  (∏_i=0^nn!(n-i)!) (∏_i=0^n(Λ+i(D-2)))
    
      As before vanishing of the determinant gives a differential equation:
    		_n(X)  B_n(X) =0
    		
    For example:
    At n=2 the derivatives are given by
    			
        Λ  G^2    = 2GΛ G 
        Λ^2 G^2    = 2GΛ^2 G + 2(Λ G)^2 = -2(D-2)GΛ G + 2(Λ G)^2 
        Λ^3 G^2    = -2(D-2)GΛ^2 G - 2(D-2)(Λ G)^2 +4Λ G Λ^2 G =
        					2(D-2)^2 G Λ G - 6(D-2)(Λ G)^2
    
    			And the matrix:
    			
        M_2=([        2        0    Λ Z_2;  -2(D-2)        2  Λ^2 Z_2; 2(D-2)^2  -6(D-2)  Λ^3 Z_2;          ])
    
    			Hence the determinant gives
    			ℰ_x^(2) = -4Λ(Λ^2 + 3(D-2)Λ +2(D-2)^2)
    			= -4Λ(Λ+(D-2)) (Λ+ 2(D-2))


 Paper Title: Real-time chiral dynamics at finite temperature from quantum simulation.


Methodological Foundations of Modern Causal Inference in Social Science Research
    Guanghui Pan[Department of Sociology, University of Oxford. 41 Park End Street, Oxford, the United Kingdom, OX1 1HH. Email: guanghui.pan@sociology.ox.ac.ukguanghui.pan@sociology.ox.ac.uk. Manuscript in progress, do not cite or redistribute without permission.]
    August 5, 2024
========================================================================================================================================================================================================================================================================





§ INTRODUCTION

In recent years, the exploration of quantum simulations for field theories <cit.> has garnered significant attention across many subfields of physics <cit.>. This growing interest is largely driven by the advancements in quantum computing technology, which promises to revolutionize our ability to tackle complex and computationally intensive problems. Traditional classical computers, despite their power, often fall short when dealing with the intricate dynamics and properties of quantum systems due to the exponential growth of computational resources required. On the other hand, quantum simulations hold the potential for breakthrough discoveries in areas such as high-energy physics, material science, and chemistry, aiding in understanding phase transitions <cit.>, solving quantum many-body problems <cit.>, and discovering new materials with unique properties <cit.>.

To explore these potential developments, we focus on the Schwinger model <cit.>, quantum electrodynamics (QED) in (1+1) dimensions. 
The Schwinger model provides an excellent testing ground for studying various phenomena, including confinement, dynamical mass generation, and anomaly-induced symmetry breaking. It offers the simplicity of fewer degrees of freedom while still encapsulating the rich physics of gauge theories, making it an ideal benchmark model for quantum simulations on both quantum and classical hardware <cit.>.


In this study, we explore the real-time dynamics of the chiral magnetic effect (CME) at finite temperature within the massive Schwinger model. The CME refers to the generation of an electric current along a magnetic field in systems where there is an imbalance between left-handed and right-handed fermions <cit.>. This imbalance is typically produced by the presence of a chiral chemical potential. To investigate the CME within the context of the Schwinger model, we employ a quench protocol as outlined in our previous study <cit.>. By introducing a chiral chemical potential through this quench process, we drive the system out of equilibrium and analyze the resulting vector currents and how they evolve over time. The Hamiltonian is adjusted to include the time-dependent chiral chemical potential, enabling the investigation of the CME within a quantum computing framework.

The interest in the study of the temperature dependence of the real-time CME stems from the following consideration. The CME is intrinsically a non-equilibrium, real-time phenomenon. However, when the external fields and the chiral chemical potential vary slowly with time, the CME response is driven by the fermion zero modes, and should thus be temperature-independent <cit.>. This independence of CME current on the temperature has been established in analytical computations <cit.>. However, under a rapid quench, the CME current receives contributions from excited fermion modes <cit.>, and can thus develop a dependence on the temperature. This temperature dependence  has not been evaluated before from first principles. In this paper, we will evaluate it using the methods of quantum simulation in massive Schwinger model at finite temperature.  

We employ the quantum imaginary time evolution (QITE) algorithm to study the thermal states and utilize the Suzuki-Trotter decomposition for real-time evolution. The QITE algorithm is essential for efficient quantum simulation on quantum circuits for several reasons, and various related algorithms have been developed <cit.>.
It enables the simulation of thermal states, crucial for studying phenomena like phase transitions at finite temperatures <cit.>. Since imaginary time evolution involves non-unitary operations, which cannot be directly implemented on quantum hardware, QITE approximates these operations with unitary ones, making it feasible on quantum circuits. Additionally, traditional classical algorithms struggle to prepare thermal states of many-body systems, while QITE allows for efficient exploration of these state spaces. Its versatility makes it applicable to a wide range of quantum systems and models, bridging theoretical models and practical implementations on quantum hardware. By iteratively applying small imaginary time steps and recalibrating the system, the QITE ensures the system gradually reaches the desired state where subsequent measurements are performed to extract physical observables.

Our motivation follows the successful quantum simulations performed on the (1+1)-dimensional Nambu–Jona-Lasinio (NJL) model <cit.>, where quantum algorithms provided insights into the chiral phase transition at finite temperature and chemical potential. Extending this line of research, our previous work on the quantum computational study of field theories has demonstrated the feasibility and efficacy of these simulations in delivering significant theoretical insights. This work on the Schwinger model bridges the study of simple models accessible by current quantum hardware and the more complex theories underpinning the standard model of particle physics. By exploring the unique properties and dynamics of the Schwinger model, we aim to provide insights into chiral dynamics in low-dimensional quantum field theories.

The rest of this article is organized as follows. In Sec. <ref>, we present the model and the quench protocol that we study in this work. In Sec. <ref>, we describe our algorithm of the quantum imaginary time evolution to implement the thermal states and subsequent expectation evaluation to extract physical observables at finite temperature. In Sec. <ref>, we show the results from our quantum simulations, where we discuss the real-time chiral dynamics of the current at finite temperatures, the phase diagram of the Schwinger model, and the temperature dependence of the electric charge (axial current). Finally, the article is concluded in Sec. <ref>. 




§ THE MASSIVE SCHWINGER MODEL WITH THE TOPOLOGICAL TERM
 


 §.§ The Schwinger Hamiltonian

The action of the massive Schwinger model <cit.> with θ term in (1+1)-dimensional Minkowski space is

    S = ∫ d^2x[-1/4 F^μν F_μν + gθ/4πϵ^μνF_μν + ψ̅(iD-m)ψ],

with D=γ^μ(∂_μ + i gA_μ).
Here, A_μ is the U(1) gauge potential, E=Ȧ_1 is the corresponding electric field, ψ is a two-component fermion field, m is the fermion mass, and γ^μ are two-dimensional γ-matrices satisfying the Clifford algebra. When g≠0, the θ-term induces the classical background electric field, which breaks the parity symmetry. The gauge field A_μ and the coupling constant g have mass dimensions 0 and 1, respectively.
By a chiral transformation, ψ→ e^iγ^5θψ and ψ̅→ψ̅e^iγ^5θ, the action is transformed to 

    S = ∫ d^2x[-1/4 F^μν F_μν  + ψ̅(iγ^μ D_μ-me^iγ^5θ)ψ].

The action is invariant under this transformation only up to the boundary term.
It is evident from eq. (<ref>) that the massive theory with a positive mass (m>0) at θ=π is equivalent to the theory at θ=0 but with a negative mass (-m).  

The Hamiltonian in the temporal gauge A_0=0 is given by

    H=∫ dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1 - me^iγ^5θ)ψ]  ,

where the space-time coordinate is labeled by x^μ=(t,z). We denote the Pauli matrices as X, Y, and Z, and use the following convention for the Dirac matrices: γ^0 = Z, γ^1 = i Y, γ^5=γ^0 γ^1 = X.
In (1+1) dimensions, the axial charge density Q_5(x)≡ψ̅(x)γ^5γ^0ψ(x) 
and the vector current density J(x)≡ψ̅(x)γ^1ψ(x) are related 
by Q_5(x) = -J(x). Moreover the vector charge density Q(x)≡ψ̅(x)γ^0ψ(x) and the 
axial current density J_5(x)≡ψ̅(x)γ^5γ^1ψ(x) are related by Q(x) = J_5(x).



 §.§ Real-time chiral dynamics at finite temperature


As we have mentioned in the introduction, the chiral magnetic effect refers to the generation of an electric current along a magnetic field in a system with an imbalance in the number of left-handed and right-handed fermions, induced by the presence of a chiral chemical potential. In the Schwinger model, when the chiral quench is applied, the system is driven out of equilibrium, creating a non-zero chiral chemical potential μ_5. This chiral chemical potential represents an asymmetry between left-handed and right-handed fermions.

Here, we consider the μ_5-quench <cit.> protocol: The system Hamiltonian with θ=0 is prepared at time t<0. Then, starting at t=0, the Hamiltonian rotates the θ angle according to θ=-2μ_5t corresponding to constant chiral chemical potential (i.e. θ̇ = -2μ_5.). The Hamiltonian after the quench is modified as

    H'= dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1-γ^1θ̇/2 - me^iγ^5θ)ψ]  .

The real-time evolution of an operator O can be evaluated by the time-ordered integral 

    O(t)=𝒯[e^i∫_0^tdt'H'(t')]O𝒯[e^-i∫_0^tdt'H'(t')],

and the thermal expectation value of O evaluated at time t at finite temperature T is

    ⟨ O(t)⟩_β=[e^-β HO(t)]/ [e^-β H],

where H is the unquenched Hamiltonian in eq. (<ref>) at θ=0 and β=1/T is the inverse of the temperature of the system.

Notably, this evolving θ = -2μ_5 t after the quench at t = 0 indicates a dynamical chiral chemical potential, driving the system into a non-equilibrium state where left-handed and right-handed fermions are imbalanced. In the presence of this chiral chemical potential, a vector current J (which is typically zero before the quench) is induced. The vector current's time evolution at a finite β, influenced by the quench, aligns with the characteristic behavior observed in the CME at a finite temperature.





 §.§ The lattice Hamiltonian on the qubits


To discretize our Hamiltonian, we use staggered fermions <cit.> on a lattice such that

    ψ(x≡ na) = 1/√(a)(
        [          χ_2n;        χ_2n+1 ]) ,

where a is the finite lattice spacing. Then the lattice Hamiltonian corresponding to eq. (<ref>) is <cit.> 

    H    = a g^2/2∑_n=1^N-1 L_n^2
     -i/2a∑_n=1^N-1[χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]
    + mcosθ∑_n=1^N (-1)^n χ^†_nχ_n
       + imsinθ/2∑_n=1^N-1 (-1)^n [χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]-θ̇/4a∑_n=1^N-1[χ^†_n+1χ_n+χ^†_nχ_n+1] ,

where L_n is the electric field operator satisfying the Gauss' law constraint

    L_n-L_n-1 =  χ_n^†χ_n-1-(-1)^n/2.

Using the Gauss' law constraint, we eliminate the link fields U_n by the gauge transformation,

    χ_n→ g_nχ_n,  χ^†_n→χ^†_n g_n^†,  
        U_n → g_n+1U_n g_n^†,

with

    g_1=1,    g_n = ∏_i=1^n-1U^†_i.


For the purpose of quantum simulation, we rewrite the lattice Hamiltonian in the spin representation using the Jordan–Wigner transformation <cit.>:

    χ_n=X_n-iY_n/2∏_i=1^n-1(-i Z_i).

The full Hamiltonian in the qubit representation considered in this study is

    H=   ∑_n=1^N-1(1/4a-m/4(-1)^nsinθ)[X_n X_n+1+Y_n Y_n+1]
         +mcosθ/2∑_n=1^N(-1)^n Z_n
       +a g^2/2∑_n=1^N-1L^2_n-θ̇/8a∑_n=1^N-1[X_n Y_n+1-Y_n X_n+1].

Moreover, the local vector and axial charge densities are, respectively <cit.>, 

    Q_n ≡    ψ̅γ^0ψ = Z_n+(-1)^n/2a,
    
        Q_5,n≡    ψ̅γ^5γ^0ψ = X_nY_n+1-Y_nX_n+1/4a .

For later convenience, we introduce the average vector and axial charge operators that sum over all lattice sites,

    Q≡a/N∑_n=1^NQ_n,     
        Q_5≡a/N-1∑_n=1^N-1Q_5,n.

Notably, Q commutes with the Hamiltonian while Q_5 does not. With the boundary condition L_0=0, the Gauss' law constraint, eq. (<ref>), leads to the solution L_n =  a∑_j=1^n Q_j.

Notably, one can project the Hamiltonian to different charge sectors to work with states with specified quantum numbers. In the case of the Schwinger model, the projection operator P to the charge neutral sector (Q=0) can be written as P_Q=0 = ∑_n|n⟩⟨n| where n are all the bitstrings with even number of 1s and 0s. On the qubits, P_Q=0 can be always written as the sum of Pauli-Z operators such that H_Q=0=P_Q=0 H P_Q=0^† is the projected Hamiltonian to the Q=0 sector[For example, at N=4, the projection operator is

    P_Q=0 = P_Q=0^† = 0.375 (1 + Z_1 Z_2 Z_3 Z_4) - 0.125 ( Z_3 Z_4 + Z_2 Z_4 + Z_2 Z_3 + Z_1 Z_4 + Z_1 Z_3 + Z_1 Z_2 )
]. Here the coefficients are determined in such a way that P_Q=0^2=P_Q=0. For the numerical simulation results presented in this paper, we always work in the Q=0 sector. This is because 
the Schwinger model spectrum does not have charged states due to confinement at all scales.
Therefore, when we say Hamiltonian, in what follows, we always mean the charge neutral Hamiltonian H=H_Q=0. 



§ QUANTUM SIMULATION OF QED_2 AT FINITE TEMPERATURE



 §.§ Quantum imaginary time evolution

Quantum imaginary time evolution (QITE) is an efficient way to investigate eigenstates and thermal states on a quantum computer <cit.>. Here, we briefly review the QITE algorithm used in this work for preparing the thermal state and extracting the finite temperature observable.

In QITE, one attempts to approximate the action of the imaginary time evolution
operator on a state | Ψ⟩ with a parameterized unitary
operation:

    c(Δβ)^-1/2 e^-ΔβĤ | Ψ⟩≈ e^-i ΔβÂ(a⃗)| Ψ⟩,

where Δβ is a small time step and c(Δβ)^-1/2 is a normalization coefficient approximated
by 1-2Δβ⟨Ψ | Ĥ | Ψ⟩. The real-time evolution operator Â(a⃗)
is given by a linear combination of N_μ Pauli string:

    Â(a⃗) = ∑_μ^N_μ a_μP̂_μ.

Here, P̂_μ=∏_lσ_μ_l,l is a Pauli string and the subscript μ of a_μ labels the various Pauli strings, and their coefficients a_μ are solved from the linear equation ( S+ S^T)  a= b, where the matrix S and vector b are defined by

    S_μν   =⟨Ψ(β)|P̂_ν^†P̂_μ|Ψ(β)⟩ ,
    
    b_μ   =-i/√(c(Δβ))⟨Ψ(β)| (ĤP̂_μ+P̂_μ^†Ĥ)|Ψ(β)⟩ .

The QITE procedure is repeated for a specified
number of time steps to reach a target total evolution time. Using this iterative approach, we are able to evolve an initial quantum state |Ψ_k(0)⟩ under the unitary operator e^-iΔβ A to any imaginary time β by trotterization:

    |Ψ_k(0)⟩→ |Ψ_k(β)⟩ +𝒪(Δβ) .

Note the sum of |Ψ_k(β)⟩ over all basis states approximates e^-βĤ/(e^-βĤ) at finite temperature T=1/β in the sense that it gives the thermal expectation of an observable Ô:

    1/Z∑_k⟨Ψ_k(β/2)|Ô|Ψ_k(β/2)⟩≈⟨Ô⟩_β=1/Z (e^-βĤÔ),

where Z= (e^-βĤ)=∑_k⟨Ψ_k(β/2)||Ψ_k(β/2)⟩ is the canonical partition function, which is also the sum of the normalizations over all the quantum states.





 §.§ Real-time evolution at finite temperature


Studying the dynamics of the quantum system at finite temperatures is usually complicated. In this work, we are interested in the real-time evolution of the thermal mixed state at finite temperatures of the Schwinger Hamiltonian. Using another evolution Hamiltonian H' that is not equal to the Hamiltonian H of the thermal system, we define a quench process by the real-time evolution of the operator. See eq. (<ref>) for our quench protocol.

To implement this protocol on the quantum circuit, we first use the QITE algorithm to obtain the collection of the |Ψ_k(β/2)⟩ states to represent the thermal state at finite temperature T=1/β. Then, we simply evolve each |Ψ_k(β/2)⟩ in real time as usual:

    |Ψ_k(t,β/2)⟩=𝒯[e^-i∫_0^tdt'Ĥ'(t')]|Ψ_k(0,β/2)⟩.

The final thermal expectation at real-time t is

    ⟨Ô(t)⟩_β=1/Z∑_k⟨Ψ_k(t,β/2)|Ô|Ψ_k(t,β/2)⟩.

Compared with eq. (<ref>), we see that thermal expectation evaluated on the quantum circuit is associated with the states instead of observables. Nevertheless, the two methods are entirely equivalent, which can be seen in our comparison between the quantum simulation and exact diagonalization results. To perform the real-time evolution, we trotterize the time into small time steps using second-order Suzuki-Trotter product formula <cit.>, which gives good convergence.




§ NUMERICAL RESULTS




 §.§ Thermal state preparation

To study the real-time evolution of physical properties at finite temperatures, we must first prepare the initial thermal state at the desired temperatures. In this work, we use the QITE algorithm developed using Qiskit <cit.> in a proceeding work <cit.> for the thermal medium preparation. Specifically, the unitary evolution is implemented using PauliEvolutionGate library with SuzukiTrotter method, and the thermal expectation is evaluated with the Sampler class on the statevector quantum simulators. In particular, we use the QITE to prepare thermal states with four qubits from β/a=0 (aT=∞) to β/a=4.0 (aT=0.25) for the three masses ma=0.25, 0.5, 4.0, and at coupling g=a^-1. A universal trotter step size Δβ = 0.01 is used.



To measure the validity of the quantum simulation, we use the fidelity between two density matrices σ and ρ:

    F(σ,ρ)=(√(√(σ)ρ√(σ)))^2.

Here, we take σ=e^-β H/Z to be the exact density matrix calculated via exact diagonalization and ρ=∑_k|Ψ_k(β/2)⟩⟨Ψ_k(β/2)|/Z for the thermal mixed state prepared by the QITE algorithm. In Fig. <ref>(left), we show the fidelity of these two density matrices as a function of the QITE imaginary evolution parameter β (i.e., inverse temperature). 
For all three cases of different masses, the fidelity is between 0.999 and 1.0, very close to 1, suggesting that the thermal state prepared by QITE is a good practical approximation of the corresponding thermal equilibrium state. In particular, we see a comparatively slight decrease of fidelity in the early β (high temperature) region for the heaviest mass ma=4.0, which is due to the insufficiently small step size Δβ = 0.01 used in the simulation for the heavy mass.

In Fig. <ref>(right), we show the purity of the mixed quantum state ρ at different temperatures and masses. Purity is defined by [ρ^2] and is an index used to determine how much a quantum state is mixed, with values equal 1 indicating a pure state, and values approaching 0 signifying a mixed state. For every mass, we expect that the purity to approach 1 as the temperature approaches 0 (equivalently, as β approaches infinity), since the state is pure at zero temperature (i.e., the ground state). For smaller mass values, such as ma = 0.25, the curve shows that the purity decreases more slowly with temperature, indicating smaller mass states transition to mixed states more gently as the temperature rises. In constrast, as the mass ma increases, the purity drops more rapidly. Despite these differences, all curves will eventually approach a lower purity limit as the temperature continues to increase, showing the natural tendency of quantum systems to shift towards mixed states under a finite temperature. 




 §.§ Real-time evolution of the charge and the current at finite temperature


Now that the thermal state is prepared using the QITE protocol, we study real-time evolution of the total electric and axial charge densities for the system at a selected temperature with both light and heavy masses. Our simulations are conducted using the exact statevector quantum simulator provided by Qiskit <cit.> and compared to the T = 0 limit obtained from exact diagonalization. Specifically, we perform our simulation using eqs. (<ref>) and (<ref>), following the quench protocol provided in Sec. <ref> at various chiral chemical potentials.


The real-time evolution is compared against the zero-temperature limit using exact diagonalization, serving as a benchmark for the results obtained from the quantum simulator at finite temperatures. We first note that the results at T=0 are consistent with the results in <cit.>, which performed the study under the periodic boundary condition at g=0 while we work with the open boundary condition at g≠0. We also observe that ⟨ Q⟩ =0 at all finite temperatures and it does not evolve over time, which is expected. The quantum simulation uses four qubits and applies the QITE algorithm for thermal state preparation. These methods allow for accurate modeling of thermal states and real-time evolution, offering a robust framework for understanding the underlying physics. The second order Suzuki-Trotter product formula <cit.> is used for the real-time evolution, with step sizes dt varying based on the masses: 0.1 for light masses and 0.01 for heavy masses. 



Fig. <ref> shows the real-time evolution of the thermal average of the axial charge ⟨ Q_5(t) ⟩_β for various temperatures, aT = 10.0,1.0,0.25. The mass takes values of ma=0.25, 1.0, 4.0 to examine how light and heavy masses affect the evolution of the vector and axial charge densities, showing how it evolves under different temperatures. The results highlight the role of a chiral chemical potential μ_5 and include both exact solutions and approximate results using quantum simulation. To confirm consistency with theoretical expectations, we show that the total vector charge is conserved and does not depend on time. 

In studying the chiral dynamics in the Schwinger model, our results share several important similarities with our previous results of nonlinear chiral magnetic waves (CMWs) <cit.>, particularly in observing how the oscillation of axial charges is influenced by mass. Here, chiral magnetic waves are collective excitations in a (1+1)-dimensional chiral medium under the influence of a magnetic field. These waves are a direct consequence of the interplay between axial (chiral) and vector (electric) currents in the presence of magnetic fields. Notably, the tendency that heavier mass results in faster oscillations of the axial charge is common <cit.>.
In both studies, it is observed that the axial charge density oscillates more rapidly with increasing mass. In the nonlinear CMWs, the nonlinearity arises because, in massive Schwinger model, the frequency of axial charge oscillations becomes significantly higher than that of the vector charge. This results in the axial charge oscillating rapidly while being nearly confined within static electric dipoles, a phenomenon referred to as the “thumper” solution  <cit.>. The interplay between the excited state (causing rapid oscillations) and the ground state (with almost static behavior) creates a non-linear dynamic in the system.




In Fig. <ref>, we present a comparison between our simulation results and exact diagonalization, observing good agreement when sufficiently small time steps are used. This figure illustrates how accurately our simulation can predict the system's behavior, validating our approach. Furthermore, the rapid oscillations observed in the simulation at a larger mass (ma = 4) also verified our expectation. These oscillations are attributed to the interplay between different energy eigenstates, and their amplitude increases as the temperature rises. The consistency between the simulation in Fig. <ref> underscores the accuracy of our model in capturing the dynamic properties of the system at varying conditions.

To study the dynamical interplay between finite temperature and real-time evolutions, we also present the density plots for ⟨ Q_5(t) ⟩_β in Fig. <ref>. The axial charge oscillates as a function of time t at different temperatures T, with increasing frequency for the heavier mass. Notably, the thermal state is first obtained using the QITE algorithm at finite β/a=1/(aT) values in the range of [0, 4], and then we obtain the density plot versus aT to the range of [∞, 0.25] with interpolation to T=0 temperature. The number of modes for the time oscillations is directly related to the mass in the Schwinger Hamiltonian. 



With the goal of extrapolating our results in the continuum, we perform the exact diagonalization with N=10 qubits in Fig. <ref>. Importantly, we have direct access to the finite temperature thermal state with exact matrix exponentiation using eq. (<ref>), and our exact results are in terms of aT and t. Here, we show the density plot of the thermal expectation value of ⟨ Q_5(t)⟩_T (with temperature in units of aT in the range of [0, 4]). Comparing to results simulated on the quantum simulator shown in Fig. <ref>, we observe the same behavior for both ma=1.0 and 4.0 cases, despite having many more qubits. Quantitatively, we can also observe good agreement between the quantum and classical simulations by counting the oscillation modes and the peak intensity values in the density plots.



§ CONCLUSION

This paper investigates the real-time dynamics of the chiral magnetic effect (CME) at finite temperature in the context of the massive Schwinger model, quantum electrodynamics (QED) in (1+1) dimensions. The study utilizes quantum simulation techniques, combining the quantum imaginary time evolution (QITE) algorithm for thermal state preparation and trotterization for real-time evolution, to examine the thermal properties of the system and the transitions induced by a chiral chemical potential via a quench process.

Our results illustrate the dynamic interplay between the fermion mass and the real-time dynamics of the axial charge. At zero temperature, the corresponding dynamics has been investigated in our previous studies   <cit.>. 
The new result in this paper is the temperature dependence of the real-time CME response. Specifically, at temperatures small compared to the fermion mass, the CME response is similar to the one found previously at zero temperature. However, at larger temperatures, the oscillations of the axial charge induced by the CME are dampened by thermal effects. This can be understood as a result of the contribution from excited fermion modes to the real-time CME response, that may depend on the temperature <cit.>. 

In addition, at large fermion mass the axial charge oscillations become very non-linear. The origin of this  non-linearity can be explained in the following way: in bosonized description of the massive Schwinger model, fermion mass induces non-linear interactions among the bosons, so the chiral magnetic waves become highly non-linear <cit.>.

Finally, our study of CME dynamics suggests that quantum simulations can be used to study other real-time phenomena at finite temperatures in gauge theories that form the basis of the Standard Model. 




§ ACKNOWLEDGEMENT

We thank Meijian Li for helpful comment on the manuscript. We are also grateful to Andrea Palermo and 
Ismail Zahed for useful discussions. This work was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under Contract No.DE-SC0012704 (KI, DK, FZ), the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Grants Nos. DE-FG88ER41450 (DK), DE-SC0012704 (DK) and DE-SC0011090 (FZ), and the  National Science Foundation under grant No. PHY-1945471 (ZK). WQ is supported by the European Research Council under project ERC-2018-ADG-835105 YoctoLHC; by the Spanish Research State Agency under project PID2020-119632GB-I00; by Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019-2022), by European Union ERDF, and by the Marie Sklodowska-Curie Actions Postdoctoral Fellowships under Grant No. 101109293.


JHEP




 Paper Title: Gravitational form factors of pion from top-down holographic QCD.






[]daisuke@rcnp.osaka-u.ac.jp
Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai, 319-1195, Japan
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan


[]iwanaka@rcnp.osaka-u.ac.jp
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan


[]tanaka@hken.phys.nagoya-u.ac.jp
Department of Physics, Nagoya University, Nagoya 464-8602, Japan

§ ABSTRACT 

The gravitational form factors (GFFs) of pions has been calculated from a top-down holographic QCD approach with momentum transfer dependence for the first time. It is important because the GFFs of hadrons have information on the internal stress distribution that may provide insight into the mechanism of how QCD forms hadrons. The forward limit value of this GFFs, i.e. the D-term, was also obtained. Furthermore, in this approach, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. 



Gravitational form factors of pion from top-down holographic QCD
    Mitsuru Tanaka
    
================================================================



Introduction.—
How forces act inside hadrons to confine quarks and gluons forming hadrons has not been solved. 
Addressing this question is inseparable from developing an insight into the non-perturbative properties of quantum chromodynamics (QCD), such as confinement and spontaneous chiral symmetry breaking, which is a crucial problem in modern physics. 

Recently, the stress inside hadrons has attracted attention (see a recent review <cit.>) due to the fact that experimentally it has been measured. The stress distribution inside nucleons, extracted from the matrix elements of the nucleon energy-momentum tensor (EMT), is obtained from deeply virtual Compton scattering <cit.>. 
The form factors that characterize the matrix elements are called gravitational form factors (GFFs), which are calculated theoretically from their coupling with gravity. 

The forces inside nucleons were analyzed experimentally in Ref. <cit.> for the quark part (2018) and in Ref. <cit.> for the gluon part (2022). Experimental results show that around the center of the nucleon, the force pushes the quarks outwards, while around the surface it reverses into a confining force, indicating an extremely anisotropic behavior. As a result, the pressure near the center of the nucleon may reach the highest value currently observed in nature. The pion GFFs have also been analyzed experimentally in Ref. <cit.>. The GFFs of nucleons <cit.> and pions <cit.> have also been measured by lattice QCD analysis, a first-principles numerical simulation of QCD. 

In this letter, we are the first to calculate the momentum transfer dependence of the pion GFFs from a top-down holographic QCD approach. We use the Sakai-Sugimoto model, a top-down holographic QCD model based on the D4-D8 brane construction <cit.>. With this approach, the spectroscopy <cit.>, decay widths and electromagnetic form factors <cit.> of mesons, baryons and glueballs have been investigated with strong coupling and large N_c limit. Recently, using the same approach, nucleon GFFs have been formulated and D-terms have been estimated <cit.>. The GFFs of pions <cit.> and other hadrons <cit.> have also been calculated using bottom-up holographic QCD, which has no direct connection with QCD through string theory. 

It is interesting that stress distributions and GFFs of pions<cit.>, which are the focus of this letter, show anomalous behavior. The stress distribution inside hadrons is very sensitive to interactions in the system. The forward limit of the GFFs giving the stress distribution is called the D-term, which takes the value -1 for spin-zero particles in the free Klein-Gordon theory. On the other hand, even an infinitesimally small interaction causes a significant deviation from -1 <cit.>. 


However, the D-term of the Nambu-Goldstone (NG) bosons in the strongly coupled system becomes -1 like the free particles, in the soft pion limit because of the restriction by chiral symmetry <cit.>. 
This fact may provide some insight into the role of chiral symmetry breaking in hadron formation since the D-term carries information about the forces inside hadrons. 
These insights are expected to be validated in more detail via analysis of hadron GFFs from future hadron experiments <cit.>. 




Formulation.—
In this section, we provide the method to derive the GFFs of pions in the top-down holographic QCD <cit.>. 
The method for calculating the GFFs of nucleons with spin 1/2 has been formulated in Ref. <cit.> and will be applied in the following. 


The matrix elements of EMT of a spin-0 particle are expressed as

    <π^a(p_2)|T̂^μν(x)|π^b(p_1)> 
       =δ^ab[2P^μ P^ν A(t)+1/2(Δ^μΔ^ν-η^μνΔ^2)D(t)]e^ix·Δ,

where a,b are the indices for isospin and the kinematic variables are defined as P^μ=(p^μ_1+p^μ_2)/2, Δ^μ=p^μ_2-p^μ_1 and t=Δ^2 with Lorentz indices μ,ν <cit.>. 
In this paper, we employ the mostly plus metric η^μν= diag(-1,1,1,1) and the normalization condition as <p⃗_⃗2⃗|p⃗_⃗1⃗>=2E(2π)^3δ^(3)(p⃗_⃗1⃗-p⃗_⃗2⃗),    E=√(p⃗_⃗1⃗^2+m^2).


The method for calculating the matrix elements of the EMT by holographic QCD is as follows. 
Within the framework of holographic QCD, perturbations to the metric caused by matter fields in the bulk propagate to the boundary, allowing us to extract the expectation values of the EMT in the boundary theory from these perturbations. The top-down holographic QCD used in this study, the Sakai-Sugimoto model, consists of N_f D8 branes embedded as a probe in black 4-brane background <cit.>, a holographic dual of N_c D4 branes. This background is a 10-dimensional curved spacetime of type IIA supergravity, derived from 11-dimensional supergravity related to M-theory on doubly Wick rotated AdS_7 black hole × S^4, 

    ds^2=   r^2/L^2[f(r)dτ^2-dx^2_0+dx_1^2+dx^2_2+dx^2_3+dx^2_11] 
       +L^2/r^2dr^2/f(r)+L^2/4dΩ^2_4 , 
    
        f(r)=    1-R^6/r^6,    R = L^2M_KK/3,    L^3=8π g_s N_c l^3_s,    
    
        R_11=   
        λ/2π N_c M_KK,    λ N_c=L^6M_KK/32π g_s l^5_s=216π^3κ,

where M_KK is the Kaluza-Klein mass in the τ direction, g_s string coupling, l_s string length, λ 't Hooft coupling, and R_11 the radius of the circle compactified in the x_11 direction.

The expectation values of EMT on the boundary, <T_μν>, can be extracted from the boundary values of the metric fluctuations in the bulk in the background spacetime (<ref>) propagated to the boundary as follows, 

    ∫_x_11,τδ g_μν∼2κ^2_7 L^5/6<T_μν>/r^4+...,

where κ^2_7 is related to the 11-dimensional gravitational constant as κ^2_11= Vol(S^4)κ^2_7=8π^2/3(L/2)^4κ^2_7.
Note that, to obtain the above, we need to apply the holographic renormalization, developed by Ref. <cit.>. 


The metric fluctuations δ g_MN of the bulk obey the following linearized Einstein equations:

    ℋ̅_MN=   ℋ_MN-g_MN/2ℋ^P_P=-2κ^2_7𝒯_MN, 
    ℋ_MN≡   ∇^2δ g_MN+∇_M∇_Nδ g^ P_P 
       -∇^P(∇_Mδ g_NP+∇_Nδ g_MP)-12/L^2δ g_MN,

where M, N and P are the Lorenz indices of the 7-dimensional space-time after integration over S^4, as in M,N,P=0,1,2,3,τ,r,11, and the source 𝒯_MN is the EMT of the matter fields which here considered only the pion field. 
We solve Eq. (<ref>) under the axial gauge, δ g_Mr = 0. 

In order to examine the EMT 𝒯_MN in the bulk, we embed the D8 branes into the 11-dimensional space, considering the action as

    S_D8= 
       -C/2π R_11g_s∫ dτ dx_11d^4xdrdΩ_4(δ(τ)+δ(τ-π/M_KK)) 
       ×√(-G_(11))/√(G_ττ)1/4G^MNG^PQ tr[F_MPF_NQ]+...,

with C=(64π^6l_s^5)^-1 and the metric tensor G_MN in Eq. (<ref>). We define the EMT as

    𝒯_MN=-2/√(-G_(11))δ S_D8/δ G^MN_(11) Vol(S^4) ,

which leads to the following expression:



    𝒯_MN(Δ⃗,r) ≡ 2π R_11∫ dτ^'𝒯_MN(τ^',k⃗,r), 
       𝒯_μν (Δ⃗,r)=L^5/r^5π^2Cr^2 L^2/3g_s √(f(r))) tr[F_μρF_ν^ρ+(1+z^2)^4/3M_KK^2F_μ zF_ν z-η_μν/4(F_ρσF^ρσ+2(1+z^2)^4/3M^2_KKF_ρ zF^ρ_z)], 
       𝒯_11,11(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[-1/4F_ρσF^ρσ-(1+z^2)^4/3M^2_KK/2F_ρ zF^ρ z], 
       𝒯_μ z(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[F_μρF_z^ρ],    𝒯_zz(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[1/2F_zρF_z^ρ-1/4(1+z^2)^4/3M^2_KKF_ρσF^ρσ],


where the radial coordinate r  (R≤ r<∞) is related to z  (-∞<z<∞) as z=±√(r^6/R^6-1).
In this study, we consider the case where the pion field is the only matter field as the source of the gravitational waves and decompose A_M(x^ν,z) as follows

    A_μ(x^ν,z)=0,     A_z(x^ν,z)=π(x^μ)ϕ_0(z), 
       ϕ_0(z)=c_0(1+z^2)^-1,

where c_0^-2=2M^2_KKκ∫ dz(1+z^2)ϕ_0(z)^2 is the normalization constant of the eigenfunction ϕ_0(z) <cit.>. Since the pion field is an external line, it can be replaced by a plane wave as π(x^ν)∼ e^ip_ν x^ν with incoming momentum p_ν.

Since we are interested in the EMT of QCD at the boundary such as to satisfy Δ^μ<T_μν>=0, we consider the metric fluctuations that are transverse Δ^μδ g_μν∼0 near the boundary, which can be parametrized as δ g_ij∼δ g^ TT_ij+(δ_ij-Δ_iΔ_j/Δ⃗^2)/5δ g^α_α with the transverse-traceless (TT) part δ g^ TT_ij, i,j=1,2,3 and α=μ,τ,11. Under the axial gauge, the Einstein equation requires the six-dimensional traceless condition δ g^α_α∼0 in the asymptotic AdS region (r→∞) labeled by (x^μ,τ,x^11). 
Therefore, we consider TT modes in the metric fluctuation, which is nothing but glueball spectra.
Among the 14 TT modes named in Ref. <cit.> (see also Ref. <cit.>), only T_4 and S_4 contribute to the EMT. Actually, substituting the pion field (<ref>) into equation (<ref>), we confirm that δ g^ other_μν falls faster than T_4 and S_4 mode at r→∞. 
By decomposing the metric fluctuations into each mode, δ g_μν=δ g^ T(2)_μν+δ g^ T(0)_μν+δ g^ S(0)_μν+δ g^ other_μν, the Einstein equations reduce to 

    r^2/L^2[1/L^2r^5∂_r((r^7-rR^6)∂_r)-L^2Δ^2/r^2]L^2/r^2δ g^ T_μν=-2κ^2_7𝒯^ T_μν, 
       r^2/L^2[1/L^2r^5∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^5-2R^6)(r^6+2R^6))) 
       -L^2Δ^2/r^2]L^2/r^2δ g^ S(0)_μν|_r→∞=-2κ^2_7𝒯^ S(0)_μν,

where the source of each mode is decomposed as 𝒯_μν=𝒯^ T_μν+𝒯^ S(0)_μν+𝒯^ other_μν (𝒯^ T_μν≡𝒯^ T(2)_μν+𝒯^ T(0)_μν), and the two T_4-modes are combined as δ g^ T_μν=δ g^ T(2)_μν+δ g^ T(0)_μν, since they obey the same equation. 

Following the same manner as in Ref. <cit.>, each part of EMT is written as 


    2κ^2_7𝒯^ T_μν=2κ^2_7𝒯_μν-2κ^2_7𝒯^ S_μν-2κ^2_7𝒯^ other_μν, 
       2κ^2_7𝒯^ S_μν=r^6+2R^6/4(r^6-R^6)(η_μν-Δ_μΔ_ν/Δ^2)[1/L^4r^3∂_r(r(r^6-R^6)∂_r(a+b)-3R^6(a+b))-Δ^2b], 
       2κ^2_7𝒯^ other_μν=1/L^4r^3∂_r[(η_μν-Δ_μΔ_ν/Δ^2)r(r^6-R^6)∂_r a-3R^6η_μνb],  
       ∂_r a=-3R^6/(5r^6-2R^6)b+9r^11/(5r^6-2R^6)R^62κ^2_7𝒯_zz,    b=-iΔ^μ/Δ^2r^6√(r^6-R^6)/R^9 2κ^2_7 𝒯_μ z,


with the ansatz, 

    δ g^ other_μν(Δ⃗,r)=r^2/L^2Δ_μΔ_ν/Δ^2a(r,Δ), 
       δ g^ other_11,11(Δ⃗,r)=r^2/L^2b(r,Δ), 
       δ g^ other_ττ(Δ⃗,r)=r^2f(r)/L^2(-b(r,Δ)).



We determine the perturbation to the metric caused by the pion field as the bulk EMT source using Green's function method as follows, 

    δ g^ T/S_μν(Δ⃗,r)∼r^2/L^2∫ dr^'√(-G_(7)(r^'))G^ T/S(Δ⃗,r,r^') 
       ×L^2/r^'^22κ^2_7𝒯^ T/S_μν(Δ⃗,r^'),

where, for S_4 mode, we can write as this form in the r→∞ limit, and ∼ implies this point. 
The Green's function G^ T/S is written as 

    G^ T/S(Δ⃗,r,r^')=∑^∞_n=1Ψ^ T/S_n(r)Ψ^ T/S_n(r^')/(m^ T/S_n)^2+Δ⃗^2,

by expanding in the complete system {Ψ^ T/S_n(r)} which obeys the eigenvalue equation, 

    1/r^3L^4∂_r((r^7-rR^6)∂_rΨ^ T_n)=-(m^ T_n)^2Ψ^ T_n, 
       1/r^3L^4∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^6-2R^6)(r^6+2R^6))Ψ^ S_n) 
       =-(m^ S_n)^2Ψ^ S_n,

where m^ T/S_n is n'th glueball mass of T/S mode and Ψ^ T/S_n(r) satisfy the completeness conditions,

    r^3/L^3∑^∞_n=1Ψ^ T/S_n(r)Ψ^ T/S_n(r^')=δ(r-r^'), 
       ∫^∞_R drr^3/L^3Ψ^ T/S_n(r)Ψ^ T/S_m(r)=δ_mn.








Results.—
Substituting the pion field into the EMT of the bulk, we read off the matrix elements of the boundary EMT from Eq. (<ref>) with the dictionary Eq. (<ref>) as follows, 

    A(t)=6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T_nΨ^ T_n(r^')/(m^ T_n)^2+Δ⃗^2r^'^3/L^3P^μν_ t𝒯_μν(Δ⃗,r^'), 
       D(t)= 
       -6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T_nΨ^ T_n(r^')/(m^ T_n)^2+Δ⃗^2r^'^3/L^3P^μν_ s(𝒯_μν+𝒯^ S_μν+𝒯^ other_μν) 
       +6/L^7∫^∞_Rdr^'∑^∞_n=1α^ S_nΨ^ S_n(r^')/(m^ S_n)^2+Δ⃗^2r^'^3/L^3P^μν_ s𝒯^ S_μν, 
       α^ T/S=(m^ T/S_n)^2L^4/6∫^∞_R dr r^3Ψ^ T/S_n(r),
 
where P^μν_ t and P^μν_ s are defined as P^μν_ t=(3P^μ P^ν/P^2-η^μν)/P^2 and P^μν_ s=(P^μ P^ν/P^2-η^μν)/Δ^2. 

To calculate infinite sums, we expand as 

    6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T/S_nΨ^ T/S_n(r^')/(m^ T/S_n)^2+Δ⃗^2=∑^∞_k=0F_k^ T/S(r^')(-Δ⃗^2)^k, 
       F^ T/S_k(r^')=∫_R^∞ drr^3/L^3∑_n=0^∞Ψ_n^ T/S(r)Ψ_n^ T/S(r^')/(m_n)^2k.

From the eigenvalue equations (<ref>), F^ T/S_k(r) satisfies the following equations:

    1/r^3L^4∂_r((r^7-rR^6)∂_rF^ T_k)=-F^ T_k-1, 
       1/r^3L^4∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^6-2R^6)(r^6+2R^6))F^ S_k) 
       =-F^ S_k-1,

where we can prove F^ T/S_0=1 by using Eq. (<ref>). 
We determine F^ T/S_k(r) by solving it as the recurrence formula with boundary conditions on ∂_r F^ T/S_k(r)|_r→ R=1,  F^ T/S_k(r)|_r→∞=0. 

We show the results of GFFs that truncate the expansion at k=10 using Eq. (<ref>) in FIG. <ref> (also including results that truncate at k=8,9), where we set κ =0.00745 and M_KK =949  MeV which are determined by ρ meson mass 776   MeV and the pion decay constant f_π=92.4  MeV <cit.>. 
In the current calculation, D(t) begins to deviate from around t∼0.5   GeV^2 while A(t) converges well in the range 0<t<0.8   GeV^2 on the expansion up to k=10 with momentum transfer. The current approach begins to deviate from QCD in the region where t is close to the M_KK, so further efforts are unnecessary. 

In the top-down holographic QCD approach, the amplitude of D(t) dumps more rapidly than that of A(t), which is qualitatively consistent with the results of lattice QCD <cit.>. 
These differences are due to the contribution of S_4 modes to D(t), which is not observed in the bottom-up holographic QCD framework <cit.>. 
Also in this approach, as in previous studies <cit.>, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. 

As you can see in FIG. <ref>, our result shows D(0)=-1. 
Note that this is a result of our calculations with loop effects in the large Nc limit, unlike the fact A(0)=1, given by the general constraints. 
This is consistent with the low-energy theorem obtained in the soft pion limit <cit.>. The result is as expected since in our present work we consider the Sakai-Sugimoto model, which is the holographic dual of QCD in the chiral limit. Several methods have been proposed on how to introduce quark masses into this model <cit.>, thus analysis beyond the chiral limit is a future work. 










Summary and outlook.—
We calculated the GFFs of pions from the top-down holographic QCD. 
We solve the Einstein equations in bulk theory with pion fields as matter fields and derived the GFFs using the dictionary (<ref>).
Our results are qualitatively compatible with those from lattice QCD <cit.> and successfully reproduce the feature where the behaviors of A(t) and D(t) are different, which is distinct from bottom-up holographic QCD <cit.>.
Additionally, we obtained the D-term as D(0)=-1 in the chiral limit. 

Finally, this study opens up several avenues for future research. First, it is possible to generalize to other hadrons, such as nucleons and vector mesons with non-zero spin. Also, it is very interesting to investigate beyond the chiral limit, because the connection between the Gell-Mann–Oakes–Renner relation and the stability of the stress distribution has been discussed in Ref. <cit.>. It is also important to investigate the behavior of the GFFs and the stress distribution inside hadrons under finite temperatures, densities, and magnetic fields, which change the nature of the interaction in the system. 




Acknowledgements.—
We are grateful to the authors of Ref. <cit.> for generously supplying the data tables utilized in FIG. <ref>.
We also thank Atsushi Hosaka, Kei Suzuki and Shigeki Sugimoto for useful discussions. This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Nos. JP24K17054, JP24KJ1620). 





*










 Paper Title: Universal Euler-Cartan Circuits for Quantum Field Theories.



 

ananda.roy@physics.rutgers.edu
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

Division of Condensed Matter Physics and Material Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA

Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA

§ ABSTRACT 
Quantum computers can efficiently solve problems which are widely believed to lie beyond the reach of classical computers. In the near-term, hybrid quantum-classical algorithms, which efficiently embed quantum hardware in classical frameworks, are crucial in bridging the vast divide in the performance of the purely-quantum algorithms and their classical counterparts. Here, a hybrid quantum-classical algorithm is presented for the computation of non-perturbative characteristics of quantum field theories. The presented algorithm relies on a universal parametrized quantum circuit ansatz based on Euler and Cartan's decompositions of single and two-qubit operators. It is benchmarked by computing the energy spectra of lattice realizations of quantum field theories with both short and long range interactions. Low depth circuits are provided for false vacua as well as highly excited states corresponding to mesonic and baryonic excitations occurring in the analyzed models. The described algorithm opens a hitherto-unexplored avenue for the investigation of mass-ratios, scattering amplitudes and false-vacuum decays in quantum field theories.




Universal Euler-Cartan Circuits for Quantum Field Theories
    David Rogerson
    August 5, 2024
==========================================================






  
*Introduction.
Quantum field theory (QFT), which reconciles quantum mechanics and the special theory of relativity, is fundamental to the description of a vast array of physical phenomena. The latter range from confinement and asymptotic freedom in theories of elementary particles to high-temperature superconductivity and topological order in condensed matter systems. The investigation of non-perturbative characteristics of generic QFTs remains an outstanding challenge with ab-initio computations likely to stay beyond the reach of the most powerful classical computers. Quantum computation <cit.> provides an alternate paradigm for the analysis of QFTs with quantum algorithms boasting exponential speedup over classical ones for the computation of scattering amplitudes in massive scalar <cit.> and fermionic <cit.> theories. 

Despite their potential, it is a daunting challenge to implement quantum algorithms on state-of-the-art quantum simulators and solve QFT problems of widespread interest. This is because the current quantum simulators contain only a few hundred qubits with modest error rates. In the current noisy quantum era, hybrid quantum-classical algorithms, which combine the power of a quantum processor with that of a classical one, are crucial in bridging the enormous divide between purely-quantum algorithms and their classical counterparts. Notable examples include the variational quantum eigensolver <cit.> and its modifications <cit.> as well as the quantum approximate optimization algorithm <cit.>, which have been successful in analyzing problems in quantum chemistry <cit.> and combinatorial optimization <cit.>. 

There are two main elements of a hybrid algorithm that govern its efficacy: i) a parametrized quantum circuit ansatz built with a predetermined pool of unitary operators, ii) an efficient optimization procedure for the parameters specifying the unitary operators. While the second element is typically based on variations of gradient descent methods <cit.>,  there is no natural prescription for the first. The circuit ansatz either relies on the Hamiltonian whose eigenstate is being sought alongside a heuristically-chosen mixer Hamiltonian <cit.> or variations of the coupled cluster theory relevant mostly for chemistry problems <cit.>. Neither of the aforementioned are suitable for generic QFT problems whose Hamiltonians, when mapped to registers of qubits in a quantum computer, could involve long-range couplings <cit.> or might not even be Hermitian <cit.>. Furthermore, many interesting properties of QFTs such as mass-ratios and scattering amplitudes remain encoded in highly-excited states <cit.>, which have largely eluded existing hybrid algorithms. Last but not least, restricting the circuit ansatz to a given pool of unitary operators that does not encompass the entire manifold of possible unitary operators necessarily implies that the result so obtained may not be the globally optimum solution which could potentially be found by considering a larger ansatz. 



In light of the above, it is natural to ask: is there a quantum-classical algorithm based on a universal parameterized quantum circuit ansatz that is tractable with established optimization strategies and applicable to a broad set of QFT problems? The goal of this work is to answer this question. The presented solution computes the optimal circuit over all possible two-qubit unitary rotations required to realize an arbitrary L-qubit unitary rotation <cit.>. The unitary operators are chosen to couple neighboring qubits, which is imposed for potential implementation on majority of quantum computing architectures and could be relaxed in a more general setting. The model under investigation affects the parameters of the circuit, but not the ansatz itself. Importantly, the proposed ansatz explores the largest possible manifold of allowed unitary operators and is closer to the globally optimal solution than what could be obtained by considering a more restrictive ansatz. The potential of the algorithm is demonstrated by computing not only ground states of strongly-interacting QFTs, but also highly excited states that correspond to bound-states of topological excitations occurring in these theories. 




  
*Algorithm. The algorithm is implemented on a quantum processor coupled with a classical one (Fig. <ref>). The former is involved in  applying the parametrized circuit and conducting measurements required for the optimization process. The classical processor receives the measurement outcomes, performs the optimization and feeds the outcome back into its quantum counterpart. 

The algorithm begins with the application of N (see Figs. <ref>, <ref> for representative choices of N) layers of unitary operators on L qubits of the quantum processor initialized in the state |ψ_ i⟩. Each of the N layers is built, in principle, out of two sublayers of two-qubit operators [orange rectangles in Fig. <ref>(b)]. Each two-qubit operator is an element of the group SU(4). It can be written, using Cartan's KAK decomposition <cit.>, as a product of four single-qubit SU(2) rotations and an SU(4)-entangler [Fig. <ref>(c)]. The latter when involving qubits i, i+1 equals e^-i(α X_iX_i+1 + β Y_iY_i+1 + γ Z_iZ_i+1) where X_i, Y_i and Z_i are the Pauli operators of the i^ th qubit. The SU(4) entangler can be realized by three controlled-NOT gates and five single-qubit rotations <cit.> [Fig. <ref>(d)]. Finally, each of the four single-qubit rotations depicted in panel Fig. <ref>(c) can be further decomposed into three single qubit rotations by the well-known Euler angles <cit.>. Using the above, it can be shown that the general circuit ansatz is constructed with two-qubit operators [orange ellipses in Fig. <ref>(b, e)] with ten undetermined angles. 


After the application of the N layers, measurements are performed to determine the relevant gradients and the cost-function L(θ⃗), where θ⃗ contains all parameters of the N layer quantum circuit. To obtain the ground state of a given Hamiltonian H, the cost-function is chosen to be L(θ⃗) = ⟨ψ_ f(θ⃗)|H|ψ_ f(θ⃗)⟩. To obtain the n^ th excited state, the cost-function is augmented to L(θ⃗)→ L(θ⃗) + ∑_m < nλ_m|⟨ψ_m|ψ_ f⟩|^2, with λ_m>0, m = 0,…, n-1. This choice amounts to simultaneously minimizing the energy and the overlap to the n-1 lower-energy states that have been already obtained.  

The measurement results are communicated to the classical processor which checks for the convergence of the cost-function for the given choice of N (iteration convergence in Fig. <ref>). If the condition is not satisfied, the parameters of the circuit are updated using an optimization routine. In this work, the quantum natural gradient optimizer <cit.> is used due to its superior performance in obtaining eigenstates of many-body Hamiltonians <cit.>. In contrast to ordinary gradient descent, this  ensures that optimization follows the steepest descent taking into account the geometry of the space of wavefunctions <cit.> and can viewed as the quantum analog of the classical information geometric approach <cit.>. The parameters at the (t+1)^ th iteration are related to the t^ th one by

    θ⃗_t + 1 = θ⃗_t - η g^-1∂ L/∂θ⃗_t,

where η is the learning rate (typically a real number < 1) and g is the Fubini-Study metric <cit.>. The latter is given by the real part of the quantum geometric tensor G:

    G = ⟨∂ψ_ f/∂θ_i|∂ψ_ f/∂θ_j⟩ - ⟨∂ψ_ f/∂θ_i|ψ_ f⟩⟨ψ_ f|∂ψ_ f/∂θ_j⟩.

Note that both the gradients need to be evaluated using measurements for both the cost-function and the Fubini-Study metric <cit.>.
Subsequently, the updated set of parameters are fed back to the quantum processor and the process is repeated until the iteration convergence criterion is satisfied. Once the latter is satisfied, a second layer convergence criterion is verified. This ensures that the number of layers chosen is sufficient to attain the desired degree of convergence. If this condition is not satisfied, the number of layers is grown by 1 and the earlier steps are repeated. Else, the algorithm terminates. When the number of layers is increased from N to N+ 1, the optimized parameters obtained for the N-layer circuit, augmented by an initial guess for the added layer, were used as the choice for the optimization process for N+ 1 layers. This improved the stability of the algorithm.

For N layers and a ring of L qubits, the total number of parameters determined by the above algorithm is 10LN, where the factor 10 originates from the number of free parameters in the two-qubit unitary operator of Fig. <ref>(e). Despite being tractable [see Fig. <ref>(c) for demonstration], conservation of symmetries present in the target Hamiltonian can drastically reduce the number of parameters to be determined. Symmetries can be conserved by modifying the general circuit ansatz of Fig. <ref> and allowing only operators that respect this symmetry. Alternatively, the cost-function can be augmented to ensure that the obtained state lies in the correct symmetry sector. For example, states which are invariant under translation by r lattice site(s) can be obtained by starting with a state and circuit ansatz respecting this symmetry. For r = 1(2), the number of undetermined parameters for N layers reduces to 10N (20N). Alternatively, in order to obtain a ±1 eigenstate of the translation operator T, the cost-function can be augmented as: L(θ⃗)→ L(θ⃗) ∓μ⟨ψ_ f(θ⃗)|T|ψ_ f(θ⃗)⟩, where μ is a positive real number. Similar considerations apply to symmetry operators related to parity/particle number conservation. 





  
*Results. The proposed algorithm is benchmarked using three paradigmatic one-dimensional QFTs. The results were obtained by modifying standard time-evolution routines for matrix product states. In all three cases, the quantum processor considered comprises a periodic ring with a fixed number of qubits with nearest-neighbor connectivity. This is independent of the model under consideration. The results obtained using the optimized quantum circuit are compared with those from density matrix renormalization group (DMRG) and exact computations. 

First, the critical Ising model in a longitudinal magnetic field is analyzed. In the continuum limit, the QFT contains eight mesonic excitations which scatter with a diagonal scattering matrix <cit.>. These mesons can be viewed as an extremal (g→ 1) limit of bound states formed by the Ising domain walls in the ferromagnetic phase g < 1. The magnetic field leads to a linearly growing energy of separation of two domain walls <cit.>, akin to what happens in quarks in quantum chromodynamics <cit.>, see Fig. <ref>(c). The QFT describing this model is integrable and the number of mesons and their mass ratios are given by the rank and the exponents of the Lie algebra E_8 respectively <cit.>. The relevant lattice Hamiltonian, imposing periodic boundary condition, is:

    H_ I = -∑_j = 1^L X_jX_j+1 - g∑_j=1^LZ_j - h∑_j=1^LX_j,

where g and h were chosen to be 1 and ≃0.156 respectively. Figs. <ref>(a) and <ref>(a) together present the results for the lowest eight translation-invariant states obtained using the proposed algorithm. The circuit ansatz was chosen to be invariant under translation by 2 sites. Furthermore, the cost-function was augmented to ensure that only +1 eigenstates of the operator for single-site translation were obtained. Fig. <ref>(a) shows the results for the ground state energy for system-sizes between L = 8 and 20 in steps of 4. The number of layers required to obtain a reasonable approximation of the ground state do not scale with the system-size, which is compatible with the observations of Ref. <cit.>. In fact, N=2 is sufficient to obtain  the ground state with error in the energy < 1% [Fig. <ref>(d)]. Fig. <ref>(a) shows the results for the first seven excited states for L = 8. For the chosen convergence criterion of 10^-3, the maximum error in the obtained energies was 0.58%. In contrast to the ground state, the number of layers required to achieve this precision ranged between L and 3L. The different translation-invariant excited states can be classified as lattice counterparts of the single and multiple meson states existing in the QFT. The mass ratio for the lowest two mesonic excitations [square and diamond markers in Fig. <ref>(a)] obtained for L = 8 obtained using the proposed algorithm is 1.39 which is close to the expected value of 1.41 obtained using exact diagonalization of  the corresponding spin-chain. This should be compared with the QFT prediction <cit.> given by 2cos(π/5)≃ 1.618. 




Next, the three-state Potts model in the presence of a magnetic field <cit.>  is analyzed. In contrast to its two-state counterpart described above, this model exhibits a richer state of excitations. The three states of each Potts spin are encoded in the triplet of states of two qubits: 

    |0⟩ = |↑↑⟩, |1⟩ = 1/√(2)(|↑↓⟩ + |↓↑⟩), |2⟩ = |↓↓⟩.

The relevant Hamiltonian is

    H_ P = -∑_j  = 1^L'(σ_jσ_j+1^† + gτ + h σ +  h.c.),

where σ and τ are the shift and clock matrices respectively <cit.>. Periodic boundary condition was imposed and L' = L/2 is the number of Potts spins encoded in L qubits. For g< 1, h = 0, the model exhibits a ferromagnetic phase with three degenerate vacua and six domain walls [the three shown in Fig. <ref>(d) and those obtained by reflection about the domain wall]. The presence of the magnetic field breaks the degeneracy of the ground space, giving rise to one true vacuum and two false vacua. The domain-walls experience a confining potential as in the Ising case. However, now there are not only mesonic excitations formed by two domain walls, but also baryonic excitations formed by three domain walls [Fig. <ref>(d)]. Notice that H_ P commutes with the ℤ_2 charge operator C which acts trivially on the state |0⟩ and interchanges |1⟩ and |2⟩. Figs. <ref>,<ref> together show the results for g = h = 0.1 for the lowest eight translation-invariant states with eigenvalue +1 for the operator C. The 2-site translation-invariant circuit ansatz was chosen with the C-eigenvalue imposed using the cost-function. As in the Ising case, the ground state is reached with depth independent of the system-size, with errors in energy less than 0.1% for two layers [Fig. <ref>(b, d)]. The excited states were obtained using between L and 3L layers with maximum error in the obtained energies being 2.26% [Fig. <ref>(b)]. The first excited state is one of the false vacua, with the rest comprising mesonic and baryonic excitations. This is verified by counting the average number of domain-walls in the corresponding state. 

Finally, results are presented for the canonical model for quantum electrodynamics in two space-time dimensions: the one-flavor massive Schwinger model <cit.>. After mapping to an array of qubits, the resulting Hamiltonian with open boundary conditions is given by <cit.>:

    H_ MS   = 1/2∑_j = 1^L  - 1(X_jX_j+1 + Y_jY_j+1) + m∑_j = 1^LP_j 
         + g^2/2∑_j = 1^L - 1[∑_k =1^j (-1)^kP_k]^2,

where P_j = [1 + (-1)^jZ_j]/2. In contrast to the Ising and Potts models, this model involves an all-to-all coupling Hamiltonian. To obtain the ground state of this model, the general circuit ansatz of Fig. <ref> is used. Note the existence of the conserved U(1) charge Q, [H, Q] = 0, where Q = ∑_j=1^LZ_j. To test the algorithm under the most challenging of conditions, the ground states were obtained for system sizes between L = 8 and 16 without conserving this charge. For these simulations, m = 0.5, g = 0.3 were chosen. Despite the long-range nature of the model Hamiltonian, reasonable approximations of the ground-state were obtained with circuit-depths that are independent of the system-size with energies within 1% accuracy obtained within application of 2 layers. Excited states were also obtained similar to the Ising and Potts models and are not shown here for brevity.



  
*Conclusion. To summarize, a hybrid quantum-classical algorithm is presented for the investigation of strongly-interacting QFTs. In contrast to existing approaches that rely on heuristic arguments for choosing the parametrized quantum circuit ansatz, a universal ansatz is described relying on Euler's and Cartan's decomposition of single and two-qubit operators. The algorithm is benchmarked by computing eigenstates of lattice models for three paradigmatic one-dimensional QFTs: the Ising, Potts and one-flavor massive Schwinger models. Quantum circuits were obtained for the realization of ground states, false vacua, mesonic and baryonic excitations occurring in these models. 

The presented results demonstrate the power of geometric control theory <cit.> for the analysis of QFTs. While geometric quantum control is known to be the central concept in the search for an optimal circuit for quantum computation <cit.>, previous proposal are centered around minimization of the distance in the Finsler space of unitary operators. In contrast, the current work presents an alternative framework based on the Fubini-Study metric and a universal circuit ansatz. The proposed framework is not only numerically-tractable, but also readily-implementable on current quantum machines, opening the door to hybrid quantum-classical analysis of a large class of QFTs. 




  
*Acknowledgements.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. 





 Paper Title: Transmission of quantum information through quantum fields in curved spacetimes.

 







mk7592@princeton.edu
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA


erickson.tjoa@mpq.mpg.de
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany


§ ABSTRACT 

We construct a relativistic quantum communication channel between two localized qubit systems, mediated by a relativistic quantum field, that can achieve the theoretical maximum for the quantum capacity in arbitrary curved spacetimes using the Unruh-DeWitt detector formalism. Using techniques from algebraic quantum field theory, we express the quantum capacity of the quantum communication channel purely in terms of the correlation functions of the field and the causal propagator for the wave equation. Consequently, the resulting quantum channel, and hence the quantum capacity, are by construction manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the field.  



Transmission of quantum information through quantum fields in curved spacetimes
    Erickson Tjoa
    August 5, 2024
===============================================================================







§ INTRODUCTION


In the spirit of it-from-(qu)bit, quantum information theory has been extensively used to understand generic physical phenomena from quantum many-body physics to high-energy physics, from applied to foundational physics. Relativistic quantum information (RQI) is one of the many attempts to realize this spirit, which thus far focuses on the role of relativity in quantum information-processing tasks, as well as a deeper understanding of fundamental theories such as relativistic quantum field theories (QFT). The role of quantum information theory in relativistic QFTs cannot be understated: a non-exhaustive list includes rigorous study of the entanglement structure, complexity, and measurement theory for QFTs <cit.>; operational approach to particle production due to non-inertial motion and gravitational fields <cit.>; connections between relativistic causality, information-theoretic causality and reference frames <cit.>;  quantum information protocols mediated by the relativistic quantum field <cit.>; etc. 

This work is motivated by a somewhat vague, ambitious, and fundamental question: 

    Problem: What is the information-carrying capacity of quantum fields?

One can think of several ways to answer this question by making the problem statement more precise. For our purposes, we will adopt an operational approach, namely, whether two parties Alice and Bob, each in possession of some quantum-mechanical probes (atoms, qudits, oscillators, etc.) are able to transmit (quantum) information using the relativistic quantum field as a medium — that is, as a relativistic quantum channel mediating the communication between Alice and Bob. The paradigmatic model of such a communication channel is through the Unruh-DeWitt (UDW) particle detector model and its generalizations <cit.>. Indeed, transmission of classical information has been analyzed by many authors both perturbatively and non-perturbatively and in flat and curved spacetimes <cit.>. Transmission of quantum information in the non-perturbative regime has been studied only in flat spacetimes in very specific examples <cit.>. Note that in all these cases, the communication is encoded through the field amplitude and not, say, through the polarization degrees of freedom of the electromagnetic field, since on its own the polarization is blind to the relativistic features and constraints. 

Here we present the most general formulation for the transmission of quantum information between Alice and Bob, each in possession of a two-level system (a `qubit detector'), by suitable coupling to a relativistic scalar field in arbitrary curved spacetime. Clearly, our work constitutes a natural extension of the flat spacetime result in <cit.>. However, we will do so by exploiting techniques from algebraic quantum field theory (AQFT) <cit.>. Crucially, what we gain from this will not be a simple rewriting of all the formulae in the literature: we will see that the quality of the relativistic quantum channel mediated by the quantum scalar field can be formulated purely in terms of smeared correlation functions of the field and the detector parameters. Our results make clear what features of the field-mediated communication channel are necessary for perfect transmission of quantum information, and they are by construction manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and (to some extent) the Hilbert space representations of the field. In particular, it automatically works for all quasifree representations that give rise to Fock representations <cit.>).

This paper is organized as follows. In Section <ref> we briefly review the scalar field theory in curved spacetimes using the AQFT framework. In Section <ref> we describe our UDW detector setup and review the quantum capacity of a quantum channel. In Section <ref> we give the protocol that allows the transmission of quantum information through the quantum field. In Section <ref> we discuss the validity of the channel and study the regimes in which it fails. In Section <ref> we conclude with some discussions of our results and outlook. We set c=ħ=1 and adopt mostly-plus signature for the metric.




§ SCALAR FIELD THEORY IN CURVED SPACETIME




In this section we review the scalar field theory in curved spacetimes within the AQFT framework, adapting the convention given in <cit.> (for more details and reviews, see also <cit.>). 



 §.§ AQFT for scalar field


Consider a Klein-Gordon (KG) scalar field ϕ in an (n+1)-dimensional globally hyperbolic Lorentzian spacetime ℳ with metric tensor g_ab, whose equation of motion is given by the KG equation: 

    (-∇_a∇^a + m^2 +ξ R)ϕ = 0 ,

where ∇_a is the covariant derivative with respect to the Levi-Civita connection, m is the mass parameter, R is the Ricci scalar curvature, and ξ≥ 0 controls the non-minimal coupling of ϕ to R. Global hyperbolicity implies that we have a foliation of spacetime Σ and the KG equation is well-posed with initial data given on Cauchy surfaces. 

The KG operator P̂ -∇_a∇^a + m^2 +ξ R admits a unique advanced and retarded Green's functions G_A(,),G_R(,) = G_A(,) which satisfies

    P̂_ G_R/A(,) = δ^n+1(-) ,

where the subscript on P̂_ means the KG operator is w.r.t. the variable . We can use these Green's functions to construct the antisymmetric causal propagator (also known as the Pauli-Jordan distribution)

    E(,)  G_R(,) - G_a(,) .

Canonical quantization of the field ϕ gives rise to a field operator ϕ̂ obeying the canonical commutation relations (CCR) 

    [ϕ̂(),ϕ̂()] = i E(,) .

As ϕ̂ is an operator-valued distribution, in the algebraic approach to QFT we consider smeared field operators, i.e., by viewing the field operator as an -linear map from the test space of smooth compactly supported functions  to a *-algebra (), i.e., as a map ϕ̂: C^∞_0 ()→𝒜() with

    ϕ̂(f)∫ V  f()ϕ̂() .

The algebra of observables () is defined to be the collection of sums of products of ϕ̂(f) over all functions in  obeying the smeared CCR relation (<ref>), i.e., a CCR algebra <cit.>. The KG equation is encoded in the fact that we identify ϕ̂(f) ≡ 0 if f=P̂h for some h∈. 



The dynamical content of the field theory is reflected in terms of the solution space of the KG equation. The solution space _() can be equipped with a symplectic form σ:_()×_()→, defined as

    σ(ϕ_1,ϕ_2) ∫_Σ_tΣ^a [ϕ_1∇_aϕ_2 - ϕ_2∇_aϕ_1] ,

where Σ^a = -t^a Σ, -t^a is the inward-directed unit normal to the Cauchy surface Σ_t, and Σ = √(h) ^n is the induced volume form on Σ_t <cit.>. As is well-known, this definition is independent of the choice of Cauchy surface. The field operator ϕ̂(f) can be expressed as symplectically smeared field operator  <cit.> 

    ϕ̂(f) ≡σ(Ef,ϕ̂) ,

and the CCR algebra can be written as 

    [σ(Ef,ϕ̂),σ(Eg,ϕ̂)] = iσ(Ef,Eg) = i E(f,g) ,

where σ(Ef,Eg) = E(f,g) in the second equality follows from Eq. (<ref>) and (<ref>). 

Since the smeared field operators are unbounded operators, for free fields it is more convenient technically to work the Weyl algebra (), whose elements are bounded operators constructed by “exponentiating” the field operators. The Weyl algebra () is a unital C^*-algebra generated by elements that formally take the form 

    W(Ef) ≡e^iϕ̂(f) ,   f∈ .

These elements satisfy Weyl relations:

    W(Ef)^†   = W(-Ef) , 
       
        W(E (P̂f) )    =  ,
    
        W(Ef)W(Eg)    = e^-i/2E(f,g) W(E(f+g))

where f,g∈. The third relation enforces relativistic causality (or microcausality).


In AQFT, a quantum state is defined as a -linear functional ω:()→ (similarly for ()) such that 

    ω() = 1 ,  ω(A^† A)≥ 0  ∀ A∈() .

The state ω is pure if it cannot be written as ω= αω_1 + (1-α)ω_2 for any α∈ (0,1) and any two algebraic states ω_1,ω_2; otherwise we say that the state is mixed. 


The way to pass from AQFT to the Hilbert space approach is through Gelfand-Naimark-Segal (GNS) reconstruction theorem <cit.>. The GNS theorem gives us, for a given algebra of observables () and a state ω, a GNS triple (ℋ_ω, π_ω,|Ω_ω⟩), where π_ω: ()→ℬ(ℋ_ω) is a Hilbert space representation with respect to state ω. In its GNS representation, any state ω is realized as a vector state |Ω_ω⟩∈ℋ_ω and  A∈() are represented as bounded operators Âπ_ω(A)∈ℬ(ℋ_ω), and we write ω(A) = ⟨Ω_ω|Â|Ω_ω|$⟩. Since there exists infinitely many unitarily inequivalent representations of the CCR algebra, the algebraic approach allows us work with all representations at once and only pick the physically relevant representation at the very end. 

For a fixed stateω, we can construct the Wightmann-point functions, defined by

    𝖶(f_1,...,f_n)ω(ϕ̂(f_1)...ϕ̂(f_n))

wheref_j∈and the RHS is computed within some GNS representation of(). The GNS representation of the Weyl algebra()allows us to calculate Eq. (<ref>) by differentiation[Strictly speaking, the state on () and on () are different, but they are in one-to-one correspondence whenever they are related by Eq. (<ref>) (see <cit.> for good exposition on this). ]: for example,  the smeared Wightman two-point function reads

    𝖶(f,g) ≡ -∂^2/∂ s∂ t|_s,t=0ω(e^iϕ̂(sf)e^iϕ̂(tg))

where the RHS is calculated in the GNS representation of()<cit.>. 

The consensus within the AQFT community is that physically reasonable states should be Hadamard states<cit.>. Very roughly speaking, these states respect local flatness and the expectation values of all observables (in particular, the renormalized stress-energy tensor) are finite <cit.>. A particularly nice subclass of Hadamard states is the family of quasifree states: for these states, all odd-point functions in the sense of (<ref>) vanish and all higher even-point functions can be written as in terms of just two-point functions. In modern terminology, the term quasifree state is synonymous with Gaussian state, where the one-point functions need not vanish and higher-point functions only depend on one- and two-point functions <cit.>.

The relevance of quasifree states lies in the fact that they are completely specified once we know the Wightman two-point functions. More precisely, any given quasifree stateωis associated with a real-bilinear inner productμ:_()×_()→that satisfies <cit.>
    |σ(Ef,Eg)|^2 ≤μ(Ef,Ef)μ(Eg,Eg) ,

where we recall thatEf,Eg∈_()for allf,g∈. The inequality is saturated ifω_μis a pure quasifree state. Any quasifree state can then defined by those that satisfy

    ω(W(Ef))  e^-μ(Ef,Ef)/2 .

However, this definition is not useful unless we can compute the induced norm||Ef|| √(μ(Ef, Ef)). 

It turns out that||Ef||is related to the Wightman two-point function as follows <cit.>: we complexify_()into the space of complex solutions_()and define the KG bilinear product by 

    ⟨φ_1,φ_2|_⟩ iσ(φ_1^*,φ_2)

forφ_1,φ_2∈_(). It can be shown that for any quasifree state, there exists a subspaceℋ⊂_()such that(ℋ,⟨·,·|_⟩)is a Hilbert space and an-linear mapK:_()→ℋsuch that for allφ_1,φ_2∈_()* K_()+i K_() is dense in ℋ;
    
  * μ(φ_1,φ_2) = ⟨Kφ_1,Kφ_2|_⟩;
    
  * σ(φ_1,φ_2) = 2⟨Kφ_1,Kφ_2|_⟩;
    
  * _()≅ℋ⊕ℋ, where ℋ is the complex-conjugate Hilbert space of ℋ and ⟨u,v|_⟩=0 for all u∈ℋ and v∈ℋ.

In a more familiar language of canonical quantization, the mapKprojects to the “positive-frequency part” of a real solution to the KG equation. The Wightman two-point function is then given by <cit.>
    𝖶(f,g)    = ⟨KEf,KEg|_⟩
       = μ(Ef,Eg) + i/2E(f,g) ,

where we have used the fact thatσ(Ef,Eg) = E(f,g). SinceE(f,g)is antisymmetric, it follows that||Ef||^2 = μ(Ef,Ef) ≡𝖶(f,f)and hence

    ω(W(Ef)) =   e^-1/2𝖶(f,f) .

Indeed, we may very well take Eq. (<ref>) as the definition of quasifree states. 




The most important example of a quasifree state is the “vacuum state”ω, where we can write the (unsmeared) vacuum Wightman function as 

    ω(ϕ̂()ϕ̂()) ≡𝖶(,)    = ∫^n  u^*_() u^*_() ,

whereu_()are called the positive-frequency modes of KG operator with respect to the KG inner product

    (ϕ_1,ϕ_2)_kg iσ(ϕ_1^*,ϕ_2) ,

whereϕ_j∈_()are complexified solutions to Eq. (<ref>). The vacuum stateωis not unique and there is a sense in which every GNS vector is a “vacuum state” in that particular representation. However, once a particular basis of modes{u_}are chosen, it fixes the vacuum state as one whose Wightman function satisfies (<ref>). In flat spacetime, the standard plane-wave basis in the inertial coordinates defines the Minkowski vacuum, which is the only state that respects the full Poincaré symmetry and minimizes the expectation value of the (renormalized) stress-energy tensor <cit.>. In curved spacetimes, there are multiple inequivalent states that qualify as a vacuum state <cit.>.




 §.§ Relationship with canonical quantization
 

The usual Fock space in canonical quantization arises from the GNS reconstruction applied to()with respect to some vacuum stateω. Technically speaking, any pure quasifree (Hadamard) state qualifies as a vacuum state, as their GNS representation gives rise to the Fock vacuum in the standard sense <cit.> (see also <cit.> for a different definition in Minkowski spacetime). This is the best one can do in generic curved spacetimes as there is no preferred vacua without further assumptions such as time translation symmetry. 

In more detail, the GNS Hilbert spaceℋ_ωfor the pure quasifree representation is the Fock space over the one-particle Hilbert spaceℋℋ_ω≡ = ⊕_n=0^∞ℋ^⊙ n ,

whereℋ^⊙0≅andℋ^⊙nmeans symmetrized direct sum (then-particle sector of the Fock space). In this representation, we can write

    π_ω(ϕ̂(f)) = â((KEf)^*) + â^†(KEf) .

In what follows we dropπ_ωif it is clear from the context that we are using a Fock representation. Note that if we consider complex smearing functionsf:→, we can write

    ϕ̂(f) ≡ϕ̂( f) + i ϕ̂( f) .

The operatorsâ(u^*),â^†(v)are smeared ladder operators[There are various conventions on how to label the smeared ladder operators (see <cit.>). We follow <cit.> in that it views â,â^† as being linear in their arguments. The convention in <cit.> does not include complex conjugation in the argument for â(·) and views â as an anti-linear map while â^† is linear.] obeying the CCR

    [â(u^*),â^†(v)] = ⟨u,v|_⟩

on a suitable dense domain of the Fock space.

The standard canonical approach is recovered by working with the unsmeared field operatorϕ̂()and considering the Fourier mode decomposition

    ϕ̂() = ∫^n â_ u_() + â_^† u_^*() ,

whereu_()andu_^*()are positive- and negative-frequency modes normalized to Dirac delta functions via the KG inner product:

    ⟨u_,u_'|_⟩   = δ^n(-') ,
    ⟨u^*_,u^*_'|_⟩   = -δ^n(-') ,
    ⟨u_,u^*_'|_⟩   = 0 .

These modes are not proper elements of the one-particle Hilbert space but they are convenient to work with. In particular, we see thatâ^*_≡â(u_^*)andâ_^†≡â^†(u_). If we were to pick any other quasifree state, the resulting one-particle structure will differ and this can give rise to Hilbert space representations that are unitarily inequivalent, e.g., by considering Kubo-Martin-Schwinger (KMS) thermal states <cit.>. 

Using these modes, we can decompose the solutionEfin the “eigenmode basis”{u_,u_^*}Ef ≡∫^n ⟨u_,Ef|_⟩u_ + ⟨u^*_,Ef|_⟩u_^* .

From the CCR and the definition of the Wightman function, we see that

    iE(,')    = ∫^n u_()u_^*(') - u_^*()u_(') ,

the positive-frequency partKEfofEfreads

    KEf    = ∫^n ⟨u_,Ef|_⟩u_≡∫^n f_ u_

where

    f_-i∫ V f()u_^*() .
 
Furthermore, sinceKEf lies inℋ, it is convenient to use the notation|KEf⟩∈ℋand|u_⟩for the (improper) basisu_. In this notation, we write

    |KEf⟩ = ∫^n f_|u_⟩ ,

This suggests that we can label the ladder operators using the “momentum space smearing function”f_rather than the “real space positive-frequency solution”KEf, i.e., we can write

    ϕ̂(f)    = â(f_^*) + â^†(f_) ≡∫^n â_^†f^*_ + â^†_f_
       ≡ϕ̂(f_) .

Essentially, we are using the elements of the one-particle Hilbert spaceℋto label the field observables rather than using the spacetime smearing functions. This notation is particularly convenient if the focus is more about which modes of the field that the detectors are coupled to <cit.>.



§ SETUP


In this section, we introduce the UDW detector model that we will use to couple Alice's and Bob's qubits to the field and briefly review the notion of quantum capacity for a quantum channel and coherent information. These are relevant measures of how much quantum information can be transmitted (on average) through the quantum channel between Alice and Bob.



 §.§ UDW detector model


In the UDW framework, Alice and Bob carry their own qubit (`detector') with a free HamiltonianĤ_ν= 1/2Ω_νσ̂_z, whereν∈A, B,Ω_νthe energy gap andσ̂_zthe Pauli-Zoperator. We denote the eigenstates ofσ̂_sby|±_s⟩withs ∈x,y,z, where we do not label the Pauli matrices withνto reduce notational clutter and the context will make it clear.  The excited and ground states of the free HamiltonianĤ_νare given by|±_z⟩with eigenvalues±Ω_ν/2respectively.

The UDW interaction between Alice's and Bob's detectors and the field is assumed to be linear in the field as it represents a simplified model of dipole interaction𝐝̂·𝐄̂in quantum optics, i.e., as a spin-boson type interaction <cit.>. The general form of such an interaction is given by a linear combination of the form[This is a slight modification of the usual spin-boson model since typically spin-boson models are assumed to have time-independent interactions <cit.> and to account for the covariant formulation of the UDW model <cit.>.]
    ĥ_I,ν() ∑_iλ_ν,i f_ν,i() m̂_ν,i(τ()) ⊗ϕ̂() ,

whereλ_ν,iis the coupling strength,m̂_ν,i(τ)is a Hermitian operator acting on the qubit, andf_ν,i()is a spacetime smearing function defining the duration of the interaction and the spatial profile of the detector. The most commonly used UDW model involves only one term inĥ_I,ν() = λ_νf_ν()m̂_ν(τ())⊗ϕ̂()<cit.>, but we will need the more general version in (<ref>) for our purposes. We drop the tensor product symbol if the context is clear. 


The time evolution operator generated by the interaction Hamiltonian (<ref>) reads

    Û = 𝒯_t exp[-i ∫_V∑_νĥ_I,ν]

whereV [n] √(-g)is the invariant volume element and the time-ordering is assumed to be given by some global time function (see <cit.> for discussions about some subtleties of time-ordering for the spatially smeared UDW model). In general, it is not possible to solve the the time evolution analytically due to the time-dependence of the interaction, thus many analyses involving the UDW detectors consider perturbative methods in the weak coupling regimes. However, our goal in this paper is to establish a perfect quantum channel between Alice and Bob and as was explained in <cit.>, this requires our model to be in the non-perturbative regime.

In order to construct the perfect quantum channel that can transmit quantum information perfectly, we need to be able to calculate the interaction unitaryÛnon-perturbatively (in the sense of not truncating the Dyson series in the weak coupling regime). For this purpose, we will need to use the so-called delta-coupling regime <cit.>, where the interaction timescale is assumed to be much faster than all the relevant timescales of the problem so that the interaction can be taken to occur at a single instant in time. Delta-coupled detectors have been used in many contexts such as entanglement extraction, relativistic communication, work extraction, and many others (see, e.g., <cit.>). As each detector has its own rest frame, we need to consider the Fermi normal coordinates (FNC) associated with each observer  <cit.>, whose center-of-mass (COM) trajectory is parametrized by their respective proper times <cit.>.

The physically reasonable assumption we make is that in the FNC, ≡(τ,), the spacetime smearing functions can be decomposed as

    f(()) = χ(τ)F() ,

where(τ,=0)labels the COM trajectory. In other words, Alice (and Bob) can in their rest frames distinguish the spatial profile (which is assumed to satisfy `Born rigidity', i.e., time-independent with respect toτ) from the switching function that controls the duration and strength of the interactions. The Hamiltonian density in this case reads


    ĥ_I,ν() ∑_iλ_ν,iχ_ν,i(τ)F_ν,i() m̂_ν,i(τ ) ϕ̂(())

and the delta coupling interaction corresponds to the case when

    χ_ν,i(τ) = δ(τ-τ_ν,i)

for some constantsτ_ν,i∈. 

For our purposes, we consider the scenario where Alice and Bob each interact with the field via delta coupling twice, so that we need to consider multiple interaction timesτ_ν,iwithτ_ν,1 < τ_ν,2. More generally, if Alice (or Bob) performs a sequence ofNdelta interactions withτ_ν,i<τ_ν,i+1, the corresponding unitary time evolutionÛ_νcan be decomposed into a sequence of simple-generated unitaries<cit.> (see also <cit.>)

    Û_ν   = ∏_j=1^NÛ_ν,j ,  Û_j = exp[-i λ_νm̂_ν(τ_ν,j)  ϕ̂(F_ν,j)]

The time-ordering𝒯is taken care of since the unitariesÛ_iare written in time-increasing order. When two or more observers are involved, we can use the global time coordinates to orderÛ_aandÛ_b: for example, if Alice and Bob are spacelike separated then the ordering does not matter, while Alice unitary acts before BobÛ_bif(f_b)is in the causal future of(f_a). The unitaryÛ_νcan be computed non-perturbatively sinceÛ_jis a controlled unitary: for any two Hermitian operatorsÂ, B̂such thatÂ=∑_i=1^ra_j|a_j⟩⟨a_j|is finite-dimensional operator in its spectral decomposition, we can write

    exp[i Â⊗B̂] = ∑_j=1^r|a_j⟩⟨a_j|⊗exp[ia_j B̂] .

As we will see, for our setup the full unitary time evolutionÛwill consist of products of Alice's two simple-generated unitaries and also Bob's two simple-generated unitaries.

Last but not least, let us recall the unitaries found in <cit.> for Minkowski spacetime. There Alice performs two simple-generated unitaries, but one of them involves coupling to a conjugate momentum.

    Û_a = e^iσ̂_x π̂(F_a)e^iσ̂_z ϕ̂(F_a) ,

where the spatially smeared operators are

    ϕ̂(F_a)    = λ_1∫^3 F_a()ϕ̂(t_a,) ,
    π̂(F_a)    = λ_2∫^3 F_a()∂_tϕ̂(t_a,) .

Note that here they taket_at_A,1andt_A,2 = t_a+ϵand consider the limit asϵ→0^+so that the unitary generated by the conjugate momentum acts later. Bob's unitary is then constructed as a “suitable inverse” of Alice's unitary to faithfully extract Alice's qubit. 

Our construction requires us to make two modifications to the above unitaries. First, we can generalize the conjugate momentum operator in curved spacetime, i.e., 

    π̂() ∂_tϕ̂()≡√(h) t^a∇_a ϕ̂() ,

wheret^ais the forward pointing unit normal, orthogonal toΣ_twith an induced metrich. Second, it is straightforward to see from integration by parts that 

    π̂(f)    = ∫ V f()∂_tϕ̂() = ϕ̂(-∂_tf) .

Thus we can always view the smeared conjugate momentum operator as a smeared field operator with a different smearing function (namely, its time derivative). In Minkowski spacetime, using the spacetime smearing notation we can thus write

    π̂(F_a)≡π̂(δ· F_a) = ϕ̂(-δ'· F_a)  ,

whereδ'(t)is the distributional derivative of the Dirac delta functionδ(t). This second step might appear unconventional sinceπ̂is not usually regarded as a spacetime-smeared operator, but this provides a simple way to recast all basic observables in terms of a single field operatorϕ̂. The connection between the modern AQFT approach based on spacetime-smeared operators (cf. Section <ref>) and the “old” (non-covariant) AQFT formulations based on two spatially smeared operators{ϕ̂|_Σ,π̂|_Σ}at a fixed Cauchy sliceΣwith equal-time CCR[ϕ̂(t,),π̂(t,)]=iδ^(n)(-)can be made more precise and rigorous  (see, e.g., <cit.>, and also <cit.> for details). 

The two modifications above suggest that what we really need for the generalization to curved spacetime is that Alice couples to two smeared field operatorsϕ̂(f_1),ϕ̂(f_2)that do not commute, i.e.,E(f_1,f_2)≠0, and not necessarily to the field and its conjugate momentum. Indeed, in the modern AQFT framework the coupling to momentum is somewhat awkward because the perfect quantum channel in <cit.> requires that Alice couples toϕ̂(F_a)more strongly thanπ̂(F_a), but the coupling strengthsλ_1,λ_2would need to have different units, so one needs an additional length scale to compare their strengths. By considering different smeared field operatorsϕ̂(f_i), we can haveλ_iwith the same units and we could use other length scales (e.g., from the effective size of the spatial smearing) if we insist on interpreting one of them as conjugate momentum. In what follows we assume thatf_i(and henceλ_i) have the same units for simplicity.




 §.§ Quantum capacity and coherent information


Here we briefly review the notion of quantum capacity of a quantum channel and quantum coherent information, following closely the exposition in <cit.> (see <cit.> for more in-depth details). We denote by𝒟(ℋ)the space of density operators associated with some Hilbert spaceℋ. LetΦ:𝒟(ℋ_a)→𝒟(ℋ_b)be a quantum channel — a completely-positive trace-preserving (CPTP) map — between Alice's and Bob's density operators. 

An ideal quantum communication channel is one that is able to send any quantum state from Alice to Bob, or equivalently, one that preserves entanglement reliably since Alice's state can always be viewed as an entangled state with some environment that purifies it.  Schematically the idea is the following. First, Alice prepares a purification of the stateρ_a∈𝒟(ℋ_a), namely,ρ_ea|ψ_ea⟩⟨ψ_ea|∈ℋ_e⊗ℋ_awithℋ_a≅ℋ_e. Then Alice applies an encoding channelℰ:𝒟(ℋ_e⊗ℋ_a)→𝒟(ℋ_e⊗ℋ_a^⊗N)to map her share of pure state intoNquantum systems, i.e.,

    ρ_ea,N (_e⊗ℰ)(ρ_ea) .

After encoding, Alice sends each quantum system throughNindependent uses of the communication channelΦ, i.e.,

    ρ_eb,NΦ^⊗ N(ρ_ea,N)∈𝒟(ℋ_e⊗ℋ_b^⊗ N) .

Bob's task is to use a decoding channel𝒟:𝒟(ℋ_e⊗ℋ_b^⊗N)→𝒟(ℋ_e⊗ℋ_b), such that

    ρ_eb (_e⊗𝒟)(ρ_eb,N) .

This is a good encoding-decoding procedure if for a givenϵwe can have

    ||ρ̃_eb-ρ_eb||_1≤ϵ

whereρ̃_ebis the state obtained ifΦis replaced with a noiseless channel_a→b:(ρ∈𝒟(ℋ_a)) ↦(ρ∈𝒟(ℋ_b)). 

The rate of the communication channel is the number of qubits transmitted per use of the channel, i.e.,

    R 1/nlog_2(ℋ_a)

and this rate is said to be achievable if for anyδ,ϵ>0, there exists an(n,R-δ,ϵ)for sufficiently largen. The quantum capacity𝒬[Φ]of the channelΦis defined by the supremum of all achievable rates (see <cit.> for more details). 

For practical calculations that we need, we consider a useful form of the quantum capacity in terms of an information-theoretic measure called coherent information. Given a quantum channelΦ:𝒟(ℋ_a)→𝒟(ℋ_b), let|ψ_ea⟩be a purification of Alice's input stateρ_aand define

    ρ_eb    (⊗Φ)(|ψ_ea⟩⟨ψ_ea|) .

Usingρ_b _e(ρ_eb) =  Φ(ρ_a), the coherent informationI_c(ρ_a,Φ)is defined to be <cit.>
    I_c(ρ_a,Φ)     S(Φ(ρ_a)) - S(ρ_eb) .

We may sometimes writeI_c(ψ_ea,Φ)to make clear the joint state onEA. The coherent information depends on the input stateρ_aand the channelΦbut is independent of its purification. From this, we can then define the coherent information of a quantum channelI_c(Φ)by maximizing over all input states:

    I_c(Φ)   max_ρ_aI_c(ρ_a,Φ) .

Note that for qubit channels we have that-1≤I_c(ρ,Φ)≤1and0≤I_c(Φ)≤1<cit.>. We can then rewrite the quantum capacity using the Lloyd-Shor-Devetak (LSD) formula <cit.>, given by <cit.>
    𝒬[Φ]lim_n→∞1/nI_c(Φ^⊗ n) ,

where the RHS is also known as regularized coherent information.  

Despite the somewhat physically intuitive expression for𝒬[Φ], the asymptotic nature of Eq. (<ref>) makes quantum capacity very difficult to compute in most cases unless the quantum channel has very specific properties (e.g., when the quantum channel is additive, namely𝒬[Φ^⊗n] = n𝒬[Φ], which is false in general <cit.>). However, for us we do not actually need this: we follow the strategy adopted in <cit.>, which is to construct a quantum channel for qubits that hasI_c(Φ) →1by appropriately adjusting the setup parameters. This works because the quantum coherent information is superadditive<cit.>:

    I_c(Φ^⊗ N) ≥ NI_c(Φ)

which immediately implies that

    𝒬[Φ] ≥ I_c(Φ) ≥ I_c(ρ_a,Φ) .

For qubit channels, we have0≤𝒬[Φ]≤1, thus we can construct an essentially perfect quantum channel if we can show thatI_c[Φ]can be made arbitrarily close to 1. 

The interpretation of the above analysis is that in the limit where𝒬[Φ]→1, the quantum channel can effectively transmit quantum information (qubits) reliably in one use of the channel. It is perhaps not surprising that this is possible if and only if the whole procedure is essentially a “swapping operation”, i.e., Alice swaps her qubit `into the field', and then Bob swaps out the (approximate) embedded qubit in the field state to recover Alice's state. In effect, this suggests that the core of the protocol comprises of two components: (1) Bob being positioned correctly in spacetime in order to recover Alice's qubit, and (2) Alice and Bob being able to implement as high quality `SWAP gates' as possible[ In <cit.>, Alice's unitary is called ENCODING gate, and Bob's unitary is called DECODING gate. While the terminology has physical justification, it does not coincide with the encoding and decoding channels ℰ,𝒟 even in the case of a single use of the communication channel Φ (N=1), as they are not part of Φ whose capacity is being computed. The ENCODING and DECODING gates were part of the definition of the communication channel.].

As a final remark, we would like to point out that the existence of many superadditive quantum channels are in some sense a `curse' to finding a good characterization of quantum capacity — we refer the readers to <cit.> for one of the more recent summary and progress on this front. From Eq. (<ref>) we see that we need0<I_c(Φ) ≤1for the channel to be able to transmit some quantum information and our construction capitalizes on this. However, the case forI_c(Φ) = 0is much more subtle, as many quantum channels exhibit strict superadditivity in both weak and strong sense <cit.>. Weak superadditivity means that there existsNsuch thatI_c(Φ^⊗N) > N I_c(Φ), and strong superadditivity meansI_c(Φ_1⊗Φ_2) > I_c(Φ_1)+I_c(Φ_2)for two channelsΦ_1,Φ_2. In both cases, the `single-letter' formula for coherent informationI_c(Φ)is not sufficient to calculate the quantum capacity,  although some exceptions are known. For instance, all degradable channels (such as some regimes of the dephasing channels) are known to be weakly additive, in which case𝒬[Φ] = I_c(Φ)and hence non-positive channel coherent information would imply zero quantum capacity <cit.>. All anti-degradable channels (such as all entanglement-breaking channels and certain regimes of quantum erasure channels) are known to have zero quantum capacity, though the argument does not rely on additivity and more towards no-cloning type arguments <cit.>. Our construction (similarly for <cit.>) exploits the structure of the detector-field coupling to make the channel coherent information tractable.





§ PROTOCOL FOR THE PERFECT RELATIVISTIC QUANTUM CHANNEL


In this section, we present the protocol for Alice and Bob to implement a perfect quantum channel through the field. While our protocol closely follows <cit.>, our approach naturally extends to arbitrary background spacetimes and choices of vacuum states of the field by recasting the protocol in the AQFT framework. 

A brief visual of the quantum channel is depicted in Fig. <ref>. Mathematically, the quantum channel from Alice to Bob generically takes the form

    Φ(ρ_a) = _a, ϕ[Û_b^†Û_a^†( ρ_a⊗0⊗ρ_b ) Û_a^†Û_b^†]  ,

where|0⟩is the field vacuum[The state |0⟩ can be replaced with any GNS vector state |Ω_ω⟩ associated with any of the quasifree representations, which accounts for the existence of many unitarily inequivalent ground states in QFT <cit.>.],ρ_bis Bob's initial state, andÛ_aandÛ_bare the unitaries implemented by Alice and Bob through the UDW interaction with the field. The task is to construct a perfect quantum channel that can faithfully transmit quantum information to arbitrary accuracy.

Essentially, the basic idea is for Alice to (approximately) perform a `SWAP' gate[Technically speaking, the qubit and the quantum field have different Hilbert space dimensions but we will call them `SWAP' gates anyway since it would be the usual SWAP gate if we were to replace the quantum field with a third qubit. The fact that we are “embedding” the qubit state into the (much higher-dimensional) field can be viewed as a form of encoding operation, which justifies calling Alice's unitary ENCODING gate in <cit.>, and similarly Bob's unitary can be viewed as decoding operation from the field to his qubit system.] on her qubit state to the field, effectively encoding it into some superposition of approximately orthogonal coherent states, so that Bob can extract the qubit state by performing another swap operation. The entire problem thus reduces to analyzing under what circumstances the UDW interaction allows for nearly perfect `SWAP' operations on both parties which will depend on both the coupling strengths and the supports of the interactions. In particular, we will see that Bob can only perform a perfect `SWAP' gate if he is causally connected to Alice and the details of the interactions determine the quality of the swap operation. 

More precisely, Alice starts with an arbitrary state|ψ_a⟩c_1 |+_z⟩ + c_2 |-_z⟩, equivalentlyρ_a=|ψ_a⟩⟨ψ_a|, and encodes that state into the field using a `SWAP' gate. Bob should then be able to decode the state from the quantum field into his own Hilbert space. If the `SWAP' gatesÛ_aandÛ_bare implemented correctly, then we should have a perfect quantum channelΦ(ρ_a) = ρ_ain the idealized limit. We start by discussing how Alice encodes her state into the field:


    Let  |ψ_a⟩ = c_1 |+_z⟩ + c_2 |-_z⟩ be Alice's initial state and let |0⟩ be the initial vacuum state of the field. Consider Alice's unitary operation at time t=t_a given by[The usual convention would be Û_a =  e^-i σ̂_x ϕ̂(f_2) e^-i σ̂_z ϕ̂(f_1), but we opt for the positive sign for the exponent for convenience as many expressions appear more symmetric (this was also implicitly done in <cit.>). This amounts to the replacement ĥ_I→ -ĥ_I, or equivalently f_i→ -f_i. ]
    Û_a = e^i σ̂_x ϕ̂(f_2) e^i σ̂_z ϕ̂(f_1) .

    Then Alice can approximately swap the state |ψ_a⟩ into the field provided the following conditions are satisfied 
    
    E(f_1, f_2)    = π/4 mod  2π ,
    
            E(f_1, f_2)^2    ≫𝖶(f_2, f_2) .

    The resulting state after Alice's operation reads 
    
    Û_a (c_1 |+_z⟩   + c_2 |-_z⟩) |0⟩
       = |+_y⟩ (c_1 |+α⟩ - i c_2 |-α⟩) ,

    where |±α⟩ are coherent states of the quantum field defined by
    
    |±α⟩ e^± i ϕ̂(f_1)|0⟩.

    In order to prove this statement, we will explicitly apply Eq. (<ref>) to Alice's qubit and the field and show that the qubit state is encoded under the conditions in Eqs. (<ref>) and (<ref>).

    The first unitary in Eq. (<ref>) gives 
    
    e^i σ̂_z ϕ̂(f_1)|ψ_a⟩|0⟩   = c_1 |+_z⟩|+α⟩  + c_2 |-_z⟩|-α⟩.

    Next, we need to apply the second unitary e^i σ̂_x ϕ̂(f_2) but we first note that 
    
    ϕ̂(f_2) |±α⟩ = ± E(f_1, f_2) |±α⟩ + e^± i ϕ̂(f_1)ϕ̂(f_2) |0⟩ ,

    which follows directly from the Baker-Campbell-Haussdorff (BCH) formula and the CCR. Now we see that if 
    
    E(f_1, f_2)^2 ≫0ϕ̂(f_2) ϕ̂(f_2)0 = 𝖶(f_2, f_2)

    then |±α⟩ become approximate eigenvectors of ϕ̂(f_2)
    ϕ̂(f_2) |±α⟩≈± E(f_1, f_2) |±α⟩.

    If we pick the eigenvalues to be E(f_1, f_2) = π/4 mod 2π, then the action of Alice's full unitary Û_a gives
    
    Û_a    (c_1 |+_z⟩ + c_2 |-_z⟩) |0⟩
       = e^i σ̂_x ϕ̂(f_2)(c_1 |+_z⟩|+α⟩  + c_2 |-_z⟩|-α⟩)
       ≈ c_1 e^i π/4σ̂_x |+_z⟩|+α⟩  
            + c_2 e^-i π/4σ̂_x |-_z⟩|-α⟩
       = |+_y⟩(c_1 |+α⟩ - i c_2 |-α⟩) ,

    where we used e^i π/4σ̂_x |+_z⟩ = |+_y⟩ and e^-i σ̂_x π/4|-_z⟩ = -i|+_y⟩ in the last equality.



The second of Alice's conditions in Eq. (<ref>) can be interpreted as a “strong coupling condition” that requires Alice's coupling through the field via the interaction profilef_1be much stronger than throughf_2. In <cit.>, this corresponds to choosingf_2such thatϕ̂_2corresponds to conjugate momentumπ̂(F_a)and the condition translates to Alice coupling to the fieldϕ̂(F_a)much more strongly than toπ̂(F_a)(cf. Eq. (<ref>)). We can interpret Eq. (<ref>) as a “fine-tuning condition” that performs near exactπ/4rotations in the Bloch sphere so that Alice's second unitary decouples her detector from the quantum field.

Now we move on to discussing how at a later time, Bob can decode Alice's state from the quantum field into his own qubit detector. We notice that the protocol in Proposition <ref> leaves Alice's system in the state|+_y⟩, therefore if Bob's qubit starts in that initial state, it seems that Bob should simply apply the inverse ofÛ_ato decode the state from the field, i.e.,

    Û_b^†Û_a^† =  e^-i σ̂_z ϕ̂(f_1) e^-i σ̂_x ϕ̂(f_2) ,

where the qubit operatorsσ̂_sare now understood to act on Bob's Hilbert space. If Bob were able to implement this unitary then we have (almost trivially) achieved our goal of creating a perfect quantum channel from Alice to Bob. The issue is that Alice's operations defined throughϕ̂(f_i)are defined at timet=t_a, while we want Bob to decode from the field using only field operatorsϕ̂(g_i)localized at timet=t_b. We need a way to write the operatorsϕ̂(f_i)using different spacetime smearing functionsg_idefined at a different Cauchy slice. To achieve this we use the following theorem:

    Suppose we have two smeared field operators ϕ̂(f) and ϕ̂(g), then the two operators are equal ϕ̂(f) = ϕ̂(g) if the smearing functions satisfy
    
    Ef = Eg.

    The proof is almost immediate from the symplectic definition of smeared operators in Eq. (<ref>). If we assume Ef = Eg[The converse statement is false. From the axioms of AQFT, we also have that ϕ̂(P̂f)=0 which is the KG equation. Thus ϕ̂(g) is only unique up to homogeneous solution of the KG equation, i.e., g∼ g+P̂f.], then 
    
    ϕ̂(f) = σ(Ef, ϕ̂) = σ(Eg, ϕ̂) = ϕ̂(g) .


Some remarks are in order in view of the above Proposition <ref>. First, intuitively Alice's state is encoded into the field amplitude that propagates into the causal future of Alice's interaction region 𝒥^+(f_i), where𝒥^+((f_i))is the causal future of(f_i); for Bob to decode that state, his qubit has to be coupled everywhere where the initial data has propagated at a given timet=t_b, which in turn forces Bob's detector to interact with the field with the correct support and coupling strengths. In particular,(g_i)should essentially beΣ_t_b ∩(Ef_i), where(Ef_i)⊆𝒥^+(f_i). For simplicity, we assume that(f_1)=(f_2)and hence(g_1)=(g_2), which in the delta-coupling scenario is easily enforced as we are fixing the spatial profile of the interaction on Alice's side — this, in turn, fixes the needed spatial profile for Bob's coupling. 

Second, despite its simplicity and simple interpretation, Eq. (<ref>) is largely impractical to use for actually calculating Bob's spatial smearing functionsG_i(x)for a given timet = t_b. Using the eigenmode basis provided in Eq. (<ref>) and the momentum space smearing functions in Eq. (<ref>), the condition in Eq. (<ref>) can be equivalently written in `momentum space' as

    f_ k = g_ k.

These equations in momentum space are often easier to work with since in many simple cases (such as all FRW spacetimes) the spatial part of the mode functionu_k() ∝e^i k ·xand we can use Fourier transforms, which is similar to what was originally done in <cit.> where Bob's spatial smearing functions were related to Alice's in momentum space. More generally, in generic curved spacetimes, we do not have a simple plane-wave basis, so it is essential that all the conditions we demand for the perfect quantum channel are given only in terms of the correlation functions of the field without reference to Fourier transforms. 

Last but not least, note that there is an asymmetry in how we quantify the quality of Alice's and Bob's respective `SWAP' gates, i.e., how close their unitaries are in achieving the ideal `SWAP' operation. On the one hand, Alice needs two smearing functionsf_1andf_2that satisfy the conditions in Proposition <ref> to approximate the ideal `SWAP'. On the other hand, Bob needs two smearing functionsg_1andg_2that are related to Alice's via Proposition <ref> and hence automatically satisfy the conditions in Proposition <ref>. However, Proposition <ref> alone is not enough for the perfect quantum channel: ifEf_i ≠Eg_i, Bob will not be able to decode Alice's state even though by construction Bob's smearing functionsg_1andg_2satisfy Eqs. (<ref>) and (<ref>). In a sense, to decode the state using the `SWAP' operation with the field Bob needs an extra condition, namely that Bob needs to interact “at the correct spacetime regions”. The reason for this asymmetry is that the Hilbert space dimension of the fieldϕ̂is (much) larger than that of the detectors: it is much simpler for Alice to encode her state in a larger system but Bob will need additional information to decode it. 

In summary, we have constructed a perfect relativistic quantum channel that allows Alice to send her qubit state to Bob through the quantum field. The quantum channel consists of two steps:

  *  At time τ= τ_a, Alice encodes her qubit into the field using the unitary Û_a such that the conditions in Proposition <ref> are satisfied.
    
    
  *  At a later time τ = τ_b, Bob decodes the state from the field using the unitary Û_b defined in Eq. (<ref>). However, to implement this protocol, Bob's detector must interact across a spacetime region that satisfies the conditions specified in Proposition <ref>.

As the proper times are not the same in both trajectories, in practice one can use the ambient global coordinate systems and foliation to help with the time-ordering of these operations. Once timesτ_aandτ_bare specified, we demand that Alice's spatial smearing functionsF_i(x)fulfill conditions (<ref>) and (<ref>), and Bob's functionsG_i(x)fulfill the condition (<ref>)[In principle, Alice or Bob could each need up to four spatial smearing functions since f_i or g_i could include both δ(t) and δ'(t) terms. For example, in flat spacetime, we can have f_1 = δ(t) F_11( x) + δ'(t)F_12( x) (similarly for f_2), but for simplicity these can be chosen to be equal. Both g_1 and g_2 are fixed by f_1 and f_2 in the ideal case.]. 

 In order to simplify our subsequent discussions, we make a physically reasonable assumption that Alice's interaction profilesf_iare completely fixed except their coupling constants, i.e., in FNC we have

    f_i = λ_i χ_i(τ)F()

whereχ_iare either the Dirac delta functionδor its distributional derivativeδ'and the spatial profile is the same for bothf_1,f_2(hence they have the same supports(f_i) = (F)⊆). In other words, Alice only varies her interaction profilef_iby tuning the coupling strengthsλ_iwithout changing the `shape' of the interaction profile, and this is sufficient to satisfy the perfect `SWAP' conditions (<ref>)-(<ref>) in Proposition <ref>.




§ ANALYTIC TESTS OF THE QUANTUM CHANNEL


In this section, we will prove that the protocol given in Section <ref> produces a perfect quantum channel in the correct limit without any approximations like Eq. (<ref>). This proves that entanglement can always be perfectly transmitted between two localized qubits through a quantum field regardless of the background spacetime geometry. We also discuss the case when Alice and Bob can in principle perfectly implement the SWAP gates with the field but are spacelike separated from each other. For this channel, the quantum and classical channel capacity are both zero, as we expect with relativistic causality. 




 §.§ Ideal case: perfect quantum channel


Recall that our task is to show that the channel coherent informationI_c(Φ)can be made arbitrarily close to 1, thus proving that the (relativistic) quantum channel can transmit quantum information perfectly. This means computing the coherent informationI_c(ρ_a,0,Φ)and maximizing over all possible input states on Alice's side. Using the definition of coherent information (<ref>), it turns out that it is sufficient to consider the case when Alice's initial state is the maximally mixed stateρ_A, 0 = 1/2 and Bob initial state is|+_y⟩state in order to show that we can attain maximum channel coherent informationI_c(Φ)→1and hence maximal quantum capacity. We stress that this does not mean Alice should send a maximally mixed state to Bob, which is “garbage” as a message and is even classical, since in practice Alice wants to be able to send pure states properly. Also, in general, it is not true thatI_c(Φ)is maximized by the maximally mixed state: for example, the completely depolarizing channelΦ(ρ) = /2hasI_c(/2,Φ) = -1but the channel coherent informationI_c(Φ)=0is attained by any pure state.


In order to show thatI_c(Φ)→1is attainable, we need to introduce a qubit of the environmentEthat purifies Alice's maximally mixed state/2, such that the joint state of Alice and the environment is the maximally entangled state|ψ_ea⟩ = 1/√(2) (|-_z⟩ |+_z⟩ + |+_z⟩ |-_z⟩ )_ea. As discussed in Section <ref>, a perfect quantum channel will preserve all the entanglement between Alice and the environmentEAto Bob and the same environmentEB. To computeI_c (/2, Φ), we must evaluate the stateρ_ebin Eq. (<ref>). 

Writing out the expression forρ_ebin terms of the unitariesÛ_aandÛ_b, we obtain:

    ρ_eb=_Aϕ [Û_bÛ_a( ψ_ea⊗0⊗+_y ) Û_a^†Û_b^†].

The simplest way of continuing is to decompose the unitaries using projections onto the eigenvectors of the qubit operators. That is, forÛ_awe write

    Û_a   = ∑_x,z ∈{±}P̂_x P̂_z ⊗ e^i x ϕ̂(f_2) e^i z ϕ̂(f_1)
       = ∑_x,z ∈{±}P̂_x P̂_z ⊗ W(xEf_2) W(zEf_1)

whereP̂_iare the projectors onto the eigenstates ofσ̂_iand in the last line we used the Weyl algebra notation introduced in Eq. (<ref>). For Bob's unitary, we write

    Û_b = ∑_x,z ∈{±}P̂_z P̂_x ⊗ W(-zEf_1) W(-xEf_2)

where the projection operators are understood to act on Bob's system. We have written Bob's unitary as the inverse of Alice's, the imperfect case whereÛ_b ≠Û_a^-1is discussed in the following sections. We also expand the Bell state as|ψ_ea⟩ = 1/√(2) ∑_j |-j_z⟩ |j_z⟩withj ∈{ ±}to get the expression

    ρ_eb   = 1/2∑_j,k,x_i, z_iω(Ô) k_zP̂_z_1P̂_x_1P̂_x_4P̂_z_4j_z
       ×|-j_z⟩_e⟨-k_z|⊗P̂_z_3P̂_x_3|+_y⟩_b⟨+_y|P̂_x_2P̂_z_2

where we write ω(Ô) ≡⟨0|Ô|0|$⟩ and

    Ô   = W(-z_1Ef_1) W(-x_1Ef_2) W(x_2Ef_2) W(z_2Ef_1) 
       × W(-z_3Ef_1) W(-x_3Ef_2) W(x_4Ef_2) W(z_4Ef_1),

x_i stands for x_1, x_2, x_3, x_4, similarly for z_i, and all sums are taken over the set {±}. Provided that we can obtain an expression for ω(Ô), a computer algebra system can straightforwardly sum over the 2^10 terms in the sum. We start by redefining summation indices x_1 ↦ -x_1, z_1 ↦ -z_1, x_3 ↦ -x_3, and z_3 ↦ -z_3 such that  

    ρ_eb   = 1/2∑_j,k,x_i, z_iω(Ô) k_zP̂_-z_1P̂_-x_1P̂_x_4P̂_z_4j_z
       ×|-j_z⟩_e⟨-k_z|⊗P̂_-z_3P̂_-x_3|+_y⟩_b⟨+_y|P̂_x_2P̂_z_2

and Ô only contains terms without minus signs

    Ô   = W(z_1Ef_1) W(x_1Ef_2) W(x_2Ef_2) W(z_2Ef_1) 
       × W(z_3Ef_1) W(x_3Ef_2) W(x_4Ef_2) W(z_4Ef_1).

In <cit.>, 0Ô0 is evaluated using the Baker-Campbell-Hausdorff (BCH) formula and Wick's theorem. In this paper, we adopt a much simpler approach using the AQFT framework. It is straightforward (albeit somewhat tedious) to apply the third Weyl condition W(Ef) W(Eg) = e^- i/2E(f,g) W(E(f+g)) seven times to obtain

    Ô   = W(E((z_1+z_2+z_3+z_4)f_1+(x_1+x_2+x_3+x_4)f_2)) 
       × e^-i/2 (x_1+x_2)(z_1-z_2-z_3-z_4) E(f_1, f_2)
       × e^-i/2 (x_3+x_4)(z_1+z_2+z_3-z_4) E(f_1, f_2)

where only one Weyl operator remains multiplied by a phase factor. 

Before we evaluate the expectation value of the remaining Weyl operator, it will be convenient to consider the decomposition of a Wightman two-point function into its real (symmetric) and imaginary (anti-symmetric) parts, i.e.,

    𝖶(f,g)    = 1/2ω({ϕ̂(f),ϕ̂(g)})+1/2ω([ϕ̂(f),ϕ̂(g)])
       = 1/2𝖧(f,g) + i/2E(f,g) ,

where we used the CCR to obtain the second term and 𝖧(f,g) ω({ϕ̂(f),ϕ̂(g)}) ≡1/2μ(Ef,Eg) is the smeared Hadamard two-point distribution of the field (cf. Eq. (<ref>)). This decomposition is important because only the symmetric part 𝖧(f,g) is state-dependent, while E(f, g) is independent of the state. This is useful as it tells us which parts of our analysis depend on the field state. Furthermore, while we have assumed that the field starts in some vacuum state for simplicity, but the protocol in Section <ref> automatically extends to all quasi-free states such as thermal states <cit.>, squeezed vacuum state <cit.>, and with slight modifications to more general Gaussian states such as coherent states (see, e.g., <cit.>). 

We can now use the definition of quasifree state ω in Eq. (<ref>) for the vacuum state to evaluate the expectation value ω(Ô) in terms of Wightman and Hadamard functions giving us

    ω(Ô)    = e^-1/2 (z_1+z_2+z_3+z_4)^2 𝖶(f_1, f_1)-1/2 (x_1+x_2+x_3+x_4)^2 𝖶(f_2, f_2)
       × e^-1/2 (x_1+x_2+x_3+x_4)(z_1+z_2+z_3+z_4)𝖧(f_1, f_2)
       × e^-i/2 (x_1+x_2)(z_1-z_2-z_3-z_4) E(f_1, f_2)
       × e^-i/2 (x_3+x_4)(z_1+z_2+z_3-z_4) E(f_1, f_2) .

Note that this expression is valid automatically for any quasifree states such as squeezed vacuum state and (squeezed) thermal state <cit.>.  The density matrix in the ordered basis ρ_eb in the |+_z+_z⟩_be, |+_z-_z⟩_be, |-_z+_z⟩_be, |-_z-_z⟩_be now reads

    ρ_eb = (P_-     0     A     C
     
        0     P_+     X     B
    
        A^*     X^*     P_+     0
    
        C^*     B^*     0     P_-)

where

    P_±   = 1/4(1 ± e^-2 𝖶(f_1, f_1) sin(2E(f_1, f_2)) )
    
        X    = 1/4sin(2E(f_1, f_2)) ( e^- 2 𝖶(f_2, f_2) +  sin(2E(f_1, f_2)) )
    
        A    = - i/4 e^ -2 𝖶(f_1, f_1) cos(2E(f_1, f_2))
       ×( sin(2E(f_1, f_2)) -e^ -2W(f_2, f_2)sinh(4𝖶(f_2, f_1)) )
    
        B    = -i/4 e^ - 2 𝖶(f_1, f_1) cos(2E(f_1, f_2)) 
       ×(sin(2E(f_1, f_2)) + e^-2𝖶(f_2, f_2)cosh(4 𝖶(f_1, f_2)))
    
        C    = 1/4 e^ - 8 𝖶(f_1, f_1)sin(2E(f_1, f_2)) 
       ×( sin(2E(f_1, f_2)) - e^-2 𝖶(f_2, f_2)cosh(4𝖧(f_1, f_2)) ) .



Now that we have an analytic expression for ρ_eb, we can use our two conditions in Proposition <ref> and verify that the coherent information can be made arbitrarily close to 1. First we use the fine-tuning condition E(f_1, f_2) = π/4 which simplifies ρ_eb  considerably to

    ρ_eb = (P̃_-     0     0    C̃
     
        0    P̃_+    P̃_+     0
     
        0    P̃_+    P̃_+     0
    C̃    0    0    P̃_- )

where

    P̃_±   = 1/4(1 ± e^-2𝖶(f_2, f_2))
    C̃   = 1/4 e^-8 𝖶(f_1, f_1)(e^-2𝖶(f_2, f_2)cosh(4𝖧(f_1, f_2)) +1).


Next, we use the strong coupling condition E(f_1, f_2)^2 ≫𝖶(f_2, f_2). Recall that we are assuming for simplicity that on Alice's side, her interaction profile will be kept fixed except for the coupling strengths, thus only λ_1,λ_2 are adjustable parameters. In this case, the strong-coupling condition essentially means that λ_1≫λ_2 and hence we have the hierarchy of scales

        
    𝖶(f_1,f_1) ∼𝒪(λ^2_1)    ≫ E(f_1,f_2)^2∼𝒪(λ^2_1λ^2_2) 
    
            E(f_1,f_2)^2∼𝒪(λ^2_1λ^2_2)    ≫𝖶(f_2,f_2)∼𝒪(λ^2_2)
    𝖶(f_1,f_1)∼𝒪(λ^2_1)    ≫𝖧(f_1,f_2) ∼𝒪(λ_1λ_2)


With this hierarchy, we can express λ_1 in units of λ_2 so that  λ_1 cλ_2 for some c≥ 0, which is also convenient since in general λ_i has physical units[The unit of λ_i is [Length]^n-3/2, and we have assumed for simplicity that λ_i have the same units since f_i are chosen to be (cf. the end of Section <ref>)]. We can thus write


    𝖶(f_1,f_1)    = c^2λ_2^2 𝗐_1 ,  𝖶(f_2,f_2) = λ_2^2 𝗐_2 ,
    
        E(f_1,f_2)    = cλ^2_2𝖾 ,    𝖧(f_1,f_2) = cλ_2^2𝗁 ,


where 𝗐_1,𝗐_2,𝖾,𝗁 are fixed by the spacetime smearings. Proposition <ref> then translates to

    λ_2^2(c^2𝖾^2)≫𝗐_2 ,  λ_2^2(c𝖾) = π/4 2π .

Note that the fine-tuning condition can still be satisfied: for example, for a fixed coupling λ_2 we find c such that 

    λ_2^2(c𝖾) = π/4 2π ,

and we can always ensure that c is sufficiently large so that

    λ^2_2(c^2𝖾^2) = c·𝖾π/4+2π n≫𝗐_2

in order to satisfy both conditions in (<ref>). 

In terms of the rescaled coupling constants, the strong-coupling condition implies that for fixed λ_2 and c≫ 1, we get

    ρ_eb(P̃_-     0     0     0
     
        0    P̃_+    P̃_+     0
     
        0    P̃_+    P̃_+     0
    
        0     0    0    P̃_- ) .

Using negativity 𝒩 as an entanglement measure <cit.>, 

    𝒩[ρ_eb] = ||ρ_eb^Γ_e||_1-1/2 = 1/2e^-2λ_2^2𝗐_2

where ρ_eb^Γ_e is the partial transpose of ρ_eb and ||·||_1 is the trace norm. This means that ρ_eb is always an entangled state for any finite coupling λ_2≪λ_1 and the channel is able to preserve some quantum entanglement. 

Finally, we obtain the perfect quantum channel in the idealized limit where c→∞ and λ_2→ 0 such that λ_1 is finite and still satisfies (<ref>), i.e., 

    ρ_eb = 1/2[ 0 0 0 0; 0 1 1 0; 0 1 1 0; 0 0 0 0 ] ,

i.e., a Bell state with maximum negativity 𝒩[ρ_eb] = 1/2. Therefore, in this limit the coherent information I_c(Φ)→ 1, which corresponds to a perfect quantum channel that can transmit a maximally entangled state perfectly. 




 §.§ Alice and Bob are spacelike separated



The preceding analysis shows that the ability to implement the approximate `SWAP' gate depends on how closely one is able to satisfy the fine-tuning and strong-coupling conditions in Proposition <ref>. While this is straightforward for Alice, Proposition <ref> tells us that for Bob to emulate ϕ̂(f_i) defined at some (global coordinate) time t(τ_a),  Bob's spacetime smearing functions g_i at time t(τ_b) has to be somewhat delocalized over some region of spacetime to catch all the signals from Alice. Since the quantum channel Φ is built out of a relativistic quantum field, we ought to be able to show that there can be no transmission of both classical and quantum information through Φ.

Formally, we start from the CCR which tells us that 

    E(f_i,g_j) = 0    ∀ i,j=1,2

when Alice and Bob are spacelike-separated, i.e., (g_i)∩𝒥^±((f_i)) = ∅. Note that the Hadamard function 𝖧(f_i,g_j) is generically nonzero for spacelike-separated regions due to entanglement contained in the field state <cit.>. The calculations proceeds similarly as in the previous section, except that Bob will have his own arbitrary smearing functions and their respective UDW interaction unitaries are



    Û_A   = ∑_x,z ∈{±}P̂_x P̂_z ⊗ W(xEf_2) W(zEf_1)
    Û_b   = ∑_x,z ∈{±}P̂_z P̂_x ⊗ W(-zEg_1) W(-xEg_2).


The condition for spacelike separation in Eq. (<ref>) will be enforced later in the calculation once everything is written in terms of two-point functions. 

We can analyse this scenario very generally by allowing Alice and the environment to start in any arbitrary state ρ_ea instead of a maximally entangled state or purifications of any initial state ρ_a,0. The expression for the density matrix ρ_eb now reads


    ρ_eb   = ∑_x_i, z_iω(Ô)  _a[ P̂_x_4P̂_z_4ρ_aeP̂_-z_1P̂_-x_1 ]
       ⊗P̂_-z_3P̂_-x_3|+_y⟩⟨+_y|_bP̂_x_2P̂_z_2

where the projection operators act on Alice or Bob and Ô is now defined to be

    Ô   = W(z_1Ef_1) W(x_1Ef_2) W(x_2Eg_2) W(z_2Eg_1)  
       × W(z_3Eg_1) W(x_3Eg_2) W(x_4Ef_2) W(z_4Ef_1).

Applying the third Weyl relation and using Eq. (<ref>), this evaluates to 

    
    ω(Ô) 
           = 
        e^-i/2 (z_1 - z_4)(x_1 + x_4) E(f_1, f_2)  
        e^-i/2 (z_2 + z_3)(x_3 - x_2) E(g_1, g_2)
        e^-i/2 (z_2 + z_3)(z_1 - z_4) E(f_1, g_1)
       × 
        e^-i/2 (z_1 - z_4)(x_2 + x_3) E(f_1, g_2)
        e^-i/2 (z_2 + z_3)(x_1 - x_4) E(f_2, g_1)
        e^-i/2 (x_2 + x_3)(x_1 - x_4) E(f_2, g_2)
       ×  e^-1/2(
          (z_1+z_4) (z_1+z_4)  𝖶(f_1, f_1)
        + (x_1+x_4) (x_1+x_4)  𝖶(f_2, f_2)
        + (z_2+z_3) (z_2+z_3)  𝖶(g_1, g_1)
        + (x_2+x_3) (x_2+x_3)  𝖶(g_2, g_2) )
       × e^-1/2(
          (x_1+x_4) (z_1+z_4)  𝖧(f_1, f_2)
        + (z_2+z_3) (z_1+z_4)  𝖧(f_1, g_1)
        + (x_2+x_3) (z_1+z_4)  𝖧(f_1, g_2) )
       × e^-1/2(
          (z_2+z_3) (x_1+x_4)  𝖧(f_2, g_1)
        + (x_2+x_3) (x_1+x_4)  𝖧(f_2, g_2)
        + (x_2+x_3) (z_2+z_3)  𝖧(g_1, g_2) ) .

This is the most general form that ω(Ô) can take in terms of Alice and Bob's two-point functions. We now enforce spacelike-separation using Eq. (<ref>), giving us

    ω(Ô) 
           = 
        e^-i/2 (z_1 - z_4)(x_1 + x_4) E(f_1, f_2)  
        e^-i/2 (z_2 + z_3)(x_3 - x_2) E(g_1, g_2)
       ×  e^-1/2(
          (z_1+z_4) (z_1+z_4)  𝖶(f_1, f_1)
        + (x_1+x_4) (x_1+x_4)  𝖶(f_2, f_2)
        + (z_2+z_3) (z_2+z_3)  𝖶(g_1, g_1)
        + (x_2+x_3) (x_2+x_3)  𝖶(g_2, g_2) )
       × e^-1/2(
          (x_1+x_4) (z_1+z_4)  𝖧(f_1, f_2)
        + (z_2+z_3) (z_1+z_4)  𝖧(f_1, g_1)
        + (x_2+x_3) (z_1+z_4)  𝖧(f_1, g_2) )
       × e^-1/2(
          (z_2+z_3) (x_1+x_4)  𝖧(f_2, g_1)
        + (x_2+x_3) (x_1+x_4)  𝖧(f_2, g_2)
        + (x_2+x_3) (z_2+z_3)  𝖧(g_1, g_2) ) .


The density matrix ρ_eb can now be calculated 

    ρ_eb = ρ_e⊗(1/2    X
     X^*    1/2)

with ρ_e = _a [ρ_ea] and 

    X    = -i/2 e^-2 𝖶(g_1, g_1)
       [e^-2 𝖶(g_2, g_2)cosh(2𝖧(g_1, g_2)) + sin(2E(g_1, g_2))].


Observe that the density matrix ρ_eb depends only on the initial condition of the environment and Bob's local two point functions, i.e. not involving Alice's smearing functions at all, and is clearly separable across B and E. 
Consequently, the quantum channel preserves no entanglement at all from the joint system EA (viewing it as ⊗Φ). In fact, since we start from any arbitrary state ρ_ea, we have effectively shown that Φ is a PPT channel <cit.>, and hence also an entanglement-breaking channel (because PPT is equivalent to separability for two qubits), and these are known to fall under the class of anti-degradable channels with zero quantum capacity 𝒬[Φ] = 0  <cit.>. 

From an information-theoretic viewpoint, zero quantum capacity says nothing about the ability of the quantum channel to transmit classical information, since the classical channel capacity 𝒞[Φ] is an upper bound for quantum capacity, i.e., 𝒞[Φ] ≥𝒬[Φ]. Quantum channels can also transmit classical bits by, for instance, considering only diagonal density matrix

    ρ_a,0 = (α    0
     0     1-α) ,  α∈ [0,1] .

and indeed it has been shown that for certain delta-coupled UDW models,  the classical capacity 𝒞[Φ] can be computed exactly and is zero if Alice and Bob are spacelike <cit.>. Our quantum channel is more complicated as Alice and Bob each perform a sequence of two simple-generated unitaries, but it is not necessary for us to find the explicit formula for 𝒞[Φ] to show that it is zero. This follows directly from the fact that the density matrix in Eq. (<ref>) does not depend on ρ_a,0 at all, therefore the quantum channel carries no information whatsoever about Alice's initial state, hence 𝒞[Φ] = 0. 

One of the main features of a relativistic quantum channel is precisely that one should be able to show that it respects the causal structure of spacetime by construction, and this requires one to be able to show, for instance, that no information transmission is allowed in spacelike separation. In fact we can say more: our formalism does not require us to specify anything about the (violation of) strong Huygens' principle <cit.>, since everything is encoded in the causal propagator. For example, in (3+1)-dimensional Minkowski spacetime, Bob can be timelike-separated from Alice but for a massless scalar field, the causal propagator has zero support in the interior of the light cone (see, e.g., <cit.>). In this case, if Bob also coupling functions to the field  g_i are strictly in the timelike interior and  (g_i)∩𝒥^+( f_j) = ∅ for any i,j=1,2, then 𝒬[Φ]=𝒞[Φ]=0.




 §.§ Intermediate scenarios


In the two preceding subsections, we have analyzed two opposite regimes: the ideal case with Ef_i = Eg_i and the worst case with Alice and Bob spacelike-separated. Let us make some comments on intermediate regimes. 

When the coherent information of a quantum channel is not close to 1, e.g., when 0 ≤ I_c(Φ) ≪ 1, it is in general very difficult to make quantitative claims about the quantum channel capacity for the reasons like superadditivity described in Section <ref>. In fact, even for quantum channels with I_c(Φ) = 0, only small subsets of these are known to have zero quantum capacity, namely the anti-degradable channels <cit.> and the PPT channels <cit.> (see also <cit.>). For intermediate scenarios, such as when Bob is only partially covering  𝒥^±((f_i)) or implementing faulty SWAP gates we are not able to give generic quantitative statements about 𝒬[Φ], as is the case for generic quantum communication channels. 

However, in some situations we may try to use no-cloning type argument to conclude that 𝒬[Φ]=0 in some scenarios as was done in <cit.> (see also <cit.>). Suppose Bob is allowed to couple only to at most half of Alice's signal at some future time t that defines the Cauchy slice for Bob's interaction. Consider another observer Charlie who now chooses to couple to the other half of the region at the same time slice so that jointly the interaction support is the full region needed for the perfect quantum channel. Since Charlie's interaction is by construction spacelike separated from Bob, Charlie's interaction is invisible to Bob (and vice versa). If Bob and Charlie were able to individually decode Alice's qubit, this would violate the no-cloning theorem, so we can conclude that the quantum capacity must be zero if the UDW interactions are set up this way[It is worth emphasizing that the protocol we have (and also in <cit.>) are generically not ideal, since for any finite interaction strengths the `SWAP' gates are never perfect and the `logical qubit' in the field is only approximate — consequently one has to be careful about using exact cloning argument. In practice, it would be better to think of the scenario in terms of approximate cloning. Indeed, in Figure 8 of <cit.> the coherent information vanishes smoothly as one varies Bob's interaction support when analyzing the cloning-type argument.]. 

The above discussion is not necessarily the only (or natural) way to view this. One way to think about this from the perspective of the quantum field is this: if Bob manages to decode Alice's state at time t, the `logical qubit' state c_1|+α⟩-ic_2|-α⟩ in Eq. (<ref>) would have been taken out by Bob, and the field would now change into a new state that has no logical qubit in it (essentially because Bob swapped out the state into his physical detector). Charlie, who couples to the field at any time t+ϵ for any arbitrary ϵ>0, would no longer have any logical qubit to extract from the field, so Charlie cannot extract another qubit for free. Yet another way to see this is that Bob's interaction is designed to implement the inverse unitary Û_a^-1, and this requires one to find g_i in 𝒥^+((f_j)) that can undo Alice's operation. Therefore, if we choose to pick g_i that does not cover the full region needed for the decoding, then Bob fails to implement Û_a^-1 and instead performs a different unitary operation. In this interpretation, we would say that Bob is unable to perform the required `SWAP' gate to decode Alice's state. Showing that Bob has zero quantum capacity would mean showing that Bob's operations somehow completely depolarize the qubit state and lose all the quantum information in it — we will not pursue this further in this paper and leave this for future work.




§ DISCUSSION AND OUTLOOK



We motivated our work through the question: What is the information-carrying capacity of quantum fields? The UDW framework allows us to formulate a precise statement of this question, in terms of quantum capacity of the relativistic quantum channel between Alice and Bob who try to transmit quantum information between their qubit detectors through the quantum field.  In this work, we have successfully generalized the results of <cit.>, in particular <cit.>, where we construct a quantum channel that can transmit quantum information arbitrarily well through a relativistic quantum scalar field in arbitrary curved spacetimes. 

   

The features of our construction is that the resulting quantum channel, and hence the quantum capacity, are manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the CCR algebra. Furthermore, in this generalization we no longer need to deal with (violation of) the strong Huygens' principle that may occur in curved spacetimes for non-conformally invariant fields (see, e.g.,<cit.>), since this is fully encoded in the causal propagator. We also showed explicitly that the quantum and classical capacity is zero for spacelike separated observers, as we expect from quantum communication channel built from a relativistic quantum field.

Crucially, in this generalized formulation it becomes clear that there are two essential features of a near-perfect quantum channel that can transmit quantum information arbitrarily well: (1) Alice and Bob's interactions need to satisfy the so-called fine-tuning and strong coupling conditions in Proposition <ref> in order to approximate an ideal `SWAP' gates; (2) Bob needs to couple to the field at appropriate spacetime regions, fixed by Proposition <ref>, so that he could execute the approximately ideal `SWAP' gates. Indeed, where Bob should be is fixed by the causal propagator that determines where Alice's information propagates in spacetime. The asymmetry between Alice and Bob can be viewed as the consequence of the asymmetry between encoding the qubit state into the field, with the logical basis being the coherent states |±α⟩, and decoding the state while taking into account relativistic causality. In this sense, our construction constitutes the most general relativistic qubit channel within the UDW framework that can transmit quantum information arbitrarily well in appropriate regimes and we also provided analyses when the quantum channel is sub-optimal.


There are several future directions that merit investigation, at least from a more fundamental standpoint:

    
  * Is it possible to construct a perfect relativistic quantum channel using higher-dimensional detector models (using qudit detector models <cit.> rather than qubits)? A priori it is not obvious how to construct logical qudits in the field using only controlled displacement operators and what kind of interaction Hamiltonians are needed to do this, if it exists at all. It is also natural to consider several emitters and receivers, exploiting the fact that one can interpret displacement operations can be viewed as Weyl generators with respect to some spacetime smearing functions <cit.>.

    
  * As mentioned in <cit.>, one may be interested in more directional coupling, which can be envisaged in (1+1)D models (see, e.g., <cit.>). Our protocol works automatically simply because only the two-point functions change to reflect that the left- and right- movers of the field modes are independent. A more interesting setup would be to study how realistic settings involving waveguides and non-trivial dispersion relations can be adapted for this purpose. 

    
  * Is the information-carrying capacity different between different fields with different spins and statistics? For example, how does one compare the scalar field with a fermionic field? The issue one has to deal with is that we have to separate the effects coming from the model dependence of the coupling (e.g., we cannot couple σ̂_x⊗ψ̂ where ψ is the fermionic field) from truly field-theoretic physics that contribute to the quantum capacity, if we wish to do it in the same way as we have done in this paper (see, e.g., some proposed fermionic couplings in RQI literature <cit.>).

    
  * Our construction relies on the fact that the quantum field is non-interacting (Gaussian). There are very few studies involving the UDW model with interacting QFTs such as the ϕ^4 theory, even perturbatively and for a single qubit (see, e.g., <cit.> for one example). The effects of interactions are unclear for (long-distance) quantum communication, both perturbatively and non-perturbatively. A possibly relevant testbed would be to consider known rigorous constructions in (1+1)D interacting QFTs (see, e.g., <cit.>). 

    
  * Is it possible to systematically study quantum capacity in the non-relativistic quantum many-body settings? Somewhat related studies to our setup are, for example, <cit.>. It would be interesting to see the similarities and differences when they are compared with setups involving relativistic quantum fields.

We leave these problems open for the future.




§ ACKNOWLEDGMENT



M. K. acknowledges support from the Princeton International Internship Program (IIP). E. T. acknowledges funding from the Munich Center for Quantum Science and Technology (MCQST), funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy (EXC2111 - 390814868). 


apsrev4-2


 Paper Title: TASI Lectures on Physics for Machine Learning.







Quantum Non-Demolition Measurements and Leggett-Garg inequality
    Stefano Gherardini
    August 5, 2024
===============================================================





        0.75
        § ABSTRACT 
            These notes are based on lectures I gave at TASI 2024 on Physics for Machine Learning. The focus is on neural network theory, organized according to network expressivity, statistics, and dynamics. I present classic results such as the universal approximation theorem and neural network / Gaussian process correspondence, and also more recent results such as the neural tangent kernel, feature learning with the maximal update parameterization, and Kolmogorov-Arnold networks. The exposition on neural network theory emphasizes a field theoretic perspective familiar to theoretical physicists. I elaborate on connections between the two, including a neural network approach to field theory.
        
        









§ INTRODUCTION




Computer science (CS) is still in its infancy.  “What!?” you incredulously exclaim, thinking back to the earliest computer scientists you can think of, perhaps Lovelace or Babbage or Turing. Since its early days, CS has steadily progressed, from cracking the Enigma code in the 40's, to personal computers in the 70's and 80's, the internet in the 90's, powerful computers-in-your-pocket in the 2000's, and finally to some version of artificial intelligence in the last decade; see <cit.> for a thorough account. It shows no signs of stopping, and one might wager that future historians will not speak of decades of progress in CS, but centuries. Computer science is still in its infancy.

It is 2024, what have we seen recently? A plethora of “experimental" breakthroughs in machine learning (ML) that constitute some of the most exciting developments in CS to date. Reinforcement learning algorithms have become world-class at Go <cit.> and Chess <cit.>, merely by playing against themselves. Diffusion models <cit.> can generate images of people that don't exist, and yet you might never know. Large language models (e.g. GPT-3 <cit.> literature, or popular accounts <cit.>) can code better than you — or me, at least —  and they can even write compelling bedtime stories for your children. These forms of artificial intelligence have already started a trillion dollar industry, and some have speculated that we've already seen sparks <cit.>  of AGI, or human-level intelligence.

We've also started to see significant advances in the sciences. For instance, AlphaFold <cit.> has achieved state-of-the-art performance for predicting the structure of folded proteins, and neural networks (NNs) give state-of-the-art models for complex objects, from quantum many-body ground states <cit.> to metrics <cit.> on the Calabi-Yau manifolds of string theory. Lack-of-rigor is a natural concern, but in certain cases ML techniques can be made rigorous or interpretable at a level that would satisfy a formal theoretical physicist or pure mathematician; see, e.g., <cit.> and references therein for examples in string theory, algebraic geometry, differential geometry, and low-dimensional topology, including uses of ML theory for science.
Of course, this is just scratching the surface, see, e.g., <cit.> for a smattering of results across physics and the physical sciences.

With this in mind, we have some explaining to do. There are a large number of impressive ML results in both pop culture and the sciences, but experiments have significantly outpaced theory over the last decade. An analogy from physics might be useful, given the original audience. The current situation in ML might be reminiscent of the 1960's in particle physics: some essentials of the theory were known, e.g. quantum electrodynamics (QED), but a deluge of newly discovered hadrons left theorists scratching their heads and demanded an organizing principle to explain the new phenomena. This was one of the motivating factors for the development of string theory, but the right theory in the end is the quark model, as described by quantum chromodynamics (QCD). It's tempting to speculate that recent results in ML are waiting for someone to discover a foundational theory, akin to QCD for the strong interactions. As theoretical physicists, we hope for this sort of mechanistic understanding. However, it might be that ML and intelligence are so complex that sociology is a better analogy, and it would be a fool's errand to expect such a precise theory. My personal guess is that detailed theory will pay off in certain areas of ML — perhaps related to sparsity in NNs, architecture design, or the topics of these lectures — but not in all. Nevertheless, given the experimental progress, one must try.


These lectures were given at TASI 2024 <cit.> to Ph.D. students in theoretical high energy physics. I was given the title “Physics4ML", which means that I'm trying to keep ML-for-Physics to a minimum. I take a physics perspective, but acknowledge that this is an enormous field to which many communities have contributed.

The central idea of the lectures is that understanding NNs is essential to understanding ML, a story I built on three pillars: expressivity, statistics, and dynamics of NNs, one for each of the lectures I gave. The content builds on many ideas explained to me over the last five years by friends, acknowledged below, who pioneered many of the developments, as well as a few of my own results. Throughout, I take a decidedly field-theoretic lens to suit the audience, and also because I think it's a useful way to understand NNs. 










 §.§ The Setup.

Understanding ML at the very least means understanding neural networks. A neural network is a function

    ϕ_θ:^d →

with parameters θ. We've chosen outputs in  because, channeling Coleman, scalars already exhibit the essentials. We'll use the lingo

    Input:      x ∈^d 
    Output:     ϕ_θ(x) ∈
    Network:     ϕ_θ∈(^d, ) 
    Data:     ,

where the data  depends on the problem, but involves at least a subset of ^d, potentially paired 
with labels y∈.

With this minimal background, let's ask our central question:
What does a NN predict?
For any fixed value of θ, the answer is clear: ϕ_θ(x). However, the answer is complicated by issues of both dynamics and statistics. 

First, dynamics. In ML, parameters are updated to solve problems and we really have trajectories in 

    Parameter Space:     θ(t) ∈^| θ |
    Output Space:     ϕ_θ(t)(x) ∈
    Function Space:     ϕ_θ(t)∈(^d, ).

governed by some learning dynamics determined by the optimization algorithm and the nature of the learning problem. For instance, in supervised learning we have data

    = {(x_α, y_α) ∈^d ×}_α=1^||,

and a loss function 

    [ϕ_θ] = ∑_α=1^||ℓ(ϕ_θ(x_α), y_α),

where ℓ is a loss function such as ℓ_MSE = (ϕ_θ(x_α)-y_α)^2. One may optimize θ by gradient descent 

    dθ_i/dt = -∇_θ_i[ϕ_θ],

or other algorithms, e.g., classics like stochastic gradient descent (SGD) <cit.> or Adam <cit.>, or a more recent technique such as Energy Conserving Descent <cit.>. Throughout, t is training time of the learning algorithm unless otherwise noted.

Second, statistics. When a NN is initialized on your computer, the parameters θ are initialized as draws

    θ∼ P(θ)

from a distribution P(θ), where  means “drawn from" in this context.
Different draws of θ will give different functions ϕ_θ, and a priori we have no reason to prefer one over another. The prediction ϕ_θ(x) therefore can't be fundamental! Instead, what is fundamental is the average prediction and second moment or variance:

    [ϕ_θ(x)]     = ∫ dθ P(θ)  ϕ_θ(x) 
    [ϕ_θ(x) ϕ_θ(y)]     = ∫ dθ  P(θ)   ϕ_θ(x) ϕ_θ(y),

as well as the higher moments. Expectations are across different initializations. Since we're physicists, we henceforth replace [·]=· and we remember this is a statistical expectation value. It's useful to put this in our language:

    G^(1)(x)     = ϕ_θ(x)
    
        G^(2)(x, y)     = ϕ_θ(x) ϕ_θ(y),

the mean prediction and second moment are just the one-point and two-point correlation functions of the statistical ensemble of neural networks. Apparently ML has something to do with field theory. 

Putting the dynamics and statistics together, we have an ensemble of initial θ-values, each of which is the starting point of a trajectory θ(t), and therefore we have an ensemble of trajectories. We choose to think of θ(t) drawn as 

    θ(t) ∼ P(θ(t)),

a density on parameters that depends on the training time and yields time-dependent correlators 

    G_t^(1)(x)     = ϕ_θ(x)_t 
    
        G_t^(2)(x, y)     = ϕ_θ(x) ϕ_θ(y)_t,

where the subscript t indicates time-dependence and the expectation is with respect to P(θ(t)). Of course, assuming that learning is helping, we wish to take t→∞ and are interested in
G_∞^(1)(x) = mean prediction of ∞-number of NNs as t→∞.
Remarkably, we will see that in a certain supervised setting there is an exact analytic solution for this quantity.

There is one more pillar beyond dynamics and statistics that I need to introduce: expressivity. Let's make the point by seeing a failure mode. Consider neural networks of the following functional form, or architecture:

    ϕ_θ(x) = θ· x.

The one-point and two-point functions of these are analytically solvable, but going that far defeats the purpose: this is a linear model, and learning schemes involving it are linear regression, which will fail on general problems. We say that model is not expressive enough to account for the data. Conversely, we wish instead to choose expressive architectures that can model the data, which essentially requires that the architecture is complex enough to approximate anything. Of course, such architectures must be non-linear.

We now have the three pillars on which we'll build our understanding of neural networks, and associated questions:

    
  * Expressivity. How powerful is the NN?
    
  * Statistics. What is the NN ensemble?
    
  * Dynamics. How does it evolve?

We will approach the topics in this order, building up theory that takes us through the last few years through a physicist's lens. 

A physicist's lens means a few things. First, it means a physicist's tools, including:


  * Field Theory as alluded to above.

  * Landscape Dynamics from loss functions on θ-space.

  * Symmetries of individual NNs and their ensembles.
 
A physicist's lens also means we will try to find the appropriate balance of rigor and intuition. We'd like mechanisms and toy models, but we won't try to prove everything at a level that would satisfy a mathematician. This befits the current situation in ML, where there are an enormous number of empirical results that need an O(1) theoretical understanding, not rigorous proof.

Henceforth, we drop the subscript θ in ϕ_θ(x), and the reader should recall that the network depends on parameters.






§ EXPRESSIVITY OF NEURAL NETWORKS



Neural networks are big functions composed out of many simpler functions according to the choice of architecture. The compositions give more flexibility, prompting
How powerful is a NN?
Ultimately this is a question for mathematics, as it's a question about functions.  Less colloquially, what we mean by the power of a NN is its ability to approximate any function.



 §.§ Universal Approximation Theorem


The Universal Approximation Theorem (UAT) is the first result in this direction. It states that a neural network with a single hidden layer can approximate any continuous function on a compact domain to arbitrary accuracy. More precisely, the origin version of the theorem due to Cybenko is

Let f: ^d → be a continuous function on a compact set K⊂^d. Then for any ϵ>0 there exists a neural network with a single hidden layer of the form

    ϕ(x) = ∑_i=1^N ∑_j=1^d w_i^(1)σ(w_ij^(0)  x_j + b^(0)_i)+b^(1),

θ = {w_ij^(0), w_i^(1), b^(0)_i, b^(1)},
where σ:→ is a non-polynomial non-linear activation function, such that

    sup_x∈ K |f(x) - ϕ(x)| < ϵ.


The architecture in Eq. (<ref>) is known by many names, including the perceptron, a fully-connected network, or a feedforward network. The parameter N is known as the width. It may be generalized to include a depth dimension L encoding the number of compositions of affine transformations and non-linearities. Such an architecture is known as a multi-layer perceptron (MLP) or a deep feedforward network.

The UAT is a powerful result, but it has some limitations. First, though the error ϵ does get better with N, it doesn't say how many neurons are needed. Second, it doesn't say how to train the network: though there is a point θ^* in parameter space that is a good approximation to any f, existence doesn't imply that we can find it, which is a question of learning dynamics.

To ask the obvious,
Why does this UAT work? 
In Cybenko's original work, he focused on the case that σ was the sigmoid function 

    σ(x) = 1/1+e^-x,

which allows us to get a picture of what's happening. The sigmoids appear in the network function of the form 

    σ(w_i^(0)· x + b_i^(0))

which approximates a shifted step function as w^(0)→∞. A linear combination can turn it into a bump with approximately compact support that gets scaled by w^(1). These bumps can then be put together to approximate any function; see, e.g., Fig. <ref>.

Cybenko's version of the UAT was just the first, and there are many generalizations, including to deeper networks, to other activation functions, and to other domains.  See <cit.> for the original paper and <cit.> for a generalization to the case of multiple hidden layers.





    


 §.§ Kolmogorov-Arnold Theorem


The UAT essentially says that any function can be approximated by a neural network, i.e. a complicated function comprised out of many building blocks. It is natural to wonder whether there are other theorems to this effect, and whether they inspire other neural network constructions.

One such result is due to Kolmogorov <cit.> and Arnold <cit.>, who showed that any multivariate continuous function can be represented (exactly, not approximated) as a sum of continuous one-dimensional functions:


Let f:[0,1]^n→ be an arbitrary multivariate continuous function. Then it has the representation 

    f(x_1,…,x_n) = ∑_q=0^2nΦ_q(∑_p=1^n ϕ_q,p(x_p))

with continuous one-dimensional functions Φ_q and ϕ_q,p.


This result is perhaps less intuitive than the UAT. If every multivariate function is to be represented by univariate functions plus addition, what then of f(x,y)=xy? This is just a pedagogical example since

    f(x,y) = xy = e^log x + log y,

but of course the general theorem is very non-trivial.

A notable feature of (<ref>) is that it is a sum of functions of functions, which one might take to inspire a neural network definition. However, one aspect that is a little unusual is that ϕ_q,p depends on both p and q, and therefore lives on the connection between x_p and x^(1)_q:= ∑_p=1^n ϕ_q,p(x_p). If this is a neural network, it's one with activations ϕ_q,p on the edges, rather than on the nodes. Furthermore, for distinct values of p,q, the functions ϕ_q,p are independent in general.

In fact, this idea led to the recent proposal of Kolmogorov-Arnold networks (KAN) <cit.>, a new architecture in which activation functions are on the edges, as motivated by KART, and are learned. Much like how MLPs may be formed out of layers that repeat the structure in the single-layer UAT, KANs may be formed into layers to repeat the structure of KART. See Figure <ref> for a detailed comparison with MLPs, including a depiction of the architectures and how they are formed out of layers motivated by their respective mathematical theorems. The KAN implementation provided in <cit.> includes visualizations of the learned activation functions, which can lead to interpretable can architectures that can be mapped directly onto symbolic formulae.




§ STATISTICS OF NEURAL NETWORKS



Let's try to understand neural networks at initialization. 
 For this, firing up your computer once is not enough, since one initialization give you one draw θ∼ P(θ) and therefore one random neural network. Instead, we want to understand the statistics of the networks:
 What characterizes the stats of the NN ensemble?
 One aspect of this is encoded in the moments, or n-pt functions,

    G^(n)(x_1,…,x_n) = ϕ(x_1)…ϕ(x_n),

which may be obtained from a partition function as 

    Z[J]    = e^∫ d^dx J(x) ϕ(x)
    
    G^(n)(x_1,…,x_n)    = (δ/δ J(x_1)…δ/δ J(x_n) Z[J])|_J=0,

where J(x) is a source.
This expectation · is intentionally not specified here to allow for flexibility. For instance, using the expectation in the introduction we have

    Z[J]     = ∫ dθ P(θ) e^∫ d^dx  J(x) ϕ(x),

reminding the reader that the NN ϕ(x) depends on θ. The partition function integrates over the density of network parameters. But as physicists we're much more familiar with function space densities according to 

    Z[J]     = ∫ϕ e^-S[ϕ] e^∫ J(x) ϕ(x),

the Feynman path integral that determines the correlators from an action S[ϕ] that defines a density on functions.

Since starting a neural network requires specifying the data (ϕ,P(θ)), the parameter space partition function (<ref>) and associated parameter space calculation of correlators is always available to us. Given that mathematical data, one might ask
What is the action S[ϕ] associated to (ϕ,P(θ))?
When this question can be answered, it opens a second way of studying or understanding the theory. The parameter-space and function-space descriptions should be thought of as a duality.



 §.§ NNGP Correspondence


Having raised the question of the action S[ϕ] associated to the network data (ϕ,P(θ)), we can turn to a classic result of Neal <cit.>.

For simplicity, we again consider a single-layer fully connected network of width N, with the so-called biases turned off for simplicity:

    ϕ(x) = 1/√(N)∑_i=1^N ∑_j=1^d w_i^(1)σ(w_ij^(0) x_j),

where the set of network parameters is θ = {w_ij^(0), w_i^(1)} independently and identically distributed (i.i.d.). 

    w_ij^(0)∼ P(w^(0))      w_i^(1)∼ P(w^(1)).
 
Under this assumption, we see
ØThe network is a sum of N i.i.d. functions.
This is a function version of the Central Limit Theorem, generalizing the review in Appendix <ref>, and gives us the Neural Network / Gaussian Process (NNGP) correspondence, 
NNGP Correspondencein the N →∞ limit, ϕ is drawn from a Gaussian Process (GP), 

    lim_N→∞   ϕ(x) ∼(μ(x), K(x,y)),

with mean and covariance (or kernel) μ(x) and K(x,y).
By the CLT, exp(-S[ϕ]) is Gaussian and therefore
S[ϕ] is quadratic in networks. Now this really feels like physics, since the infinite neural network is drawn from a Gaussian density on functions, which defines a generalized free field theory.

We will address generality of the NNGP correspondence momentarily, but let's first get a feel for how to do computations.
To facilitate then, we take P(w^(1)) to have zero mean and finite variance, 

    w^(1)=0     w^(1)w^(1) = μ_2,
 which causes the one-point function to vanish
G^(1)(x) = 0. Following Williams <cit.>, we compute the two-point function in parameter space (with Einstein summation)

    G^(2)(x,y) 
                = 1/Nw_i^(1)σ(w_ij^(0) x_j)  w_k^(1)σ(w_kl^(0) y_l)
       1/N w_i^(1) w_k^(1)σ(w_ij^(0) x_j) σ(w_kl^(0) y_l)
        = μ_2/Nσ(w_ij^(0) x_j) σ(w_il^(0) y_l),

where the last equality follows from the ones being i.i.d., w_i^(1)w_k^(1) = μ_2 δ_ik. The sum over i gives us N copies of the same function, leaving us with 

    G^(2)(x,y) = μ_2  σ(w_ij^(0) x_j) σ(w_il^(0) y_l),

where we emphasize there is now 
no summation on i. This is an exact-in-N two-point function that now requires only on the computation of the quantity in bra-kets. One may try to evaluate it exactly by doing the integral over w^(0). If it can't be done, Monte Carlo estimates may be obtained from M samples of w^(0)∼ P(w^(0)) as

    G^(2)(x,y) ≃μ_2/M∑_samples^M σ(w_ij^(0) x_j) σ(w_il^(0) y_l).

In typical NN settings, parameter densities are easy to sample for convenience, allowing for easy computation of the estimate. If the density is more complicated, one may always resort to Markov chains, e.g. as in lattice field theory.

With this computation in hand, we have the defining data of this NNGP,

    lim_N→∞   ϕ(x) ∼(0, G^(2)(x,y)).

The associated action is 

    S[ϕ] = ∫ d^dx d^dy ϕ(x)  G^(2)(x,y)^-1 ϕ(y),

where 

    ∫ d^dy   G^(2)(x,y)^-1 G^(2)(y,z) = δ^(d)(x-z).

defines the inverse two-point function. In fact, this allows us to determine the action of any NNGP with μ(x)=G^(1)(x)=0, by computing the G^(2) in parameter space and inverting it.



So certain large neural networks are function draws from generalized free field theories. But at this point you might be asking yourself
How general is the NNGP correspondence?
Neal's result — that infinite-width single-layer feedforward NNs are drawn from GP — stood on its own for many years, perhaps (I am guessing) due to focus on non-NN ML techniques in the 90's and early 2000's during a so-called AI Winter. As NNs succeeded on many tasks in the 2010's after AlexNet <cit.>, however, many asked whether architecture X has a hyperparameter N such that the network is drawn from a Gaussian Process as N→∞. Before listing such X's, let's rhetorically ask
Didn't Neal's result essentially follow from summing N i.i.d. random functions? Maybe NNs do this all the time?
In fact, that is the case. Architectures admitting an NNGP limit include

    
  * Deep Fully Connected Networks, N=width,
    
  * Convolutional Neural Networks, N=channels,
    
  * Attention Networks, N=heads,

and many more. See, e.g., <cit.> and references therein.



 §.§ Non-Gaussian Processes


If the GP limit exists due to the CLT, then violating any of the assumptions of the CLT should introduce non-Gaussianities, which are interactions in field theory. From Appendix <ref>, we see that the CLT is violated by finite-N corrections and breaking statistical independence. See <cit.> for a systematic treatment of independence breaking, and derivation of NN actions from correlators.

We wish to see the N-dependence of the connected 4-pt function. William's technique for computing G^(2) extends to any correlator. To avoid a proliferation of indices, we will compute it using the notation

    ϕ(x) = ∑_i w_i φ_i(x)

where w_i is distributed as w^(1) was in the single layer case, and φ_i(x) are i.i.d. neurons of any architecture. The four-point function is 

    G^(4)   = ϕ(x)ϕ(y)ϕ(z)ϕ(w)
       = ∑_i,j,k,lw_i w_j w_k w_lφ_i(x)φ_j(y)φ_k(z)φ_l(w)
       = ∑_i w_i^4φ_i(x)φ_i(y)φ_i(z)φ_i(w)
               + ∑_i≠ jw_i^2w_j^2φ_i(x)φ_i(y)φ_j(z)φ_j(w) + perms.

One can see that you have to be careful with indices. The connected 4-pt function is <cit.>

    G^(4)_c(x_,y,z,w) = G^(4)(x,y,z,w) - (G^(2)(x,y)G^(2)(z,w) + perms),

and watching indices carefully we obtain 

    G^(4)_c(x_,y,z,w) = 1/N(μ_4  φ_i(x)φ_i(y)φ_i(z)φ_i(w)
     - μ_2^2  (φ_i(x)φ_i(y)φ_i(z)φ_i(w) + perms)),

with no Einstein summation on i.
We see that the connected 4-pt function is non-zero at finite-N, signalling interactions. We will see that in some examples G^(4)_c can be computed exactly.




 §.§ Symmetries


We have gotten some control over the statistics of the ensemble of neural networks. It natural 
at this point to ask 
Is there structure in the ensemble?
By this I mean properties that the ensemble realizes that an individual network might not see. We will return to this question broadly in Section <ref>, but for now we will focus on one type of structure: symmetries.

To allow for symmetries at both input and output, in this section we consider networks

    ϕ: ^d →^D.
 
with D-dimensional output. Sometimes the indices will be implicit.

A classic example is equivariant neural networks <cit.>.
We say that ϕ is G-equivariant with respect to a group G if 

    ρ_D(g) ϕ(x) = ϕ(ρ_d(g) x)     ∀ g ∈ G,

where 

    ρ_d ∈Mat(^d)    ρ_D ∈Mat(^D)

are matrix representations of G on ^D and ^d, respectively. The network is invariant if ρ_D = 1, the trivial representation.
Equivariance is a powerful constraint on the network that may be implemented in a variety of ways. For problems that have appropriate symmetries, building them into the network can improve the speed and performance of learning <cit.>, e.g., at the level of scaling laws <cit.>. For instance, in Phiala Shanahan's lectures you'll learn about SU(N)-equivariant neural networks from her work <cit.>, which are natural in lattice field theory due to invariance of the action under gauge transformations.

But this is the statistics section, so 
we're interested in symmetries that arise in ensembles of neural networks, which leave the statistical ensemble invariant. In field theory, we call them global symmetries. Let the network transform under a group action as 

    ϕ↦ϕ_g,      g ∈ G.

We say that ensemble of networks has a global symmetry group G if the partition function is invariant,

    Z_g[J] = Z[J],     ∀ g ∈ G.

At the level of expectations, this is 

    e^∫ d^d x J(x) ϕ_g(x)  =e^∫ d^d x J(x) ϕ(x)    ∀ g ∈ G,

where one can put indices on ϕ and J as required by D. By a network redefinition on the LHS, this may be cast as having a symmetry if · is invariant. 
In the usual path integral this is the statement of invariant action S[ϕ] and measure ϕ. In parameter space, the redefinition may be instituted <cit.> by absorbing g into a redefinition of parameters as θ↦θ_g, with symmetry arising when 

    ∫ dθ_g P(θ_g) e^∫ d^d x J(x)ϕ_θ(x) = ∫ dθ P(θ) e^∫ d^d x J(x)ϕ_θ(x),

i.e. the parameter density and measure must be invariant. We will give a simple example realizing this mechanism in a moment.

It is most natural to transform the input or output of the network. Our mechanism allows for symmetries of both types, which are analogs of spacetime and internal symmetries, respectively. It may also be interesting to study symmetries of intermediate layers, if one wishes to impose symmetries on learned representations. Equivariance fits into this picture because it turns a transformation at input into a transformation at output. The ensemble of equivariant NNs is invariant under ρ_d action on the input if the partition function is invariant under the induced ρ_D action on the output.


Example.  Consider any architecture of the form 

    ϕ(x) = ∑_i w_i φ_i(x),      w_i ∼ P(w)  even,

for any neuron φ, which could itself be a deep network. The _2 action 
ϕ↦ -ϕ may be absorbed into parameters w_g = -w_i with dw P(w) invariant by evenness, which is a global symmetry provided that the domain is invariant. This theory has only even correlators. 

Less trivial examples will be presented below, when we present specific networks and compute correlators.



 §.§ Examples


In the statistics of neural networks, we have covered three topics: Gaussian limits, non-Gaussian corrections from violating CLT assumptions, and symmetries. We will present this essential data in some examples and then discuss similarities and differences. For canonical cases in ML, like networks with ReLU activation or similar, see <cit.> and references therein.

Gauss-net.
The architecture and parameter densities are 

    ϕ(x) = w^(1)_iexp(w^(0)_ij x_j + b^(0)_i)/√(exp[2(σ_b_0^2 + σ^2_w_0/x^2 )]),

for parameters drawn i.i.d. from 

    w^(0)∼𝒩(0,σ^2_W_0/),     w^(1)∼𝒩(0, σ^2_w_1/2N),     b^(0)∼𝒩(0, σ^2_b_0).

The two-point function is

    G^(2)(x_1, x_2) = σ_w_1^2/2 e^-1/2σ_w_0^2 (x_1-x_2)^2

and we see that the theory has a correlation length 

    ξ = √(/σ_w_0^2).

The connected four-point function is

    G^(4)_c =   1/4 Nσ_w_1^4 [ 3e^4σ_b_0^2 e^-σ_w_0^2/2d∑_i x_i^2 -2(x_1 x_2 +  6  perms)
       - (e^-1/2dσ_w_0^2
        ((x_1-x_4)^2+(x_2-x_3)^2) +  2  perms 
        ) ].

We see that the 2-pt function is translation invariant, but not the 4-pt function.


Euclidean-Invariant Nets. 
We are interested in constructing input layers that are sufficient to ensure invariance under SO(d) and translations, i.e., under the Euclidean group. Consider an input layer of the form 

    ℓ_i(x) = F( w^(0)_i) cos(∑_j w^(0)_ij x_j + b^(0)_i),     i ∈1,…,N,

where the sum has been made explicit since i's are not summed over, with 

    w^(0)_ij∼ P(w^(0)_ij)      b^(0)_i ∼Unif[-π,π].

It is easy to see that x_i→ R_ij x_j for R∈ SO(d) can be absorbed into a redefinition of the w^(0)_ij's, and the partition function is invariant under this action provided that F( w^(0)_i) and P(w^(0)_ij) are invariant. Furthermore, arbitrary translations x_j → x_j+ϵ_j give a term w^(0)_ijϵ_j that can be absorbed into a redefinition of the b^(0)_i that leaves the theory invariant. See <cit.> for the 2-pt and 4-pt functions.

We emphasize two nice features:


  * Larger Euclidean Nets. Any network that builds on ℓ without reusing its parameters is Euclidean-invariant.

  * Spectrum Shaping. In computing G^(2)(p), b_i gets evaluated on p, and F may be chosen to shape the power spectrum (momentum space propagator) arbitrarily.

We refer the reader to <cit.> for general calculations of ℓ-correlators and to the specializations below for simple single-layer theories. We turn the ℓ_i(x) input layer into a scalar NN by

    ϕ(x) = ∑ w^(1)_i ℓ_i(x)

with drawn w^(1)_i ∼ P(w^(1)).

Cos-net.
The architecture is defined by specifying (<ref>) to F=1 

    ϕ(x) =  w^(1)_i cos( w^(0)_ij x_j + b^(0)_i),

and specific parameter densities

    w^(1)∼(0,σ_w_1^2/N)      w^(0)∼(0,σ_w_0^2/d)      b^(0)∼Unif[-π,π].

The two-point function is equivalent to that of Gauss-net

    G^(2)(x,y) = σ_w_1^2/2 e^-1/2σ_w_0^2 (x_1-x_2)^2.

The connected four-point function is 

    G^(4)|_c =    1/8 Nσ_w_1^4 [ 3  (e^-1/2dσ_w_0^2
       (x_1+x_2-x_3-x_4)^2+  2 perms) 
       -(2 e^-1/2dσ_b^2
       ((x_1-x_4)^2+(x_2-x_3)^2)+  2 perms) ].

The 2-pt function and 4-pt function are fully Euclidean invariant.

Equipped with both Cos-net and Gauss-net, we'd like to compare the two:


  * Symmetry. Cos-net is Euclidean-invariant ∀ N, by construction, while Gauss-net enjoys this symmetry only when N→∞; in that limit it only depends on G^(2)(x,y).

  * Large-N Duality. As N→∞, both theories are drawn from the same Gaussian Process, i.e., with the same G^(2)(x,y).


Scalar-net.
Here is one final example you might like, specializing (<ref>)

    ϕ(x) =  √(2 vol(B_Λ^d)/(2π)^d σ_w_1^2)1/√( w^(0)_i^2 + m^2)  w^(1)_i cos( w^(0)_ij x_j + b^(0)_i),

and specific parameter densities

    w^(1)∼(0,σ_w_1^2/N)      w^(0)∼Unif(B_Λ^d)      b^(0)∼Unif[-π,π],

where B_Λ^d is a d-ball of radius Λ. The theory is translation invariant by construction, and so we compute the power spectrum of the two-point function G^(2)(x-y) to be 

    G^(2)(p) = 1/p^2+m^2.

We see that we have a realization of the free scalar field theory in d Euclidean dimensions. See Sec. <ref> for an extension to ϕ^4 theory.
    



§ DYNAMICS OF NEURAL NETWORKS



Having covered expressivity and statistics, we turn to dynamics. Focusing on the most elementary NN dynamics, we ask
How does a NN evolve under gradient descent? First we will study a simplification known as the neural tangent kernel (NTK), and then will use it in the case of MSE loss to solve a model exactly. We'll discuss drawbacks of the NTK, and then improve upon them with a scaling analysis that ensures feature learning.

We will study the dynamics of supervised learning with gradient  descent, with data 

    = {(x_α,y_α)}_α=1^||,

and loss function

    [ϕ] = 1/||∑_α=1^||ℓ(ϕ(x_α), y_α),

We optimize the network parameters θ by gradient descent,

    dθ_i/dt = -η∇_θ_i[ϕ].

It is also convenient to define 

    Δ(x) = - δℓ(ϕ(x),y)/δϕ(x),

where y is to be understood as the label associated to x, which yields 

    dθ_i/dt= η/||∑_α=1^||Δ(x_α) ∂ϕ(x_α)/∂θ_i.

as another form of the gradient descent equation, by chain rule. Δ(x) is the natural object of gradient descent in function space.

We use Einstein summation throughout this section unless stated otherwise (which will happen).



 §.§ Neural Tangent Kernel


We now arrive at a classic dynamical result in ML theory, the neural tangent kernel <cit.>. We train the network by gradient descent 

    dϕ(x)/dt   = ∂ϕ(x)/∂θ_idθ_i/dt
       = η/||∑_α=1^||Δ(x_α)  Θ(x,x_α),

where 

    Θ(x,x_α) = ∂ϕ(x)/∂θ_i∂ϕ(x_α)/∂θ_i

is the neural tangent kernel (NTK).

Given the short derivation, this is clearly a fundamental object, but it seems terrible to work with, because it is:


  * Parameter-dependent. For modern NNs, there are billions of parameters to sum over.

  * Time-dependent. Of course, the learning trajectory is θ(t), and therefore the NTK time evolves.

  * Stochastic. Since θ(t) begins at θ(0) sampled at initialization, the NTK inherits the randomness.

It's also non-local, communicating information about the loss at train points x_α to the test point x. In summary, the NTK is unwieldy.

The reason it is a classic result, however, is that it simplifies in the N→∞ limit. In this limit, neural network training is in the so-called

    Lazy regime:     |θ(t) - θ(0)| ≪ 1,

i.e. the number of parameters is large but the evolution keeps them in a local neighborhood. In such a regime the network is approximately a linear-in-parameters model <cit.>

    lim_N→∞ϕ(x)≃ϕ_lin(x) := ϕ_θ_0(x) + (θ-θ_0)_i ∂ϕ(x)/∂θ_i|_θ_0,

and we have 

    lim_N→∞Θ(x,x') ≃Θ(x,x')|_θ_0.

That is, the infinite-width NTK is the NTK at initialization, provided that the network evolves as a linear model. Furthermore, in the same limit the law of large numbers often allows a sum to be replaced by an expectation value, e.g., 

    lim_N→∞Θ(x,x')|_θ_0 = β_θ(x,x') =: Θ̅(x,x'),

for computable β(x,x'), yielding network dynamics governed by 

    dϕ(x)/dt= -η/||∑_α=1^||δ l(ϕ(x_α),y_α)/δϕ(x_α)  Θ̅(x,x_α),

where Θ̅ is the so-called frozen NTK, a kernel that may be computed at initialization and fixed once-and-for-all. This is a dramatic simplification of the dynamics.

However, you should also complain.
ComplaintThe dynamics in (<ref>) simply interpolates between information at train points x_α and test point x,
according to a fixed function Θ̅. This isn't “learning" in  the usual NN sense, and there are zero parameters. In particular, since the NN only affects (<ref>) through Θ̅, which is fixed, nothing happening dynamically in the NN is affecting the evolution. We say that in this limit the NN does not learn features in the hidden dimensions (intermediate layers), since their non-trivial evolution would cause the NTK to evolve.

Example. 
Let's compute the frozen NTK for a single-layer network, to get the idea. The architecture is 

    ϕ(x) = 1/√(N)∑_i=1^N ∑_j=1^d w_i^(1)σ(w_ij^(0) x_j).

We make the sums explicit here because one is very important.
The NTK is 

    Θ(x,x')    = ∑_i∂ϕ(x)/∂ w_i^(1)∂ϕ(x')/∂ w_i^(1) + ∑_ij∂ϕ(x)/∂ w_ij^(0)∂ϕ(x')/∂ w_ij^(0)
       = 1/N∑_i=1^N (∑_j,l=1^d σ(w_ij^(0)x_j)σ(w_il^(0)x'_l)
            + ∑_j=1^d x_j x'_j  w_i^(1)w_i^(1) σ'(w_ij^(0) x_j) σ'(w_ij^(0)x'_j) ) 
       =: 1/N∑_i β_i(x,x').

If you squint a little, you'll see that the i-sum is a sum over the same type of object, β_i(x,x'), whose i dependence comes from all these i.i.d. parameter draws in the i-direction. By the law of large numbers, we have that in the N→∞ limit

    Θ̅(x,x') = β_i(x,x'),

with no sum on i.
We emphasize 
ØThe NTK in the N→∞ limit is deterministic (parameter-independent), depending only on P(θ).
Sometimes, the expectation may be computed exactly, and one knows the NTK that governs the dynamics once and for all.



 §.§ An Exactly Solvable Model


Let us consider a special case of frozen-NTK dynamics with MSE loss,

    ℓ(ϕ(x),y) = 1/2(ϕ(x)-y)^2.

Then the dynamics (<ref>) becomes 

    dϕ(x)/dt = -η/||∑_α=1^|| (ϕ(x_α)-y_α) Θ̅(x,x_α).

The solution to this ODE is 

    ϕ_t(x) = ϕ_0(x) + 1/||Θ̅(x,x_α) Θ̅(x_α, x_β)^-1(1-e^-ηΘ̅t)_βγ(y_γ-ϕ_0(x_γ)),

where computational difficulty is that Θ̅(x,x_α) is a ||× || matrix and takes O(||^3) time to invert. 
The solution defines a trajectory through function space from ϕ_0 to ϕ_∞. The converged network is 

    ϕ_∞(x) = ϕ_0(x) +Θ̅(x,x_α) Θ̅(x_α, x_β)^-1(y_β-ϕ_0(x_β)).

This is known as kernel regression, a classic technique in ML. 
In general kernel regression, one chooses the kernel. In our case, gradient descent training in the N→∞ limit is kernel regression, with respect to a specific kernel determined by the NN, the NTK Θ̅. 

On train points we have memorization

    ϕ_∞(x_α) = y_α    ∀α.

On test points x, the converged network is performing an interpolation, communicating residuals R_β on train points β through a fixed kernel Θ̅ to test points x. The prediction depends on ϕ_0, but may be averaged over to obtain 

    μ_∞(x) := ϕ_∞(x) = Θ̅(x,x_α) Θ̅(x_α, x_β)^-1y_β,

provided that ϕ_0=0, as in many initializations for the parameters.
Let's put some English on the remarkable facts,

    
  * μ_∞(x) is the mean prediction of an ∞ number of ∞-wide NNs trained to ∞ time.
    
  * If ϕ_0 is drawn from a GP, then ϕ_∞ is as well. The mean is precisely μ_∞(x), see <cit.> for the two-point function and covariance.




 §.§ Feature Learning


The frozen NTK is a tractable toy model, but it has a major drawback: it does not learn features.
In this section we perform a more general study of learning dynamics, with a focus on choosing wise N-scaling such that features are non-trivially learned during gradient descent. To do this we will engineer three properties: that features (pre-activations) are finite at initialization, predictions evolve in finite time, and features evolve in finite time. 

This section is denser than the others, so I encourage the reader to remember that we are aiming to achieve these three principles if they become overwhelmed with details. Throughout, I am following lecture notes of Pehlevan and Bordelon <cit.>  with some changes in notation to match the rest of the lectures; see those lectures for more details, <cit.> for original literature from those lectures that I utilize, and <cit.> for related work on feature learning.

We study a deep feedforward network with L layers and width N, which in all is a map

    ϕ:^D →

(note the input dimension D; d is reserved for below) defined recursively as 

    ϕ(x)    = 1/γ_0 N^dz^(L)(x) 
     
    z^(L)(x)    = 1/N^a_L w_i^(L) σ(z_i^(L-1)(x))
    
    z^(ℓ)_i(x)    = 1/N^a_ℓ W^(ℓ)_ijσ(z_j^(ℓ-1)(x)) 
    
    z_i^(1)(x)    = 1/N^a_1√(D) W^(1)_ijx_j

where Einstein summation is implied throughout this section   (unless stated otherwise) and all Latin indices run from {1,…,N} except in the j-index in the first layer, when they are {1,…,D}. The parameters are drawn 

    w_i^(ℓ)∼(0,1/N^b_L)      W^(ℓ)_ij∼(0,1/N^b_ℓ).

We scale the learning rate as 

    η = η_0 γ_0^2 N^2d-c

with γ_0,η_0 O(1)  constants. We use a parameterization that will be convenient where d has already been introduced but c is a new parameter. For notational brevity we will sometimes use a Greek index subscript in place of inputs, e.g. z^(ℓ)_α:= z^(ℓ)(x_α). The z's are known as the pre-activations, as they are the inputs to the activation functions σ.

We have a standard MLP but have parameterized our ignorance of N-scaling, governed by parameters (a_ℓ,b_ℓ,c,d). We will use this freedom to set some reasonable goals:

    
  * Finite Initialization Pre-activations. z^(ℓ)∼ O_N(1)   ∀ l.
    
  * Learning in Finite Time. dϕ(x)/dt∼ O_N(1).
    
  * Feature Learning in Finite Time. dz^(ℓ)/dt ∼ O_N(1)    ∀ l.

These constraints have a one-parameter family of solutions, which becomes completely fixed under an additional learning rate assumption. We take each constraint in turn.

Finite Pre-activations. Since the weights are zero mean,  trivially

    z^(ℓ)_α = 0     ∀ l.

We must also compute the covariance. For the first pre-activation, we have 

    z_iα^(1)z_jβ^(1) = 1/D N^2a_1w^(1)_imw^(1)_jn  x_mαx_nβ = δ_ij1/DN^2a_1+b_1 x_mαx_mβ.

A similar calculation for higher layers l  gives

    z_iα^(ℓ)z_jβ^(ℓ)   = 1/ N^2a_ℓw^(1)_imw^(1)_jn σ(z^(ℓ-1)_mα)σ(z^(ℓ-1)_mβ)
       = δ_ij1/N^2a_ℓ+b_ℓ-11/Nσ(z^(ℓ-1)_mα)σ(z^(ℓ-1)_mβ)
       = δ_ij1/N^2a_ℓ+b_ℓ-1Φ^(ℓ-1)_αβ,

where 

    Φ^(ℓ)_αβ := 1/Nσ(z^(ℓ)_mα)σ(z^(ℓ)_mβ)

is a feature kernel. We are constructing a proof-by-induction that the pre-activations are O_N(1), so at this stage we may assume that the pre-activations z^(ℓ-1)∼ O_N(1) and therefore Φ^(ℓ-1)∼ O_N(1) since it is the average of N O_N(1) quantities σ(z^(ℓ-1)_mα). With this in hand, 

    2a_1+b_1=0      2a_ℓ + b_ℓ = 1   ∀ l > 1,

ensures that the pre-activations z^(ℓ) are O_N(1), as is empirically required for well-behaved training.

As an aside: in the N→∞ limit feature kernels asymptote to deterministic objects, akin to the frozen NTK behavior we have already seen, and intermediate layer pre-activations z_α^(ℓ) are also Gaussian distributed. Therefore in that limit, the statistics of a randomly initialized neural network is described by a sequence of generalized free field theories where correlations are propagated down the network according to a recursion relation; see, e.g., <cit.>.

Learning in Finite Time. Similar to our NTK derivation, we have 

    dϕ(x)/dt = η/||∑_α=1^||Δ(x_α) ∂ϕ(x_α)/∂θ_i∂ϕ(x)/∂θ_i,

We have that 

    dϕ(x)/dt∼ O_N(1)   ↔   γ_0^2 N^2d/N^c∂ϕ(x_α)/∂θ_i∂ϕ(x)/∂θ_i∼ O_N(1).

A short calculation involving a chain rule, keeping track of the different types of layers, and tedium yields

    η_0γ_0^2 N^2d/N^c∂ϕ(x_α)/∂θ_i∂ϕ(x)/∂θ_i   = 1/N^c[ 1/N^2 a_L - 1Φ_μν^(L-1)
       + ∑_ℓ=2^L-11/N^2 a_ℓ - 1 G_α x^(ℓ)Φ_α x^(ℓ-1) + 1/N^2 a_1 G_α x^(1)Φ_α x^(0)]

where 

    G_μν^(ℓ) = 1/N g^(ℓ)_iμ  g^(ℓ)_iν,
          g^(ℓ)_iμ = √(N) ∂ z^(L)_μ/∂ z^(ℓ)_iμ

arises naturally when doing a chain rule through the layers and x in the subscript is a stand-in for an arbitrary input x. Some additional work <cit.> shows that g, G ∼ O_N(1). Since we've shown that the feature kernels are also O_N(1), then if 

    2a_1 + c = 0,      2a_ℓ + c = 1    ∀ l > 1,

we have that dϕ(x)/dt∼ O_N(1), i.e., predictions evolve in finite time, and therefore learning happens in finite time.

Features Evolve in Finite Time. 
Finally, we need the features to evolve. For the first layer we compute

    dz_iμ^(1)/dt   =1/N^a_1√(D)dW_ij/dt x_jμ 
     = 1/N^2a_1+c-d+1/2η_0γ_0/||∑_ν=1^||Δ_ν g_iν^(1)Φ_μν^(0),
 
where the only N-dependence arises from the exponent. Given (<ref>), 

    d=1/2

is required to have  dz^(1)/dt ∼ O_N(1), a constraint that in fact persists for all dz^(ℓ)/dt. We note that we have
No feature learningif d< 1/2,
which applies in particular to the NTK regime d=0.

Summarizing the Constraints. Putting it all together, by enforcing finite pre-activations and that predictions and features evolve in finite time, we have constraints given by

    2a_1 + b_1    = 0 
    
        2a_ℓ + b_ℓ   = 1   ∀ l > 1 
    
        2a_1 + c    = 0 
    
        2a_ℓ + c    = 1   ∀ l > 1 
    
        d    = 1/2.

These constraints are solves by a one-parameter family depending on a∈

    (a_ℓ, b_ℓ, c_ℓ)    = (a,1-2a,1-2a)   ∀ l > 1 
     
    (a_1, b_1, c_1)     = (a-1/2, 1-2a, 1-2a).

If one makes the addition demands that η∼ O_N(N^2d-c) is O_N(1), so that you don't have to change the learning rate as you scale up, then c=1 is fixed and there is a unique solution. This scaling is known as the maximal update parameterization, or μ P <cit.>.







§ NN-FT CORRESPONDENCE



Understanding the statistics and dynamics of NNs has led us naturally to objects that we are used to from field theory. The idea has been to understand ML theory, but one can also ask the converse, whether ML theory gives new insights into field theory. With that in mind, we ask
What is a field theory?
At the very least, a field theory needs 
    
  * Fields, functions from an appropriate function space, or sections of an appropriate bundle, more generally. 
    
  * Correlation Functions of fields, here expressed as scalars
    
    G^(n)(x_1,…,x_n) = ϕ(x_1)…ϕ(x_n).


You might already be wanting to add more beyond these minimal requirements – we'll discuss that in a second. For now, we have 
Answera FT is an ensemble of functions with a way to compute their correlators.
In the Euclidean case, when the expectation is a statistical expectation, one my say
Euclidean Answera FT is a statistical ensemble of functions.
Our minimal requirements get us a partition function

    Z[J] = e^∫ d^dx J(x)ϕ(x)

that we can use to compute correlators, where at this stage we are agnostic about the definition of ·. In normal field theory, the · is defined by the Feynman path integral 

    Z[J] = ∫ϕ  e^-S[ϕ] + ∫ d^dx J(x)ϕ(x),

which requires specifying an action S[ϕ] that determines a density on functions exp(-S[ϕ]). But that's not the data we specify when we specify a NN. The NN data (ϕ_θ, P(θ)) instead defines 

    Z[J] = ∫ dθ P(θ) e^∫ d^dx J(x)ϕ_θ(x).

These are two different ways of defining a field theory, and indeed given (ϕ_θ,P(θ)) one can try to work out the associated action, in which case we have dual description of the same field theory, as in the NNGP correspondence.
The parameter space description is already quite useful, though, as it enables the computation of correlation functions even if the action isn't known. In certain cases it enables the computation of exact correlators in interacting  theories.

Okay, you get it, this is a different way to do field theory. Now I'll let you complain about my definition. You're asking 
Shouldn't my definition of field theory include X?
I'm writing this before I give the lecture, and my guess is you already asked about a set of X's, e.g. 

    X ∈{Quantum ,
         Lagrangian,
         Symmetries,
         Locality, …}.

The problem is that with any such X, there's usually some community of physicists that doesn't care. For instance, not all statistical field theories are Wick rotations of quantum theories; not all field theories have a known Lagrangian; not all field theories have symmetry; not all field theories are local. So I'm going to stick with my definition, because at a minimum I want fields and correlators.

Instead, if your X isn't included in the definition of field theory, it becomes an engineering problem. Whether you're defining your specific theory by S[ϕ], (ϕ_θ,P(θ)), or something else, you can ask
Can I engineer my defining data to get FT + X?
For X=Symmetries you've already seen ways to do this at the level of actions in QFT1 and at the level of (ϕ_θ,P(θ)) in these lectures.


For a current account of recent progress in NN-FT, see the rather long Introduction of <cit.> and associated references, as well as results.



 §.§ Quantum Field Theory


We've been on Euclidean space the whole time[This can be relaxed, see e.g. for a recent paper defining an equivariant network in Lorentzian signature <cit.>.], so it's natural to wonder in what sense these field theories are quantum. In a course on field theory, we first learn to canonically quantize and then at some later point learn about Wick rotation, and how it can define Euclidean correlators. The theory is manifestly quantum.

But given a Euclidean theory, can it be continued to a well-behaved quantum theory in Lorentzian signature, e.g. with unitary time evolution and a Hilbert space without negative norm states? If we have a nice-enough local action, it's possible, but what if we don't have an action? We ask:
Given Euclidean correlators, can the theory be continued to a well-behaved Lorentzian quantum theory?
This is a central question in axiomatic quantum field theory, and the answer is that it depends on the properties of the correlators. The Osterwalder-Schrader (OS) theorem <cit.> gives a set of conditions on the Euclidean correlators that ensure that the theory can be continued to a unitary Lorentzian theory that satisfies the Wightman axioms.
The conditions of the theorem include 


  * Euclidean Invariance. The correlators are invariant under the Euclidean group, which after continuation to Lorentzian signature becomes the Poincaré group.

  * Permutation Invariance of the correlators G^(n)(x_1,…,x_n) under any permutation of the x_1,…,x_n.

  * Reflection Positivity. Having time in Lorentzian signature requires picking a Euclidean time direction τ. Let R(x) be the reflection of x in the τ=0 plane. Then reflection positivity requires that

    G^(2n)(x_1,…,x_n,R(x_1),…,R(x_n)) ≥ 0.

Technically, this is necessary but not sufficient. An accessible elaboration can be found in notes <cit.> from a previous TASI.

  * Cluster Decomposition occurs when the connected correlators vanish when any points are infinitely far apart.

If all of these are satisfied, then the pair (ϕ_θ, P(θ)) that defines the NN-FT actually defines a neural network quantum field theory <cit.>.
In NN-FT, permutation invariance is essentially automatic and Euclidean invariance may be engineered as described in Section <ref>. Cluster decomposition and reflection positivity hold in some examples <cit.>, but systematizing their construction is an important direction for future work.

There is at least one straightforward way to obtain an interacting NN-QFT. Notably, if (ϕ_θ, P(θ)) is a NNGP that satisfies the OS axioms (this is much easier <cit.>) with Gaussian partition function

    Z_G[J] = ∫ dθ P(θ) e^∫ d^dx J(x)ϕ_θ(x),

then one may insert an operator associated to any local potential V(ϕ), which deforms the action in the expected way and the NN-FT to 

    Z[J]    = ∫ dθ P(θ) e^∫ d^dx V(ϕ_θ(x)) e^∫ d^dx J(x)ϕ_θ(x)
       =: ∫ dθP̃(θ) e^∫ d^dx J(x)ϕ_θ(x),

where the architecture equation ϕ_θ(x) lets us sub out the abstract expression for a concrete function of parameters, defining a new density on parameters P̃(θ) in the process. The interactions in V(ϕ) break Gaussianity of the NNGP that was ensured by a CLT. This means a CLT assumption must be violated: it is the breaking of statistical independence in P̃(θ). The theory Z[J] defined by (ϕ_θ, P̃(θ)) is an interacting NN-QFT, since local potentials that deformed Gaussian QFTs still satisfy reflection positivity and cluster decomposition.



 §.§ ϕ^4 Theory


With this discussion of operator insertions, it's clear now how to get ϕ^4 theory. We just insert the operator

    e^∫ d^dx ϕ_θ(x)^4

into the partition function associated to the free scalar of Section <ref>; this operator with the architecture (<ref>) technically requires an IR cutoff, though other architectures realizing the free scalar may not. The operator insertion deforms the parameter densities in (<ref>) and breaks their statistical independence, explaining the origin of interactions in the NN-QFT. See <cit.> for a thorough presentation.



 §.§ Open Questions


In this section we've discussed a neural network approach to field theory in which the partition function is an integral over probability densities of parameters, and the fields (networks) are functions of these parameters. We have summarized some essential results, but there are many outstanding questions:


  * Reflection Positivity. Is there a systematic way to engineer NN-FTs (ϕ_θ, P(θ)) that satisfy reflection positivity?

  * Cluster Decomposition. Can  systematically understand the conditions under which cluster decomposition holds in NN-FT?

  * Engineering Actions. How generally can we define an NN-FT that is equivalent to a field theory with a fixed action?

  * Locality in the action can be realized at infinite-N, but is there a fundamental obstruction at finite-N?

On the other hand, an NN-FT approach to CFT and to Grassmann fields is well underway and should appear in 2024.



§ RECAP AND OUTLOOK


Neural networks are the backbone of recent progress in ML. In these lectures we took the perspective that to understand ML, we must understand neural networks, and built our study around three pillars: the expressivity, statistics, and dynamics of neural networks.

Let's recap the essentials and then provide some outlook. In Section <ref> on expressivity, we asked “How powerful is a NN?" We presented the Universal Approximation Theorem (UAT) and gave a picture demonstrating how it works. We also presented the Kolmogorov-Arnold Representation Theorem (KART), which recently motivated a new architecture for deep learning where activations are learned and live on the edges. We wonder:
OutlookCybenko's UAT was around for over 20 years before the empirical breakthroughs of the 2010's. Existence of good approximators wasn't enough, we needed better compute and optimization schemes to find them dynamically.  However, given tremendous progress in dynamics, should we systematically return to the approximation theory literature, to help motivate new architectures, as e.g., in KART and KAN?

In Section <ref> on statistics, I reminded the reader that a randomly initialized NN is no more fundamental than a single role of the dice. It is a random function with parameters that is drawn from a statistical ensemble of NNs, and we asked “What characterizes the statistics of the NN ensemble?” I reviewed a classic result of Neal from the 90's that demonstrated that under common assumptions a width-N feedforward network is drawn from a Gaussian Process — a Gaussian density on functions — in the N→∞ limit. I explained that work from the last decade has demonstrated that this result generalizes to many different architectures, as a result of the Central Limit Theorem. Non-Gaussianities therefore arise by violating an assumption of the Central Limit Theorem, such as N→∞ or statistical independence, and by violating them weakly the non-Gaussianities can be made parametrically small. In field theory language, NNGPs are generalized free field theories and NN Non-Gaussian processes are interacting field theories. I also introduced a mechanism for global symmetries and exemplified many of the phenomena in the section.

In Section <ref> I presented some essential results from ML theory on NN dynamics, asking “How does a NN evolve under gradient descent?” When trained with full-batch gradient descent and a particular normalization, the network dynamics are governed by the Neural Tangent Kernel (NTK) that in general is an intractable object. However, in N→∞ limits it becomes a deterministic function and the NTK at initialization governs the dynamics for all time. An exactly solvable model with mean-squared error loss is presented that includes the mean prediction of an infinite number of infinitely wide neural networks trained to infinite time. The mean network prediction is equivalent to kernel regression, and the kernel communicates information from train points to test points in order to make predictions. However, from this description of the NTK a problem is already clear: nothing is being learned, and in particular late-time features in the hidden dimensions are in a local neighborhood of their initial values.  I showed how a detailed N-scaling analysis allows one to demand that network features and predictions update non-trivially, leading to richer learning regimes known as dynamical mean field theory or the maximal update parameterization.
OutlookThese recent theories of NN statistics and dynamics sometimes realize interesting toy models, but with known shortcomings from a learning perspective, though the N-scaling analysis in the feature learning section is promising. Surely there is still much to learn overall, e.g., we have said very little in these lectures about architecture design principles. For instance, some architectures, such as the Transformers <cit.> central to LLMs, are motivated by structural principles, whereas others are motivated by theoretical guarantees. 
 
 What are the missing principles that combines statistics, dynamics, and architecture design to achieve optimal learning?




Finally, in Section <ref> I explained how neural networks provide a new way to define a field theory, so that one might use ML theory for physics and not just physics for ML theory. This NN-FT correspondence is a rethink of field theory from an ML perspective, and I tried to state clearly about what is and isn't known about obtaining cherished field theory properties in that context. For example, I explained how by engineering a desired NNGP, such as the free scalar, one may do an operator insertion in the path integral and interpret the associated interactions as breaking the statistical independence necessary for the CLT. By such a mechanism, one may engineer ϕ^4 theory in an ensemble of infinite width neural networks.
OutlookCan NN-FTs motivate new interesting physics theories or provide useful tools for studying known theories?

Our discussion in this Section is directly correlated with the topics of these lectures. It reviews recent theoretical progress, but the careful reader surely has a sense that theory is significantly behind experiment and there is still much to do. It is a great time to enter  the field, and to that end I recommend carefully reading the literature that I've cited, as well as the book Geometric Deep Learning <cit.> that covers many current topics that will be natural to a physicist, including grids, groups, graphs, geodesics, and gauges. Many aspects of that book are related to principled architecture design, which complements the statistical and dynamical approach that I've taken in these lectures. I also recommend the book Deep Learning <cit.> for a more comprehensive introduction to the field.

I end with a final analogy between history and the current situation in ML. We are living in an ML era akin to the pre-PC era in classical computing. Current large language models (LLMs) or other deep networks are by now very powerful, but are the analogs of room-sized computers that can only be fully utilized by the privileged. Whereas harnessing the power of those computers in the 60's required access to government facilities or large laboratories, only billion dollar companies have sufficient funds and compute to train today's LLMs from scratch. Just as the PC revolution brought computing power to the masses in the 80's, a central question now is whether we can do the analog in ML by learning how to train equally powerful small models with limited resources. Doing so likely requires a deeper understanding of theory, especially with respect to sparse networks.



Acknowledgements: I would like to thank the students and organizers of TASI for an inspiring scientific environment and many excellent questions. I am grateful for the papers of and discussions with friends whose works have contributed significantly to these lectures, especially Yasaman Bahri, Cengiz Pehlevan, and Greg Yang.  I'd also like to thank my collaborators on related matters, including Mehmet Demirtas, Anindita Maiti, Fabian Ruehle, Matt Schwartz, and Keegan Stoner. I would like to thank Sam Frank, Yikun Jiang, and Sneh Pandya for correcting typos in a draft. Finally, thanks to GitHub Copilot for its help in writing these notes, including the correct auto-completion of equations and creation of  figures! Two-column landscape mode was chosen to facilitate boardwork, but I think it also makes a pleasant read, HT @ <cit.>. I am supported by the National Science Foundation under CAREER grant PHY-1848089 and
Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions).



Disclaimers: In an effort to post these notes shortly after TASI 2024, there are more typos and fewer references than is ideal. Other references and topics may be added in the future to round out the content that was presented in real-time. I also struck a playful tone throughout, because lectures are supposed to be fun, but may have overdone it at times. I welcome suggestions on the above, but updates will still aim for clarity and brevity, targeted at HET students.



§ CENTRAL LIMIT THEOREM


Let us recall a simple derivation of the Central Limit Theorem (CLT), in order to better understand the statistics of neural networks. Consider a sum of random variables

    ϕ = 1/√(N)∑_i=1^N X_i,

with X_i = 0. The moments μ_r and cumulants κ_r are determined by the moment generating function (partition function) Z[J] = e^Jϕ and cumulant generating function W[J] = log Z[J], respectively, as 

    μ_r     = (d/dJ)^r Z[J]|_J=0
    κ_r     = (d/dJ)^r W[J]|_J=0.

If the X_i are independent random variables, then the partition function factorizes 
Z_∑_i X_i[J] = ∏_i Z_X_i[J], and the cumulant generating function of the sum is the sum of the cumulant generating functions, yielding

    W_∑_i X_i[J] = ∑_i W_X_i[J] 
    κ_r^∑ X_i = ∑_i κ_r^X_i.
 
If the X_i are identically distributed, then the cumulants κ_r^X_i are the same for all i and we account for the 1/√(N) appropriately, we obtain

    κ_r^ϕ = κ_r^X_i/N^r/2-1.

This yields 

    lim_N→∞κ_r>2^ϕ = 0,

which is sufficient to show that ϕ is Gaussian in the large-N limit. In physics language, cumulants are connected correlators, and (<ref>) means that Gaussian (free) theories have no connected correlators.

In neural networks we will be interested in studying certain Gaussian limits. From this CLT derivation, we see two potential origins of non-Gaussianity:

    
  * 1/N-corrections from appearance in κ_r^ϕ.
    
  * Independence breaking since the proof relied on (<ref>).








utphys




 Paper Title: Comparative of light propagation in Born-Infeld, Euler-Heisenberg and ModMax nonlinear electrodynamics.


Fontaine-Laffaille Theory over Power Series Rings
    Christian Hokaj
    August 5, 2024
=================================================





§ INTRODUCTION


Nonlinear electromagnetic effects become significant when the electromagnetic field strengths approach the critical fields E_cr≈ m^2_ec^3/(eħ)≈ 10^18 Volt/m or B_cr≈ 10^9 Tesla; B_cr represents the field at which the cyclotron energy equals m_ec^2, where m_e is the mass of the electron, and it defines the field scale at which the external field becomes significant on quantum processes; it is also known as Schwinger critical field and it marks out the threshold of nonlinear QED scale <cit.>. Moreover, anyone familiar with quantum field theory is led to the  conclusion that a classical phenomenological theory that pretends a description of polarization effects (light-light interaction or pair production in vacuum excited by an electric field) has to be nonlinear.

The nonlinear electromagnetic (NLEM) interactions can be described in a classical way using an effective Lagrangian that  depends nonlinearly on the two Lorentz and gauge invariants, F and G, of the Faraday tensor F_μλ=∂_μA_λ-∂_λA_μ,


    F=F^μλF_μλ= 2(B^2-E^2),    G=F^*μλF_μλ= -4 B⃗·E⃗,
 
where F^*μλ=1/2ϵ^μλαβF_αβ.  

Maxwell's electrodynamics cannot describe nonlinear interactions between two electromagnetic waves, for example, the interaction light-by-light; these phenomena are well known in QED, and considering classical nonlinear electrodynamics (NLED) it is possible to describe them <cit.>. 
Several NLED proposals exist to describe such nonlinear effects, but three of them stand out:  Born-Infeld (BI), Euler-Heisenberg (EH), and ModMax theories. 

  
In 1934, Born and Infeld <cit.> presented a theory with nonlinear corrections to Maxwell electrodynamics with the aim of solving the singularity of the electric field self-energy of charged particles;  they propose the existence of a maximum attainable electromagnetic field, denoted by the BI parameter b, with a magnitude b= e / r_0^2 = 10^20 Volt/m, where r_0 is the classical electron radius. The interpretation of this absolute field is also that the classical self-energy of the electron is equal to its mass-energy at rest <cit.>. 
The Lagrangian for this NLED is established as 

    L_BI=-4b^2 (1-√(1+F/2b^2-G^2/16b^4)).


This classical theory effectively models vacuum polarization as a material medium.  
From the viewpoint of QED, the BI modifications of the Maxwell theory result in a change in the photon-photon scattering signal. This is the reason why it has been proposed that experiments that measure the QED process could constrain the maximum field b of the BI theory <cit.>. Another interesting feature is that it presents neither birefringence nor shock waves; the absence of birefringence leads to the idea that vacuum birefringence experiments could set bounds on the BI parameter b  <cit.>; also experimental constrains have been proposed through interferometry <cit.>.
It is also worth noting that there is a connection of BI-NLED with string theory; in bosonic open string theory
the BI action arises as an effective low-energy action <cit.>.


On the other hand, the EH theory was derived from QED principles by W. Heisenberg and H. Euler in 1936 <cit.>  from the perspective of QED  <cit.>. It considers vacuum polarization to one loop and is valid for electromagnetic fields that change slowly compared to the inverse of the electron mass.  By treating the vacuum as a medium, EH effective action predicts rates of nonlinear light interaction processes.
The Lagrangian for EH NLED is 

    L_EH=-F/4+α^2/90 m_e^4(F^2+7/4G^2),

where α is the fine structure constant and m_e the electron mass.

An interesting aspect of NLED are the symmetries that are preserved.
The BI NLED is not conformally invariant while the EH NLED is neither conformal invariant or dual invariant. The question then arose if a NLED fulfilling the same symmetries as Maxwell's existed; i.e. a NLED with conformal and electric-magnetic duality invariances. It indeed exists. Recently  was derived a NLED theory endowed with the two symmetries <cit.>.  It is characterized by a dimensionless parameter γ and it reduces to Maxwell theory if γ=0. This theory, known as Modified Maxwell (ModMax, MM) NLED, has stimulated research in several aspects, from classical solutions <cit.> to supersymmetric analysis <cit.>. The Lagrangian for the ModMax NLED is 

    L_MM=-F/4coshγ +sinhγ/4√(F^2+G^2).
    

In this work, we discuss the phase velocities of light moving in the three previously described NLEDs; although the parameter of each NLED is different we would like to know which one is the most effective in slowing down light velocity in a magnetic or electric background. Additionally, 
there is not a criterion to constrain the values of the ModMax parameter γ, apart from being positive, as energy conditions of a MM black hole demand.   We use BI and EH theories to set bounds on γ through the comparison of the phase velocities of light propagating in a magnetic (or electric) background.  In a second part of the paper, regarding NLED coupled to gravity, we compare the shadows of the respective black hole solutions; confronting their shadows with the observational evidences from Sagittarius A^* and the M87^* galaxy
allows to delimit the ranges of the nonlinear  parameters (μ for EH, b for BI, and γ for ModMax) that would fit with charged astrophysical supermassive black holes.


The paper is organized as follows: in Section <ref> it is presented the effective or optical metric, that is the background metric modified by the nonlinear electromagnetic effects; the null geodesics of the effective metric are the actual trajectories of light propagation; then the phase velocity general expression is derived. In Section <ref>, the effective metric and phase velocity of light propagation in an intense magnetic  background are given for each one of the NLEDs under study. Through phase velocities for the BI and EH NLEDs we set bounds on the  ModMax nonlinear parameter γ.
In Section <ref>, we consider the static spherically black hole solutions of the three analyzed NLEDs and calculate the  corresponding shadows that are then confronted versus the Sagittarius A^* and the M87 galaxy observed shadows. These tests set bounds on the possibility of a charged supermassive black hole. Finally, the conclusions are given in the last section.





§ NLED EFFECTIVE METRIC AND PHASE VELOCITY



It is well known that intense EM fields, where the Maxwell theory is no longer valid, can resemble a curved spacetime, in the sense that light trajectories are not straight lines but undergo deflection. Deviations from the straight trajectories in vacuum are described in NLED by the null trajectories of an effective or optical metric. The concept of effective metric in nonlinear electrodynamics has its origin in the works by  G. Boillat <cit.>, Bialynicki-Birula <cit.>,   Plebański <cit.> and more recently by Novello <cit.>. In this work, we take the approach developed by Plebański and later on by Novello and collaborators. 

The equations of the propagation of the field discontinuities in nonlinear electrodynamics by a Lagrangian L(F, G)  can be derived by analyzing the propagation of linear waves associated with the discontinuity of the field, f_αβ,  in the limit of geometrical optics <cit.>.

 Considering a null vector k_μ=(ω,k⃗) that is normal to the characteristic surfaces or wavefronts, the effective  metric g_ eff^μν must be the metric in which the wave vector is null, k^μk_μ=0,


    g_ eff^(a) μνk_μk_ν=0,    a=1,2,

where the (a) superscript corresponds to the two metrics that can arise in NLED when birefringence occurs.
The propagation vector can be expressed as  



    k_(a)^2= - λ_aF^τμF^ν_τk_μk_ν,   a=1,2,

where  λ_a can be interpreted as the  birefringence or refractive index of a material medium and is  given in terms of the NLED Lagrangian and its derivatives by,


    λ_a=-4L_FF+Ω_(a)L_FG/L_F+G(L_FG+Ω_(a) L_GG),   a=1,2,

where  L_X=d L/dX, X=F, G, and Ω_(a) in Eq. (<ref>), depends on the derivatives of the Lagrangian with respect to the invariants <cit.>.  The expression for the effective metric can be identified from the dispersion relation, Eq. (<ref>) with Eq. (<ref>) is, 


    { g^μν + λ_at^μν} k_μk_ν= 
     g_ eff  (a)^μνk_μk_ν=0,   a=1,2,

where t^μν=F^μλF_λ^ν, is defined by the particular (intensive) fields F_μν of the system, and g^μν  is the background metric that can be the Minkowski metric or a general curved spacetime, for instance, a black hole or a cosmological spacetime. 
Then  the effective metrics for general nonlinear electrodynamics with Lagrangian L(F,G) are given by,

    g_ eff (a)^μν =  g^μν+ λ_a t^μν,    a=1,2.




If the Lagrangian is such that L_FG=0, the effective metrics become


    g_ eff (1)^μν   =   (L_F-2L_GG F )g^μν-4L_GGt^μν, 
    
    g_ eff (2)^μν   =     L_Fg^μν-4L_FFt^μν,


Additionally, when we turn off either the electric or the magnetic field, i.e., E⃗=0 or B⃗=0, then G=4E⃗·B⃗=0, and the effective metrics become


    g_ eff (1)^μν   =     L_F g^μν, 
    
    g_ eff (2)^μν   =     L_Fg^μν-4L_FFt^μν,

one of  the effective metrics, g_ eff (1), is conformal to the background metric; in this case, one of the optical paths is the null geodesic of the background metric and the other one is the null geodesic of the effective metric g_ eff (2). 

In this work, we analyze three different NLEDs: BI, EH and ModMax, and here we present their respective effective metrics, as listed below. 

Note that the BI effective metric does not present birefringence, then there is only one,

    g_ eff, BI^μν=(b^2+F/2) g^μν+F^μ_λF^λν.


On the other hand, the EH NLED has two effective metrics, given by Eqs. (<ref>) and (<ref>), explicitly, 


    g_ eff(1), EH^μν = - 1/4( 1 + μ F ( 5/2 F^2 - 7/4 G^2)/(F^2 + 7/4 G^2)^3/2) g^μν
     - 7/4μ F^2/(F^2 + 7/4 G^2)^3/2 F^μ_λF^λν,

and


    g_ eff(2), EH^μν= - 1/4( 1 - μ F /(F^2 + 7/4 G^2)^1/2) g^μν
     - 7/4μ G^2/(F^2 + 7/4 G^2)^3/2 F^μ_λF^λν,

where  we  denote μ = 2 α^2/(45 m_e^4).

Finally, ModMax NLED does present birefringence but one of the two effective metrics is conformal to the background metric; such that the effective metrics are given by


    g_ eff(1), MM^μν=g^μν+4tanhγ/√(F^2+G^2)+Ftanhγt^μν,



    g_ eff(2), MM^μν=g^μν.

Note that the electromagnetic tensor F_μν and its invariants that appear in the previous equations correspond to the intense electromagnetic background field that the light wave traverses with the phase velocity derived from the corresponding g_ eff^μν.
In the Maxwell case  L=-F/4,   L_F=-1/4,    L_FF= 0 and L_G= 0, then both effective metrics become conformal to the background metric, g_ eff^(1) μν = g_ eff^(2) μν= g^μν, and the null geodesics coincide with the light trajectories in the background spacetime.

See <cit.> for a study on the Fresnel equation in nonlinear electrodynamics and <cit.> for a classification of the effective metrics.




 §.§ Phase velocity of light propagation in the Effective Metric


Here we explain how the phase velocity of light can be calculated
from the dispersion relation.  Let us consider a  null vector k_μ={ω, k⃗}. If k_i is the light propagation  direction, from the dispersion relation,  Eq. (<ref>), 


    g_ eff(a)^ttω^2+2g_ eff(a)^itω k_i+g_ eff(a)^ij k_i k_j=0,

where i,j subscripts (or superscripts) denote the spatial coordinates and t denotes the time coordinate. Defining the normalized wave vector k̃_i=k_i/|k⃗|, Eq. (<ref>) can be written  as 


    g_ eff(a)^ttω^2/|k⃗|^2+2g_ eff(a)^itω/|k⃗|k̃_i+g_ eff(a)^ijk̃_ik̃_j =0.


Then the light's phase velocity,  v=ω/|k⃗|, for propagation in the direction k̃_i,  v^i is given by 

    (v^i)_a= ω/|k⃗|k̃_i=g_ eff(a)^itk̃_i/g_ eff(a)^tt±√((g_ eff(a)^itk̃_i/g_ eff(a)^tt)^2-g_ eff(a)^ijk̃_ik̃_j/g_ eff(a)^tt),      a=1,2.


The two possible effective metrics g_ eff(a)^μν, a=1,2, render two
dispersion relations that correspond to two modes of polarization, this is known as the birefringence effect <cit.>. Note that these velocities differ from the phase velocities of a massless test particle that follows the null geodesics of the background metric g_μν and  is given by analogous expressions making g_ eff(a)^μν↦ g^μν, 


    v_ massless = ±√(-g^ijk̃_ik̃_j/g^tt).




§ COMPARATIVE OF THREE NONLINEAR ELECTRODYNAMICS: LIGHT PROPAGATION



In this Section, our goal is to study light propagation in the three NLED theories and to evaluate how strong electromagnetic fields impact its phase velocity. We will compare the phase velocities of light propagating in a magnetic  background in a Minkowski spacetime, and then in an electric background. 



 §.§ NLED phase velocities in a magnetic background 

We examine the phase velocities of light propagating in the z direction, while the magnetic background field is oriented making an angle with the propagation direction; we consider the magnetic field with one component along the propagation direction and a second component as transversal or perpendicular,  B⃗=  B_z ẑ + B_xx̂ + B_yŷ =B_z ẑ + B_⊥ (x̂ + ŷ). We calculate the phase velocities for the case of a static observer or as measured in the Laboratory.
The phase velocities corresponding to the three NLEDs are derived from Eq. (<ref>),  considering a Minkowski background,
g^μν= η^μν =  diag [-1,1,1,1]; while the uniform magnetic field components are F^xy=-B_z,   F^xz=B_y,    F^yz=-B_x, and the invariants, F= 2B^2,    G=0.

For the EH NLED, the expressions for the phase velocities are

    v_EH1^2=1-14 μ  B_⊥^2/10 B^2 μ +1,    v_EH2^2=1-8 μ  B_⊥^2/1-4 B^2 μ.

Considering terms up to μ ( α^2), the EH phase velocities are given by,


    v_EH1,B^2=1- 14 μ B_⊥^2,     v_EH2,B^2=1- 8μ B_⊥^2.

And performing an average over polarization and direction,

    v_EH,B^2=1- 11 α/270 π( B_⊥/B_cr)^2,

where we substituted μ in terms of the critical field  μ= α/(90 π B_cr^2); that gives a light velocity decreasing respect the one in vacuum c=1 of

    v_EH,B^2=1- (9.5 × 10^-5) ( B_⊥/B_cr)^2.



For  the BI NLED, there is no birefringence, and the phase velocity is 

    v_BI^2=1-B_⊥^2/b^2+B^2,

with b being the maximum attainable electric or magnetic field.
Up to first order in 1/b^2, neglecting terms 1/b^4 and higher, 


    v_BI,B^2=1-B_⊥^2/b^2.

Such that light velocity is reduced the most, for fields with a component perpendicular to the light propagation (B_⊥) of intensities of the order of b= 10^11 Tesla.

For the ModMax NLED, there is birefringence and one of the light trajectories is 
the one in vaccum, with phase velocity v^2=c^2=1. While the other one is

    v_MM^2=1-2 B_⊥^2/B^2( tanh(γ)/1 + tanh(γ )).


The phase velocities as  functions of the corresponding adimensional parameters: μ B^2 for EH; B^2/b^2 for BI are illustrated in Fig. <ref>. For the ModMax NLED the phase velocity depends on B_⊥/B; in Fig. <ref>  is shown how the phase velocity decreases in ModMax case for B_⊥/B =0.1 .    The differences in the decreasing of the phase velocities  is qualitatively the same for BI and EH, while  the Mod Max case is illustrated as a function of B_⊥/B in Fig. <ref>. 





 §.§ NLED phase velocities in an electric background 

The phase velocities when considering an electric background field in a Minkowski spacetime are presented. The setting is analogous to the one for the magnetic background: light propagating in the ẑ direction and the electric background of the form E⃗= E_z ẑ + E_⊥ (x̂ + ŷ). 

The two possible phase velocities for the EH NLED are given by 

    v_EH1,E^2=1-14 E_⊥^2 μ/4 E^2 μ +1,    v_EH2,E^2=1-8 E_⊥^2 μ/12 E^2 μ +1.

For the BI NLED, since there is no birefringence, the phase velocity reads

    v_BI,E^2=1-E_⊥^2/b^2.

While  phase velocity in a ModMax NLED is 

    v_MM,E^2=1-2 E_⊥^2/E^2( tanh(γ)/1 + tanh(γ )).

The behavior at the order μ E^2, E^2/b^2∼ 10^-2 is very similar the one for the magnetic background field. Moreover, the phase velocities for the wave in the presence of ModMax magnetic and electric fields in Eqs. (<ref>) and (<ref>) have the same dependence on the magnitude of the fields, a consequence of the duality invariance required by ModMax theory.  




 §.§ Bounds on the ModMax parameter γ from BI and EH


The phase velocity of light propagation in a strong magnetic field background obtained from the ModMax theory is given by,

    v_MM^2=1-2 B_⊥^2/B^2 ( tanh (γ )/1 + tanh (γ )).

As a consequence that there are no bounds for the allowed values of γ, and that for large values of γ, tanh(γ) ↦ 1, in the expression for the phase velocity, the factor with γ tends to 1/2 and then v_MM can be made arbitrarily small, even for small values of the magnetic field if the component that is perpendicular to the wave propagation is of the order of the total magnetic field, B_⊥≈ B. 
Since EH describes the nonlinear electromagnetic effect of vacuum polarization and BI the effect of photon splitting, the phase velocity of light propagation derived from these effects compared with the one from the ModMax theory  can give  insight of the ranges of γ. For this reason we search for bounds on γ from the phase velocities derived for BI and EH.



  §.§.§ Bounds on the NLED ModMax parameter  from BI


Comparing the phase velocity for light propagation that arises in the BI theory  in a magnetic background, Eq. (<ref>) with the phase velocity  from the ModMax theory, Eq. (<ref>),  it is obtained for the parameter γ


    tanhγ = B^2/2 b^2- B^2≈B^2/2 b^2,

if we consider a magnetic field background of the order of the maximum attainable magnetic field b, i.e. B/b ≈ 1, we obtain the bound 
γ= 0.549306. While if  B is a tenth of b, B/b ≈ 0.1 then the bound on the ModMax parameter is γ= 0.005. Smaller magnetic fields give smaller bounds for γ.
Then BI constrains γ to the interval 0 < γ≤ 0.549306.





  §.§.§ Bounds on the NLED ModMax parameter  from EH


The two phase velocities for light propagation in a strong magnetic background that arise in the EH theory, Eqs. (<ref>), are compared with the MM phase velocity.  From the first one, v_EH1,B  compared with the phase velocity for the ModMax theory, Eq. (<ref>), we obtain for the parameter γ the constraint


    tanhγ = 7 α/90 π(B/B_cr)^2 = 1.8 × 10^-4(B/B_cr)^2,

Such that for fields near the critical one, B ≈ B_cr
the nonlinear MM parameter is γ = 1.8 × 10^-4.

Following the same procedure for the second velocity  v_EH2,B in Eqs.  (<ref>) we obtain a second bound of γ = 10^-4,
for fields B ≈ B_cr = 10^9 Tesla.
Such that the EH theory sets 
γ in the interval 0.0001 < γ < 0.00018.




Therefore the  EH theory delimits  the nonlinear ModMax parameter to the interval 0.0001 < γ < 0.00018, while BI places  γ in a wider interval, 0 ≤γ≤ 0.549306. We should take into account that EH and BI apply to different nonlinear electromagnetic effects that arise at different orders of magnitude of the background fields
( B_cr≈ 10^9 Tesla for EH and b ≈ 10^11 Tesla for BI), 
but both serve to the purpose of delimiting the value of γ, i.e. the effect of the ModMax electrodynamics in light propagation 
should resemble in a certain way the ones of the BI or EH electrodynamics. The BI constrain 0 < γ < 0.549306  includes the constraints derived from EH,  therefore the greater field intensity allows a larger γ in the ModMax theory.




§ COMPARATIVE OF THREE NONLINEAR ELECTRODYNAMICS: BLACK HOLE SHADOWS



It is well known that the coupling of Nonlinear Electrodynamics (NLED) with Einstein's gravity can yield black hole solutions. In this section, we focus on the static black holes corresponding to the three NLEDs discussed above and we compare the metric functions for these BHs with the Reissner-Nordstrom one, that is the linear limit; in the comparison is also included the Schwarzschild BH as a reference. Additionally, we compare the resulting shadow radii  with the observed shadows of Sagittarius A^* and M87^* black holes <cit.>. 



 §.§  NLED BH Metric functions

We consider the corresponding BH metrics as the background metrics. It is interesting to compare the metric functions for the BHs of the three NLEDs in consideration with the one of  Reissner-Nordstrom, the static spherically symmetric solution of the coupled Einstein-Maxwell equations. For the static spherically symmetric line element, that is the background metric g_μν,

    ds^2= - f(r) dt^2 + dr^2/f(r) + r^2 d Ω,

the corresponding metric functions f(r) for Schwarzschild (Schw), Reissner-Nordstrom (RN), Euler-Heisenberg (EH),  Born-Infeld (BI), and ModMax (MM), are, respectively, 

    f_Schw.(r)    =    1-2M/r,
    
    f_RN(r)    =    1-2 M/r+Q^2/r^2, 
    
    f_EH(r)    =    1-2 M/r+Q^2/r^2 - μ  Q^4/20 r^6,


    f_BI(r) = 1-2 M/r+2/3 b^2 r^2 (1-√(Q^2/b^2 r^4+1))+2 Q^2 /3r√(b/Q) F[ ArcCos(b r^2/Q-1/b r^2/Q+1),1/√(2)],


    f_MM(r) = 1-2 M/r+e^- γ Q^2/r^2.
 

In Fig. <ref>, we plot the metric functions for the Schwarzschild, RN, EH, BI and ModMax black holes, for a fixed  BH charge Q=0.9. In the BI case  for b>0.7, f_BI(r) behaves similarly near the origin than the one for RN, while if b<0.7, the metric function f_BI(r) resembles the  Schwarzschild behavior near r=0 <cit.>.





 §.§ The shadow of the NLE black holes


Considering the equations of motion of a test particle in the external field of a static BH, the radial motion is governed by

    ṙ^2 
       + L^2/g_rr( g_ϕϕ/r^4
       +g_tt/f^2 (r)1/b^2 
       - δ/L^2)
       = 0,

where δ= 1,0,-1 for space-like, null, or time-like geodesics, respectively; the test particle angular momentum is L,  and  the impact parameter b= L/E, where  E is the particle energy.
In NLED photons travel through null geodesics of the effective metric, then  in Eq. (<ref>) δ=0 and g ↦ g^ eff(a), such that the radial motion of the light ray obeys,

    ṙ^2 
       + L^2/g^ eff(a)_rr( g^ eff(a)_ϕϕ/r^4
       +g^ eff(a)_tt/f^2 (r)1/b^2)
       = 0,    a=1,2,

where a=1,2 denote the two effective metrics, given by  Eq. (<ref>) for BI, Eqs. (<ref>)- (<ref>) for EH, and Eqs.   (<ref>)- (<ref>) for ModMax. The electromagnetic tensor components  F_μν for the construction of the effective metric  correspond to the ones of  a charged static BH, that in case of both charges, electric Q_e and magnetic Q_m  are given by 


    F^tr = Q_e/r^2 L_F,    F_θϕ = Q_m sinθ,

we shall consider only electric charged BHs, Q_e=Q.  

From 
ṙ^2 + V^(a)_ eff =0, we identify the effective potential as 

    V^(a)_ eff=L^2/g^ eff(a)_rr( g^ eff(a)_ϕϕ/r^4
       +g^ eff(a)_tt/f^2 (r)1/b_a^2),    a=1,2.

The radius of the  circular orbits r_c 
corresponds to an extreme of the effective potential,

    V^(a)_ eff |_r_c =0,   dV^(a)_ eff/dr |_r_c  =0,   a=1,2.



The critical impact parameters can be obtained in terms of the effective metrics as 

    (b^2_c)_a= ( -r_c^4 g_tt^ eff (a)/f^2 (r_c)g_ϕϕ^ eff (a))_r_c,   a=1,2.

The NLED theories that present birefringence have two values for the critical impact parameter, corresponding to the two effective metrics;  for the NLED-BHs of our study, EH, BI and ModMax, respectively, are


    b_c,EH1^2=r_c^2/f_EH(r_c)( 1-10μQ^2/r_c^4/1+4μQ^2/r_c^4),    b_c,EH2^2=r_c^2/f_EH(r_c)( 1+4μQ^2/r_c^4/1+12μQ^2/r_c^4),



    b_c,BI^2=r_c^2/f_BI(r_c)( 1-Q^2/b^2 r_c^4),



    b_c,MM1^2= e^-2γr_c^2/f_MM(r_c),    b_c,MM2^2= r_c^2/f_MM(r_c).


We calculate the radii of the shadows for the three NLED-BHs and compare them with the RN BH. The shadow can be determined by considering a light ray from the observer position r_o to the unstable circular orbit (UCO) making an angle ψ_sh with respect to the radial direction. The expression for the deviation angle associated with the shadow is <cit.>

    sin^2(ψ_sh)_a=-b^2_ca(g_ϕϕ^ eff (a)/g_tt^ eff (a)f^2 (r_o)/r_o^4)_r_o,    a=1,2.

where  b_c is the impact parameter of the UCO with radius r_c.

The NLED theories that present birefringence have two values for the shadow radius  r_sh, corresponding to the two effective metrics. The radius of the BH shadow is approximated in terms of the observer position r_o and the angle ψ_sh as <cit.>

    r_sh=r_otanψ_sh≈ r_osinψ_sh,

then, we obtain that for the EH BH there are two radii, then two shadows, 

    r_sh,EH1^2=r_c^2 (f_EH(r_o)/f_EH(r_c)) ( r_c^4 -10 Q^2 μ/r_c^4 +4Q^2 μ) ( r_o^4 +4 Q^2 μ/r_o^4 -10Q^2 μ),



    r_sh,EH2^2=r_c^2 (f_EH(r_o)/f_EH(r_c)) (r_c^4 +4 Q^2 μ/r_c^4 +12Q^2 μ) ( r_o^4 +12 Q^2 μ/r_o^4 +4 Q^2 μ).

The static BI BH has one shadow radius, 

    r_sh,BI^2 = r_c^2 ( f_BI(r_o)/f_BI(r_c)) ( 1 - Q^2/b^2 r_c^4)/(1-  Q^2/ b^2 r_o^4).

And although there are two critical impact parameters for the ModMax BH, an observer located at infinity observes only one shadow,

    r_sh,MM^2= r_c^2 ( f_MM (r_o)/f_MM (r_c)).

Considering that the observer is at infinity r_o↦∞, then f(r_o ↦∞)=1 and at first order in the nonlinear parameters, the radii of the shadows are, for EH BH up to first order in μ,


    r_sh,EH1^2= (r_c^2/f_EH(r_c)) (1 - 14 Q^2 μ/r_c^4),    r_sh,EH2^2= (r_c^2/f_EH(r_c)) (1 - 8Q^2 μ/r_c^4).

For the BI BH up to first order in 1/b^2,

    r_sh,BI^2= (r_c^2/f_BI(r_c)) (1 - Q^2 /b^2 r_c^4).

While for the ModMax BH the shadow radius is

    r_sh,MM^2= (r_c^2/f_MM(r_c)).

It is worth to note that r_c is different for each BH; r_c is the radius of the photosphere,  precisely the radius at which is maximum  the effective potential for the null geodesics of the effective metric;
i.e. r_c is calculated from the conditions V_ eff^' =0 and  V_ eff^''  < 0. For RN the radius of the UCO is given by


    r_c,RN= 3M/2( 1 + √(1- 8 Q^2/9M^2)),

Moreover, the impact parameter and shadow radius at infinity of the RN BH can be derived from any of the above expression in the limit of the vanishing nonlinear parameter, and are given by,

    b_c,RN^2=r_c^2/f_RN(r_c) =  r_sh,RN^2= 3^3 M^2  ( 1+ √(1- 8Q^2/9M^2))^4/8 ( 1 - 2Q^2/3M^2 +  √(1- 8Q^2/9M^2)),

while for Schwarzschild it is well known that the radius of the photosphere r_c = 3 M, while the shadow radius is r_sh,Schw= 5.196 M. 




 §.§ Confronting the NLED BHs shadow with observations


In this section we test the NLED BH shadows with the observations of the shadow reported by the EHT for the supermassive BHs in the center of the Milky Way (Sgr A^*) and the M87^* galaxy. This gives insight about is a supermassive BH can bear electric or magnetic fields of high intensities.


Fig. <ref> shows the shadow radii for the BHs in consideration as a function of the charge Q, contrasted with the σ intervals given by the EHT for Sgr A^* <cit.>. As the charge increases the radius of the shadow decreases. 

The radius of the shadow for BI and the second solution of EH behave similarly for values of the nonlinear parameters b=0.7, μ=0.5; for Q/M ≤ 0.2 both BHs are in excellent agreement with the shadow of Sgr A^* and up to Q/M ≤ 0.7 their shadows remain in the 1 σ band of Sgr A^*. The ModMax BH is the one that allows  the  largest charge/mass ratio Q/M keeping its shadow radius in the 1σ region for γ = 0.5.




In Fig. <ref> the shadow radii for the NLED BHs  is contrasted with the observations of  M87^* <cit.>. Note that the 1σ constraint reaches values above the radius for the Schwarzschild BH, nonetheless the BHs in consideration do not exceed the Schwarzschild radius. 
The confidence intervals of M87^*  are less restrictive than the ones for Sgr A^*; that allows larger values of Q/M BH in the 1 σ band.
For Q/M ≤ 0.35 EH and BI BHs are in excellent agreement with the shadow of M87^*and up to Q/M ≤ 0.8 their shadows remain in the 1 σ band. While the ModMax BH remains in the 1 σ band for  Q/M ≤ 1 for γ = 0.5.




In Fig. <ref> we compare the 1σ bands for the SgrA^* and M87^* being the light gray the 1σ band  for M87^* and the dark gray the one for SgrA^*. As noted before, the confidence intervals for M87^* include the ones for SgrA^*.





Since tighter confidence intervals come from SgrA^*, 
in Fig. <ref> we confront  the shadow radii of EH1, BI and MM BHs for different values of their respective nonlinear parameters μ, b and γ. For the EH BH, we only plot the first solution as the behaviors are similar; as the value of μ increases, the allowed values of the charge that fall in the 1σ and 2σ regions, are more restricted, however μ=0.3 admits charge-mass ratios of 0.8 ≤ Q/M in the 1 σ interval and 0.9 ≤ Q/M in the 2 σ interval. 

For the BI BH increasing b the radius of the shadow tends to the one of the RN BH, as expected, since in the limit b→∞ Maxwell electrodynamics is recovered. It is important to highlight that Figure <ref> shows that BI and EH nonlinear theories do not allow higher values of the charge  in agreement with the observations; then, if one looks for the existence of a charged black hole, neither BI or EH NLEDs are the theories that would help to achieve that, at least in the range of supermassive black holes; however, up to a relation charge mass Q/M ≈ 0.3 charged BHs are not ruled out for any of the examined NLEDs.

In contrast, for the MM BH, if γ→ 0, we recover the radius of the shadow of the RN BH, and  as γ increases higher values of the BH charge are admitted; for γ =0.5 the whole range of Q/M ≤ 1 is in the 1 σ band.
Moreover, the MM BH fits better the shadow observations than RN.





 
  
   
    
     
 






§ CONCLUSIONS



We considered three nonlinear electrodynamics (NLED) theories: Born-Infeld (BI), Euler-Heisenberg (EH), and modified Maxwell (ModMax, MM) to study light propagation in a strong magnetic (electric)  background; the effect of these intense fields in light propagation is the slowing down of the phase velocity. Another effect that arises due to the high intensities of the fields is birefringence in vacuum. For the NLED theories in consideration, only the BI theory does not present birefringence; and in the case of the ModMax theory, as a consequence of the conformal symmetry, one of the light trajectories coincides with the null geodesics of the Minkowski space.  The NLED theory that diminishes the phase velocity the most is the EH theory and the effect is enhanced in proportion to the magnetic (electric) component that is orthogonal to the propagation direction. 
 Our treatment is valid for very strong fields; if we consider, for instance, that B^2/b^2 ≈ 10^-2, then the BI field is of the order of 10^11 Tesla that is one hundred times the critical Schwinger field or  B_ cr≈ 10^9 Tesla. It is important to note that for fields of the orders B^2/b^2≈10^-2 and μ B^2≈ 10^-2 the diminishing in the velocities for BI and EH is qualitatively similar, as illustrated in  Figure <ref>. The plots that serve the comparison are in terms of dimensionless parameters μ B^2 for EH and B^2/b^2 for BI. Still, we should keep in mind that the critical fields of both nonlinear theories are different by two orders of magnitude, being the BI theory the one with the larger critical field. 
 
 Additionally, the comparison between the phase velocities is useful to set bounds on the ModMax nonlinear parameter γ since so far it is not constrained apart from the restriction of being positive that is derived from the energy conditions in the ModMax BH. BI constrains γ to the interval  0 < γ < 0.549306; the larger bound for BI fields B ≈ b with b = 10^11Tesla  (b = 10^20 Volt/m)  being the maximum attainable electromagnetic BI field. Then
we derive that the EH theory is more stringent constraining  γ to the interval 0.0001 <γ < 0.00018 for fields of the order of B_cr∼ 10^9 Tesla  ( E_cr=10^18 Volt/m ).  

In the second part of the paper we contemplate light propagation in the vicinity of NLED BHs as the background metric and calculate the corresponding shadows 
to compare the NLED effects with the ones for the linear solution of the Einstein-Maxwell equations, the Reissner-Nordstrom (RN) metric and, as a reference, we include in the comparison the Schwarzschild BH. Then these shadows are
confronted  with the observational shadows from Sagittarius A^∗ and M87^*.  The situation is different for each electrodynamics.  Due to birefringence, EH presents two shadows and BI only one. For ModMax one of the effective metrics is the background metric as a consequence of the conformal invariance.

We can determine ranges for the nonlinear parameters μ for EH, b for BI, and γ for ModMax, for their shadows to be in agreement with the evidence of Sagittarius A^* and M87^*.
 If one is looking for a charged BH, the most effective theory is ModMax, which coupled with Einstein equations results in a metric function similar to the one for RN but with a screened charge; the ModMax BH allows  values of the BH charge-mass Q/M ≤ 1 still in the 1 σ confidence interval with the observations of Sgr. A^* and M87^*. Meanwhile, by increasing the BI and EH nonlinear parameters, we are not able to reach higher values of the BH charge that keep the shadow in the 1σ interval of Sgr.A^* that sets the most stringent astrophysical limits for the NLED parameters. 

 In summary, up to a relation charge mass Q/M ≤ 0.3  astrophysical charged BH are not ruled out for any of the examined NLEDs; their shadows remain in the 1σ interval of Sgr.A^*.  However neither BI-BH or EH-BH point to a possible astrophysical supermassive charged BH with Q/M > 0.3. In contrast, the Mod Max BH shadow remains in the 1 σ interval for BH charge-mass  0  ≤ Q/M ≤ 1, fitting data even better than the Einstein-Maxwell solution RN.




EGH acknowledges financial support by Conahcyt-Mexico through the Ph. D.  scholarship No. 761878.
The work of AM has been sponsored by Conahcyt-Mexico through the Posdoc Project I1200/311/2023. 
NB acknowledges partial support by Conahcyt-Mexico 
project CBF2023-2024-811. 






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