# Holographic phenomenology via overlapping degrees of freedom

Oliver Friedrich1,2, ChunJun Cao3,4,5, Sean Carroll6,7, Gong Cheng5,8, Ashmeet
Singh9 1 University Observatory, Faculty of Physics, Ludwig-Maximilians-
Universität, Scheinerstr. 1, 81677 Munich, GER\EU  2 Excellence Cluster
ORIGINS, Boltzmannstr. 2, 85748 Garching, GER\EU  3 Joint center for Quantum
information and Computer science, University of Maryland, College Park, MD
20740, USA  4 Institute for Quantum Information and Matter, California
Institute of Technology, Pasadena, CA 91125, USA  5 Department of Physics,
Virginia Tech, Blacksburg, VA 24061, USA  6 Departments of Physics & Astronomy
and Philosophy, Johns Hopkins University, Baltimore, MD 21218, USA  7 Santa Fe
Institute, Santa Fe, NM 87501, USA  8 Maryland Center for Fundamental Physics,
University of Maryland, College Park, MD 20740, USA  9 Department of Physics,
Whitman College, 345 Boyer Ave, Walla Walla, WA 99362, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The holographic principle suggests that regions of space contain fewer
physical degrees of freedom than would be implied by conventional quantum
field theory. Meanwhile, in Hilbert spaces of large dimension $2^{n}$, it is
possible to define $N\gg n$ Pauli algebras that are nearly anti-commuting (but
not quite) and which can be thought of as “overlapping degrees of freedom”. We
propose to model the phenomenology of holographic theories by allowing field-
theory modes to be overlapping, and derive potential observational
consequences. In particular, we build a Fermionic quantum field whose
effective degrees of freedom approximately obey area scaling and satisfy a
cosmic Bekenstein bound, and compare predictions of that model to cosmic
neutrino observations. Our implementation of holography implies a finite
lifetime of plane waves, which depends on the overall UV cutoff of the theory.
To allow for neutrino flux from blazar TXS 0506+056 to be observable, our
model needs to have a cutoff $k_{\mathrm{UV}}\lesssim
500\,k_{\mathrm{LHC}}\,$. This is broadly consistent with current bounds on
the energy spectrum of cosmic neutrinos from IceCube, but high energy
neutrinos are a potential challenge for our model of holography. We motivate
our construction via quantum mereology, i.e. using the idea that EFT degrees
of freedom should emerge from an abstract theory of quantum gravity by finding
quasi-classical Hilbert space decompositions. We also discuss how to extend
the framework to Bosons. Finally, using results from random matrix theory we
derive an analytical understanding of the energy spectrum of our theory. The
numerical tools used in this work are publicly available within the
`GPUniverse` package, https://github.com/OliverFHD/GPUniverse .

###### Contents

  1. 1 Introduction & summary of results
    1. 1.1 Summary of results
  2. 2 The Weyl field as a collection of qubits
    1. 2.1 Weyl field basics
    2. 2.2 Decomposition into qubits
  3. 3 Fermion field with overlapping qubits
    1. 3.1 Chao-Reichert-Sutherland-Vidick embedding
    2. 3.2 Fiducial parameters & satisfying the Bekenstein bound
    3. 3.3 Degrees of freedom of other quantum fields
    4. 3.4 Breaking of Lorentz symmetry
  4. 4 Dynamics of the holographic Weyl field
    1. 4.1 Energy spectrum and re-formulation as non-local Heisenberg model
    2. 4.2 Lifetime of plane waves
    3. 4.3 Consistency with the cosmic ray spectrum
  5. 5 Real-space anti-commutator
  6. 6 Alternative modelling choices
    1. 6.1 non-isotropic overlaps
    2. 6.2 alternative Hamiltonian
  7. 7 Why these random overlaps? Context within quantum mereology
  8. 8 Discussion
  9. A Simulated realisations of the CRSV-embedding
  10. B Random vectors on the unit sphere and the Johnson-Lindenstrauss theorem
    1. B.1 Moments on the unit sphere
    2. B.2 Moments of orthogonal vectors
    3. B.3 The Johnson-Lindenstrauss theorem
    4. B.4 Angles between complex vectors
    5. B.5 Moments of the overlaps $\braket{z_{\bm{q}}}{z_{\bm{p}}}$
  11. C Our construction as a Heisenberg model
    1. C.1 Energy spectrum of the holographic field
    2. C.2 Lifetime of plane waves
    3. C.3 Eigenvalue structure of Hermitian and anti-persymmetric matrices
  12. D Jordan-Wigner strings to construct anti-commuting creation and annihilation operators

##  1 Introduction & summary of results

The holographic principle has emerged as one of the most prominent conjectured
properties of an eventual theory of quantum gravity [1, 2, 3, 4]. Its most
precise formulation has been given by [5] (generalising results of [6]), and
it states that the entropy $S$ that can be accumulated on a light-sheet111In a
$3+1$ dimensional spacetime a light-sheet emanating from a space-like surface
$\mathcal{B}$ has been defined by [5] as a 3 dimensional null hypersurface
generated by a family of light-rays that are orthogonal to $\mathcal{B}$ and
whose expansion $\theta$ is $\geq 0$ everywhere (i.e. the light-sheet stops
where the expansion $\theta$ becomes negative). emanating from a space-like
surface $\mathcal{B}$ is bounded by a quarter of the area $|\mathcal{B}|$ of
that surface (measured in units of the Planck area). In the weak gravity
regime and for spherically symmetric, space-like volumes $\mathcal{R}$, a
simpler version of the holographic principle states that the maximum entropy
that can be localised within $\mathcal{R}$ equals the boundary area
$|\delta\mathcal{R}|$ of that region divided by four times the Planck area,

$S(\mathcal{R})\leq\frac{|\delta\mathcal{R}|}{4\ell_{\mathrm{P}}^{2}}\ .$ (1)

Such a bound is incompatible with standard quantum field theory, because the
local degrees of freedom of the latter scale extensively with volume and not
with area222In particular, in a cosmological context $S(\mathcal{R})$ would be
the maximum entropy that can be localised on a light-sheet inside the cosmic
horizon, which is strictly different from the cross-horizon entanglement
entropy (which in fact does scale like area, cf. [7]; the fact that those two
different quantities are of the same order of magnitude remains to be
explained, but is not addressed here). [8, 9, 3, 10, 4]. The above bound also
seems to imply that the Hilbert space of quantum gravity is locally finite
[11], which calls the validity of bosonic quantum field theory into question
even further [12, 13]. Even within an AdS-framework, the emergence of sub-AdS
locality remains an open problem where a large separation of the string scale
and the AdS scale is needed [14, 15].

An area scaling of entropy requires the degrees of freedom of quantum gravity
to be de-localised. But how is this reconciled with the fact that local
degrees of freedom seem to be a ubiquitous part of nature? So far, the most
concrete example of a mapping between local, bulk degrees of freedom and
holographic, boundary degrees of freedom is given by the AdS/CFT
correspondence [16, 17] which connects a theory of quantum gravity on $d+1$
dimensional anti-de Sitter space to the degrees of freedom of a conformal
field theory living on the boundary of that space. Within that correspondence,
information present in the bulk theory can indeed be recovered from the
boundary theory [18]. But generalising these results to more realistic
spacetimes has so far proven to be challenging. Anti-de Sitter space seems to
make the implementation of holography uncharacteristically easy. Put into
simplistic terms, this is because in hyperbolic space (as are the constant
time sheets of anti-de Sitter space) the boundary area and the volume of a
spherical region grow at the same rate in the large-scale limit. And that
significantly reduces the amount of de-localisation required to achieve
holography. One can see this clearly in discretised versions of anti-de Sitter
space in the form of tensor networks [19]: it is the hyperbolic nature of the
tessellation in such a network, that makes its boundary legs numerous enough
to encode all information present in the bulk factors of the network.

The search for generalisations of AdS/CFT to more realistic spacetimes remains
one of the most promising routes to finding a holographic description of our
own Universe. But it is worthwhile considering complementary test beds to
study the consequences of holography. A constructive way to find such test
beds is to modify quantum field theory in such a way that it obeys Equation 1
and its ramifications (such as local finite dimensionality). This has e.g.
been explored by [10, 20], who simply removed the modes of a scalar field
which upon excitation would form black holes, and more recently by [12, 13],
who replaced the canonically conjugate field variables of a scalar field by
finite-dimensional generalised Pauli operators [21]. In this paper we present
a new addition to this program which realises holography in quantum fields
with the help of overlapping degrees of freedom. (Our approach to holographic
phenomenology is thus distinct from works such as [22, 23], which use
holography to infer features of a four-dimensional QFT.)

In quantum computing, it is customary to assume that $N$ independent qubits
live in $N$ separate tensor factors of a $2^{N}$ dimensional Hilbert space.
However, such independence is only approximate in practice. This is
particularly pronounced, for instance, in various experimental setups
involving long range interactions such as Rydberg systems where actions
performed on one physical qubit influences other nearby qubits. To this end,
Chao et. al. [24] introduced the concept of overlapping qubits, such that
Pauli operators $P_{i},P_{j}$ supported on different qubits $i,j$ satisfy

$||[P_{i},P_{j}]|\psi\rangle||<\epsilon.$ (2)

Although these overlapping qubits are good approximations of independent
qubits from the lens of commutation relations, they showed that the Hilbert
space dimension associated with such systems can be as small as $poly(N)$.
This surprising result follows from a measure concentration argument [25]
which states that one can identify exponentially many near-orthogonal vectors
in a vector space (the Johnson-Lindenstrauss theorem). Related ideas have been
applied to areas of computer science [26, 27], compressed sensing[28, 29], and
also lately in the context of quantum gravity (e.g. [30, 31, 32, 33, 34, 35,
36]) to decrease the number of degree of freedom of the system.

Here we want to apply the same idea as a simple model of potential holographic
effects on quantum field theories and study its physical consequences. For
instance, one can relax the causal locality condition $[O,O^{\prime}]=0$ for
some spacelike separated operators $O,O^{\prime}$ such that
$[O,O^{\prime}]\approx 0$. Although this may appear strange at first sight,
[36] suggests that similar behaviours are expected for gauge invariant
operators coupled to long range interactions such as the electromagnetic or
gravitational field [37, 38], where the operator dressing renders an operator
non-local. A similar relaxation for fermions is also possible, e.g. [39],
where one considers the anti-commutator instead of the commutator. More
generally, violation of microcausality is also expected at the non-
perturbative level for quantum gravity [40].

###  1.1 Summary of results

For the impatient reader we are summarising our main results already here,
with details and derivations given in subsequent sections as well as our
appendices. We consider a massless, left-handed Weyl fermion which, when
constrained to a box of finite side length $L$, can be decomposed into Fourier
modes as

$\hat{\psi}(\bm{x},t)=\sum_{\bm{p}}\frac{1}{(|\bm{p}|L^{3})^{\frac{1}{2}}}\left\\{\hat{c}_{\bm{p}}(t)\
u(\bm{p})\ e^{i\bm{p}\bm{x}}+\hat{d}_{\bm{p}}(t)^{\dagger}\ u(\bm{p})\
e^{-i\bm{p}\bm{x}}\right\\}\ ,$ (3)

where the sum is over all momenta $\bm{p}\in\\{\
2\pi/L\cdot(n_{1},n_{2},n_{3})\ |\ n_{i}\in\mathbb{Z}\ \\}$, and where we
define the mode functions $u(\bm{p})$ in more detail in Section 2. The
operator $\hat{d}_{\bm{p}}^{\dagger}$ can be thought of as creating an anti-
spinor of momentum $\bm{p}$ while $\hat{c}_{\bm{p}}$ is destroying a spinor
with momentum $\bm{p}$. In standard QFT these operators would satisfy the
anti-commutation relations

$\displaystyle
0=\\{\hat{c}_{\bm{p}},\hat{c}_{\bm{q}}\\}=\\{\hat{d}_{\bm{p}},\hat{d}_{\bm{q}}\\}=\\{\hat{c}_{\bm{p}},\hat{d}_{\bm{q}}\\}=\\{\hat{c}_{\bm{p}},\hat{d}_{\bm{q}}^{\dagger}\\}$
(4)
$\displaystyle\\{\hat{c}_{\bm{p}},\hat{c}_{\bm{q}}^{\dagger}\\}=\\{\hat{d}_{\bm{p}},\hat{d}_{\bm{q}}^{\dagger}\\}=\delta_{\bm{p},\bm{q}}\
.$ (5)

These relations can only be satisfied if the individual momenta $\bm{p}$
represent independent Hilbert space factors $\mathcal{H}_{\bm{p}}$ (which
further split into particles and anti-particles as
$\mathcal{H}_{\bm{p}}^{c}\otimes\mathcal{H}_{\bm{p}}^{d}$). In particular, the
overall Hilbert space in which the field $\hat{\psi}$ is defined needs to be
given by

$\displaystyle\mathcal{H}=\bigotimes_{\bm{p}}^{\mathrm{JW}}\mathcal{H}_{\bm{p}}\
,$ (6)

where the JW above the tensor product symbol indicates that operators of the
individual Hilbert spaces have to be embedded into the product space via
Jordan-Wigner strings in order to achieve anti-commutativity between different
factors (as opposed to commutativity; cf. our D). We will explicitly break
this product structure, and describe the field $\hat{\psi}$ using a Hilbert
space that is strictly smaller than indicated by Equation 6.

We follow the lines of [24], who in the context of quantum computing have
studied imperfect qubit registers that suffer from qubit overlap (cf. the
sketch in our Figure 1 for an illustration). They considered $N$ qubits, but
assumed them to be embedded in a Hilbert space of dimension $2^{n}<2^{N}$. The
Pauli algebra and the raising and lowering operators of an individual qubit
can be represented in a Hilbert space of dimension $2^{n}$ as follows (though
see our Section 3 for more details):

  * 1.

Choose generators $\bm{C}_{1}\,,\,\dots\,,\,\bm{C}_{2n}$ for the Clifford
algebra in the Hilbert space of dimension $2^{n}<2^{N}\,$.

  * 2.

Choose a pair $\bm{v},\bm{w}$ of orthonormal vectors in $\mathbb{R}^{2n}\,$.

  * 3.

Define

$\displaystyle\bm{\sigma}_{x}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{v}\bm{C}_{j}\
,\ \bm{\sigma}_{y}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{w}\bm{C}_{j}\ ,\
\bm{\sigma}_{z}=-i\bm{\sigma}_{x}\bm{\sigma}_{y}\ ,$

where $\braket{e_{j}}{v}$ and $\braket{e_{j}}{w}$ are the $j$th components of
$\bm{v},\bm{w}$ in some orthonormal basis $\\{\bm{e}_{j}\\}$ of
$\mathbb{R}^{2n}\,$.

  * 4.

Finally, define raising and lowering operators as

$\displaystyle\bm{c}=\frac{1}{2}\left(\bm{\sigma}_{x}+i\bm{\sigma}_{y}\right)\
,\ \bm{c}^{\dagger}=\frac{1}{2}\left(\bm{\sigma}_{x}-i\bm{\sigma}_{y}\right)\
.$

To embed $N$ qubits into the Hilbert space of dimension $2^{n}$, we simply
choose $N$ orthonormal pairs
$(\bm{v}_{1},\bm{w}_{1})\,,\,\dots\,,\,(\bm{v}_{N},\bm{w}_{N})$ at random. We
then construct the operators defining the $j$th qubit with the help of the
pair $\bm{v}_{j},\bm{w}_{j}$ following the steps outlined above. This
procedure raises (at least) two questions:

  * A)

Since $n<N$, the operators we obtain for different qubits cannot satisfy the
anti-commutation relations from Equations 4. How large are the deviations from
those relations?

  * B)

The vector pairs $\bm{v}_{j},\bm{w}_{j}$ are chosen at random. Where does this
randomness come from? Which part of an underlying, more fundamental theory is
represented by this randomness?

We discuss question B) in detail in Section 7 . To answer question A), let us
look at the number $n$ and $N$ that would be required to describe the Weyl
field.

Figure 1: The upper sketch illustrates a perfect register of eight non-
overlapping qubits. The Hilbert space of the entire register is the tensor
product of the individual qubit Hilbert spaces. The lower sketch illustrates
the situation when the total Hilbert space is strictly smaller than such a
tensor product. In this case the qubits must be overlapping, by which we mean
that the operator algebras defining the different qubits are not
(anti-)commuting.

Figure 2: The ($c$-particle part of the) Weyl field can be decomposed into a
collection of qubits, which are represented by points $\bm{p}$ in Fourier
space (upper sketch). In 3 dimensions, the number of qubits in a small Fourier
space shell $s$ (red crossed in upper sketch) scales as $k_{s}^{2}\Delta_{s}$
where $k_{s}$ is the radius and $\Delta_{s}$ is the width of the shell. We
want to embed these qubits into a Hilbert space that is strictly smaller then
the tensor product of the individual qubit Hilber spaces (cf. Figure 1) in
order to achieve a holographic scaling of the effective degrees of freedom in
our field.

We had already said that the Weyl field can be considered as a collection of
qubits: one for each wave mode $\bm{p}$ in the particle sector and one for
each wave mode $\bm{p}$ in the anti-particle sector (cf. Equation 3 as well as
our sketch in Figure 2, which applies to either one of the two sectors). For a
thin Fourier space shell $s$ of radius $k_{s}$ and width $\Delta_{s}$ the
number density of modes $\bm{p}$ with
$|\bm{p}|\in[k_{s}-\Delta_{s}/2,k_{s}+\Delta_{s}/2]$ per radius will scale as

$\displaystyle\frac{N_{s}}{\Delta_{s}}\propto k_{s}^{2}\ ,$ (7)

where $N_{s}$ is the total number of modes inside shell $s$. The above scaling
is the Fourier space manifestation of the volume extensive scaling of QFT
degrees of freedom. We will now squeeze the qubits present at these modes into
a Hilbert space of dimension $2^{n_{s}}<2^{N_{s}}$ (cf. again Figure 2). To
approximately achieve holography the radial number density of effective qubits
in the shell should scale as [12] 333According to [8, 41, 42] the effective
degrees of freedom should deplete even more quickly than this naive area
scaling - cf. our comment in Section 8 .

$\displaystyle\frac{n_{s}}{\Delta_{s}}\propto k_{s}\ ,$ (8)

where $n_{s}$ is now the effective number of qubit degrees of freedom in shell
$s$. At a first glance this procedure may seem infeasible, because for high
momenta $k_{s}$ it may lead to a vast overpopulation of the physical Hilbert
space with our unphysical qubits. But the authors of [43] have shown that -
using the embedding procedure we outlined above - the physical Hilbert space
can indeed be exponentially overcrowded with qubits (i.e. $n_{s}\sim\ln
N_{s}$) while still keeping the operator norm of the anti-commutator of any
pair of qubits smaller than some small parameter $\epsilon$.

Note that, in order to achieve holography, we don’t even need this exponential
overcrowding. We only need polynomial overcrowding,

$\displaystyle N_{s}\propto n_{s}^{2}\ .$ (9)

This leads us to our first result: in Section 3 we show that in order to make
the Weyl field obey a holographic scaling of the effective degrees of freedom
and also satisfy the Bekenstein bound for the box of size $L$, we need to
allow overlaps of the creation and annihilation operators for different field
modes that are bounded by

$\displaystyle|\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}|\,<\,\epsilon(k)\,$
$\displaystyle\approx\,\sqrt{\frac{48\log\left(k_{s}L\right)}{\alpha\pi(k_{s}L)^{2}}}\,\frac{\Lambda_{\mathrm{UV}}}{\Lambda_{\mathrm{Planck}}}$
(10) $\displaystyle\qquad\mathrm{for\ all}\ \ |\bm{p}|\approx
k\approx|\bm{q}|,\ \bm{p}\neq\bm{q}\ .$

Here $\alpha=\Delta_{s}/k_{s}$ is the relative width of the Fourier space
shells, and we have explicitly allowed a UV-cutoff $\Lambda_{\mathrm{UV}}$ of
our theory that is different from the Planck scale
$\Lambda_{\mathrm{Planck}}$. The above result holds when $\hat{\psi}$ is the
only quantum field in the Universe. In the presence of other fields, it
changes to

$\displaystyle\epsilon(k)\approx\sqrt{\frac{48\log\left(k_{s}L\right)}{\alpha\pi(k_{s}L)^{2}}}\,\sqrt{\frac{N_{\mathrm{dof,total}}}{N_{\mathrm{dof},\psi}}}\,\frac{\Lambda_{\mathrm{UV}}}{\Lambda_{\mathrm{Planck}}}\
,$ (11)

where $N_{\mathrm{dof,total}}$ are the total number of degrees of freedom
present in the Universe and $N_{\mathrm{dof},\psi}$ are the degrees of freedom
contributed by $\hat{\psi}$ .

We show the bound of Equation 11 for different values of the ratio
$N_{\mathrm{dof},\psi}\ /\ N_{\mathrm{dof,total}}$ in Figure 3, fixing
$\alpha=0.01$ and $\Lambda_{\mathrm{UV}}=\Lambda_{\mathrm{Planck}}$. Note that
it is in fact decreasing with increasing wave number $k$ , i.e. at higher
energies the overlap between different modes becomes more and more suppressed.
At energies probed by the Large Hadron Collider it falls down to $\sim
10^{-43}$, indicating that any pair of Fourier modes in that regime behave as
almost independent degrees of freedom (though see the end of Section 3.2 for a
more nuanced discussion).

Figure 3: Bound on the anti-commutator
$\\{\hat{c}_{\mathbf{p}}^{\dagger},\hat{c}_{\mathbf{q}}\\}$ induced for modes
$\mathbf{p}\neq\mathbf{q}$, $|\mathbf{p}|\approx k\approx|\mathbf{q}|$, due to
squeezing the degrees of freedom of our field into too small Hilbert spaces
via the Chao-Johson-Lindenstrauss embedding (cf. Section 3). At energies
probed by the Large Hadron Collider or at energies of everyday life (cf. small
arrows) the bound is vastly smaller than the operator norms of
$\hat{c}_{\mathbf{p}},\hat{c}_{\mathbf{p}}^{\dagger}$, indicating that any
pair of Fourier modes behave as almost independent degrees of freedom.
Different lines assume different ratios between the overall degrees of freedom
of our field and the total degrees of freedom present in the Universe.

To investigate potential dynamical consequences of the non-vanishing Fourier
mode overlaps, we consider a Hamiltonian for our field that takes the same
form as for the standard non-overlapping Weyl field,

$\displaystyle\hat{H}$
$\displaystyle=\sum_{\bm{p}}|\bm{p}|\left\\{\left(\hat{c}_{\bm{p}}^{\dagger}\hat{c}_{\bm{p}}-\frac{1}{2}\right)+\left(\hat{d}_{\bm{p}}^{\dagger}\hat{d}_{\bm{p}}-\frac{1}{2}\right)\right\\}$
(12) $\displaystyle\approx\sum_{\mathrm{shell}\ s}k_{s}\sum_{\bm{p}\in
s}\left\\{\left(\hat{c}_{\bm{p}}^{\dagger}\hat{c}_{\bm{p}}-\frac{1}{2}\right)+\left(\hat{d}_{\bm{p}}^{\dagger}\hat{d}_{\bm{p}}-\frac{1}{2}\right)\right\\}\
,$

where in the second line we made explicit our decomposition of Fourier space
into a set of shells. Within each shell the occupation number operators
$\hat{N}_{\bm{p}}=\hat{c}_{\bm{p}}^{\dagger}\hat{c}_{\bm{p}}$,
$\hat{N}_{\bm{q}}=\hat{c}_{\bm{q}}^{\dagger}\hat{c}_{\bm{q}}$ of different
Fourier modes $\bm{p}$ and $\bm{q}$ will not commute anymore. This also means
that $[\hat{H},\hat{N}_{\bm{p}}]\neq 0$ and the occupation number in a given
Fourier mode is not conserved, despite Equation 12 formally looking like the
free field Hamiltonian. To determine the severity of this effect we first
define plane wave states in our theory through the following steps:

  * •

consider the states $\ket{\psi(t)}$ which at an initial time $t_{0}$ start in
the eigenspace of $\hat{N}_{\bm{p}}$ with eigenvalue $+1\,$, i.e.
$\hat{N}_{\bm{p}}\ket{\psi(t_{0})}=+\ket{\psi(t_{0})}$;

  * •

in this subspace, find the states that minimize
$\left|\left[\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\bra{\psi(t)}\hat{N}_{\bm{p}}\ket{\psi(t)}\right]_{t=t_{0}}\right|$
;

  * •

this condition is still satisfied by an entire subspace of states, so our
final selection step is to choose the candidate state that initially has the
lowest energy expectation value $\bra{\psi(t_{0})}\hat{H}\ket{\psi(t_{0})}$ .

In the following we will use the notation
$\ket{\bm{p}}\equiv\ket{\psi(t_{0})}\,$, where $\ket{\psi(t)}$ is chosen as
explained above, and we will consider $\ket{\bm{p}}$ as the closest equivalent
our theory has to the standard plane wave state
$\hat{c}_{\bm{p}}^{\dagger}\ket{0}$ for the non-overlapping Weyl field. In the
standard theory, plane wave states would be eigenstates of the Hamiltonian
operator and thus have a trivial time evolution. In our holographic version of
the Weyl field this is not the case anymore. Operators $\hat{N}_{\bm{p}}$ and
$\hat{N}_{\bm{q}}$ for different modes $\bm{p},\bm{q}$ do not commute any
longer, and thus the Hamiltonian (which is a sum over all the different
occupation number operators) does not commute with any of the
$\hat{N}_{\bm{p}}$ either. As a consequence, time evolution will move the
states $\ket{\bm{p}}$ out of the $+1$-eigenspace of $\hat{N}_{\bm{p}}\,$. Note
however that we have assumed different Fourier space shells to be non-
overlapping, so the initial probability amplitude in $\ket{\bm{p}}$ will only
be re-distributed with the shell that contains $\bm{p}\,$. Since in our
fiducial construction the overlaps between different modes is independent of
their relative direction, we interpret this modified time evolution as an
isotropic scrambling of plane waves within their Fourier space shells. This
indicates a breaking of Lorentz symmetry (in particular, of momentum
conservation), and we discuss implications of that in Section 3.4.

To quantify the severity of the scrambling effect, we estimate a
characteristic lifetime of plane waves as

$\displaystyle\frac{1}{T_{\mathrm{scramble}}^{2}}=-\mathbb{E}\left\\{\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\bra{\bm{p}}\hat{N}_{\mathbf{p}}\ket{\bm{p}}\right\\}\
,$ (13)

where the expectation value $\mathbb{E}\\{\cdot\\}$ is taken with respect to
the random vectors that are used in the CRSV-embedding to construct
representations of the different field modes (cf. Section 3). In C.2 we
demonstrate analytically that the following relation between
$T_{\mathrm{scramble}}$ and the momentum $|\bm{p}|$ holds in the large-$n_{s}$
limit (cf. more details also in Section 4):

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\bra{\bm{p}}\hat{N}_{\bm{p}}\ket{\bm{p}}\propto\frac{|\bm{p}|^{2}}{|\bm{p}|L}$
(14) $\displaystyle\Rightarrow
T_{\mathrm{scramble}}\propto\sqrt{\frac{L}{|\bm{p}|}}\ .$ (15)

The exact amplitude of this scaling depends on the values of different
parameters of our construction, and in particular on ultraviolet cutoff
$k_{\mathrm{UV}}\,$: choosing a higher $k_{\mathrm{UV}}$ requires larger mode
overlaps in order to still satisfy the cosmic Bekenstein bound, which in turn
leads to a stronger scrambling. Figuratively speaking, the mode overlaps
generate a cosmic fog and that fog has to become thicker when
$k_{\mathrm{UV}}$ is increased, i.e. when more modes are allowed to exist. In
Figure 4 we illustrate the impact of choosing different UV-cuts, fixing the
other parameters to fiducial values detailed in Sections 3.2 and 3.3. In that
figure we also indicate the typical energies and travel distances of neutrinos
created by a number of known and resolved sources (see also Table 1 for a
summary; we assume that the neutrino dispersion relation is still close to
light-like in order to translate lifetimes to distances). We provide a more
detailed discussion of these results in Section 4, but the highly energetic
neutrino emission observed from blazar TXS 0506+056 requires
$k_{\mathrm{UV}}<470\,k_{\mathrm{LHC}}\,$, where we take
$k_{\mathrm{LHC}}=14\,$TeV. So electroweak theory should be modified only a
few orders of magnitude above modern particle physics experiments. In Section
4.3 we argue that this is still consistent with current measurements of the
cosmic neutrino spectrum by IceCube (cf. Figure 7 in that section). The
breakdown of our effective theory beyond the energy $k_{\mathrm{UV}}$ should
happen either as a sharp cutoff or as a transition to super-holographic mode
overlaps, because any modes present beyond $k_{\mathrm{UV}}$ need to be
squeezed into a very small part of the total Hilbert space in order to still
satisfy the Bekenstein bound.

Figure 4: Lifetimes of plane waves in the holographic Weyl field depend on the
UV-cut of our theory. We display the corresponding free path as a function of
plane wave energy according to our calculations in C.2 (see also Section 4.2)
for $k_{\mathrm{UV}}=k_{\mathrm{Planck}}$ (blue, dotted line), for
$k_{\mathrm{UV}}=10^{-3}\,k_{\mathrm{Planck}}$ (orange, dash-dotted line), and
for $k_{\mathrm{UV}}=470\,k_{\mathrm{LHC}}$ (green, dashed line). All of these
lines assume a static Universe, a light-like dispersion relation and that our
holographic field makes up 1% of the total effective degrees of freedom
present in the Universe (cf. Section 3.3 for a discussion of that choice). We
also indicate the energies and path lengths for a number of known and resolved
sources of neutrino emission - cf. Table 1 for more details. To explain how
all of these events could have been observed, we need to assume a break down
of neutrino EFT at $\lesssim$ 500 times the collision energies reached at the
LHC.

Apart from the behaviour of plane waves, it is also possible to understand the
dynamics of our field more generally. In C.1 we use results from random matrix
theory to derive that the set of eigenvalues of the Hamiltonian in each
individual shell of Equation 12 is given by

$\displaystyle\\{\lambda\\}$
$\displaystyle=\left\\{\left.\frac{k_{s}}{2}\sum_{i=1}^{n_{s}}\lambda_{i}s_{i}\
\right|\ \\{s_{i}\\}\in[-1,+1]^{n_{s}}\right\\}\ ,$ (16)

where $\lambda_{i}$ are the positive eigenvalues of a generalised Wigner
matrix. Using a rigidity property for the eigenvalues of Wiger matrices
derived by [44] the $i$th of these eigenvalues will be given with high
accuracy by the $i/n_{s}$-quantile of a Wigner semi-circle distribution. In
particular we are able to derive the vacuum energy in a Fourier space shell of
width $\Delta_{s}$ and radius $k_{s}$ as

$\displaystyle E_{\min,s}$
$\displaystyle\approx-\frac{N_{s}k_{s}}{2}\cdot\left(\frac{8}{3\pi}\sqrt{\frac{\displaystyle
n_{s}}{\displaystyle N_{s}}}\right)\ .$ (17)

In Figure 5 we have used simulated realisations of overlapping qubits to show
that this result is accurate already for quite low numbers of field modes. At
energies (and hence mode numbers) relevant for high-energy physics, it
represents a strong suppression with respect to the non-overlapping Weyl
field, whose vacuum energy in each shell would simply be $-N_{s}k_{s}/2$. When
choosing $\Lambda_{\mathrm{UV}}=\Lambda_{\mathrm{Planck}}$ the total vacuum
energy of the field is lowered by a factor of $\sim 10^{-30}$ (which is
however not enough to alleviate the cosmological constant problem).

Figure 5: Suppression of minimum eigenvalue of the Hamiltonian (compared to the case of non-overlapping degrees of freedom) in a Fourier space shell $s$ of our holographic Weyl field as a function of its overcrowding $N_{s}/n_{s}$ ($=N_{\mathrm{qubit}}/\log_{2}(\dim\mathcal{H})$ in the figure label). Note that $N_{s}\propto k_{s}^{2}\Delta_{s}$ and $n_{s}\propto k_{s}\Delta_{s}$, where $\Delta_{s}$ is the width of the shell and $k_{s}$ its radius. Thus, $N_{s}/n_{s}\propto k_{s}$, i.e. the $x$-axis of the figure is proportional to shell radius. The different symbols represent measurements of vacuum energy suppression in simulated realisations of the Fourier space shells for $n_{s}=4,8,12$. Overall $640$ independent realisations of those simulations have been averaged to obtain these measurements. The transparent bands indicate the standard deviation of the individual realisations. The dashed line displays our theoretical prediction for the suppression of vacuum energy, as derived in C. Note that due to numerical limitations we were only able to simulate very low dimensional shells. Our theoretical prediction will become even more precise at higher values of $n_{s}$ and $N_{s}$ because it describes an asymptotic, high-dimensional limit. Neutrino sources |  characteristic energy scale |  travel distance $d_{\mathrm{source}}$ |  UV-scale where Fermion EFT must break down in order to ensure $d_{\mathrm{source}}<T_{\mathrm{scramble}}/c$ |  vacuum energy suppression (due to holography & UV-cut)
---|---|---|---|---
|  |  |  |
Atlas detector |  $\lesssim$ 14 TeV |  50 m |  $1.04\,k_{\mathrm{Planck}}\,=\,9.1\cdot 10^{14}\,k_{\mathrm{LHC}}$ |  $1.4\cdot 10^{-31}$ (for $k_{\mathrm{UV}}=k_{\mathrm{Planck}}$)
|  |  |  |
CERN $\rightarrow\ $ Gran Sasso ([45]) |  $\sim$ 30 GeV |  731 km |  $4.0\cdot 10^{-2}\,k_{\mathrm{Planck}}\,=\,3.5\cdot 10^{13}\,k_{\mathrm{LHC}}$ |  $1.9\cdot 10^{-36}$
|  |  |  |
Solar (e.g. [46]) |  $<$ 18 MeV |  $1.5\cdot 10^{12}$ m |  $1.8\cdot 10^{-4}\,k_{\mathrm{Planck}}\,=\,1.6\cdot 10^{11}\,k_{\mathrm{LHC}}$ |  $1.1\cdot 10^{-44}$
|  |  |  |
SN 1987A ([47, 48]) |  $\lesssim$ 20 MeV |  $1.6\cdot 10^{18}$ m |  $1.7\cdot 10^{-7}\,k_{\mathrm{Planck}}\,=\,1.5\cdot 10^{8}\,k_{\mathrm{LHC}}$ |  $2.9\cdot 10^{-55}$
|  |  |  |
TXS 0506+056 ([49]) |  $\sim$ 290 TeV |  $4.1\cdot 10^{25}$ m $\approx 1.3$ Gpc |  $5.4\cdot 10^{-13}\,k_{\mathrm{Planck}}\,=\,4.7\cdot 10^{2}\,k_{\mathrm{LHC}}$ |  $1.7\cdot 10^{-74}$
Table 1: Known and resolved sources of neutrinos, their typical energies and
their travel distances. In order for the travel time of these particles to be
shorter than the plane wave scrambling time of our holographic construction,
we need to impose UV cutoffs that are significantly below the Planck scale
(but still safely above energy scales probed at the LHC; cf. 4th column).
Holography and the UV cutoffs combine to suppress vacuum energy density
significantly compared to a standard Fermion with
$k_{\mathrm{UV}}=k_{\mathrm{Planck}}$ (but not enough to solve the
cosmological constant problem; cf. 5th column). The table employs our fiducial
assumption that about $1\%$ of the Universes degrees of freedom are
contributed by our holographic Fermion. Whether this is a realistic assumption
depends on the degree to which different quantum fields overlap with each
other (cf. our discussion in Section 3.3).

Finally, in Section 5 we investigate the impact of the overlapping Fourier
modes on the real space propagator
$\\{i\psi_{\alpha}(\bm{x})^{\dagger},\psi_{\beta}(\bm{y})\\}$ . After tracing
out the spin degree of freedom labelled by $\alpha,\beta$ we obtain

$\sum_{\alpha}\\{i\psi_{\alpha}(\bm{x})^{\dagger},\psi_{\alpha}(\bm{y})\\}=2i\delta_{D}(\bm{x}-\bm{y})+2iC(\bm{x},\bm{y})\
.$ (18)

The first term on the right hand side of Equation 18 is the result one would
obtain for the standard Weyl field and the second term is a correction induced
by the mode overlap. Since we squeeze the degrees of freedom of our field into
the true, physical Hilbert space of the field via a stochastic procedure, the
function $C(\bm{x},\bm{y})$ is also stochastic. It vanishes on average,
$\mathbb{E}[C(\bm{x},\bm{y})]=0$ , but it fluctuates with an amplitude

$\displaystyle\sqrt{\mathbb{E}\left[|C(\bm{x},\bm{y})|^{2}\right]}$
$\displaystyle=\frac{1}{\sqrt{128\pi^{5}}}\frac{\Lambda_{\mathrm{UV}}^{2}}{L}\
,$ (19)

where $L$ is again the IR scale corresponding to the size of the box to which
we confine our field, and $\Lambda_{\mathrm{UV}}$ is a UV cutoff. Note that
the amplitude of the correction term is in fact independent of $\bm{x}$,
$\bm{y}$ and in particular their distance. So, within our construction, a
holographic scaling of the effective field degrees of freedom manifests itself
as a stochastic, long-range contribution to the field propagator. In
particular, the correction can be interpreted as a source of non-locality
because the field and its conjugate momentum do not anti-commute at non-zero,
equal-time distances.

Since the correction term $2iC(\bm{x},\bm{y})$ in Equation 18 has a scale
independent amplitude, it is tempting to assume that the standard term
$2i\delta_{D}(\bm{x}-\bm{y})$ will always dominate the propagator at
sufficiently small distances $\bm{x}-\bm{y}$ . To make this assumption more
precise, and in order to test it, we smooth our field with a Gaussian test
function of width $R$ (standard deviation). The resulting smoothed field
$\hat{\psi}_{R}$ has the propagator

$\displaystyle\sum_{\alpha}\\{i\psi_{R,\alpha}(\bm{x})^{\dagger},\psi_{R,\alpha}(\bm{y})\\}\
$ $\displaystyle=\ 2iC_{R}(\bm{x},\bm{y})+\frac{2i}{\sqrt{(4\pi)^{3}R^{6}}}\
e^{-\frac{1}{4R^{2}}|\bm{x}-\bm{y}|^{2}}$ (20) $\displaystyle\equiv\
2iD_{R}(\bm{x},\bm{y})\ ,$

where for simplicity we have again taken the trace over spin degrees of
freedom, and where $C_{R}(\bm{x},\bm{y})$ is an appropriately smoothed version
of $C(\bm{x},\bm{y})$. We can then compare the average fluctuation of the
long-range propagator (which is dominated by $C_{R}(\bm{x},\bm{y})$) to the
average local propagator (which is given by the standard local propagator
because $\mathbb{E}[C_{R}]=0$). In particular, we consider the ratio of these
two and in Section 5 we obtain

$\displaystyle\frac{\mathbb{E}\left[|C_{R}(\bm{x},\bm{y})|^{2}\right]}{\mathbb{E}[D_{R}(\bm{x},\bm{x})]\
\mathbb{E}[D_{R}(\bm{y},\bm{y})]}\ =\
\frac{N_{\mathrm{dof,total}}}{N_{\mathrm{dof},\psi}}\left(\frac{2R}{\pi
L}\right)^{2}\ .$ (21)

We display the square root of this ratio in the lower panel of Figure 8 for
different values of the ratio $N_{\mathrm{dof},\psi}/N_{\mathrm{dof,total}}$ .
Even for cosmologically large smoothing scales ($R\,\sim 30$ Mpc) the typical
amplitude of the long-range propagator remains at a level of $\sim 1\%$ of the
local anti-commutator. So the non-locality induced by our holographic scaling
of the effective field degrees of freedom seems to become significant only on
very large scales. In a sense, this is just the real space equivalent of our
result regarding the mode anti-commutators
$\\{\hat{c}_{\mathbf{p}}^{\dagger},\hat{c}_{\mathbf{q}}\\}$ from Figure 3.

The rest of this paper is structured as follows. In Section 2 we review basics
of the Weyl field and how to decompose it into a set of qubits. Section 3 then
introduces modifications to make the degrees of freedom of this field obey an
approximate area law scaling. We study the dynamics of our holographic Weyl
field in Section 4. In particular, results about its energy spectrum are
presented in Section 4.1 and we discuss our definition of plane waves and
their lifetime in Section 4.2. Section 5 investigates the real space anti-
commutator of our field, while Section 6 repeats some of our analysis for two
possible modifications of our fiducial construction. In Section 7 we put our
construction into the context of quantum mereology and discuss why classical
stochasticity like the one present in our qubit overlaps is likely to arise in
a dynamical emergence of classical degrees of freedom from an abstract quantum
theory. Finally, Section 8 summarises our paper and provides an outlook.

Note that many of our technical results are derived in our appendices (in
particular C) and only quoted in the main text.

##  2 The Weyl field as a collection of qubits

For the sake of simplicity we consider a Weyl fermion, which corresponds to
one of the two helicity sectors of a massless Dirac fermion. This section
serves as a reminder of the Weyl formalism. We also make explicit in which
sense the Weyl field can be decomposed into a collection of qubits. In Section
3 we will use this decomposition to define a modified version of the Weyl
field, in which certain degrees of freedom are overlapping.

###  2.1 Weyl field basics

The (left-handed) Weyl spinor is a two-component field $\psi$ with the
Lagrangian

$\mathcal{L}=i\psi^{\dagger}\sigma^{\mu}\partial_{\mu}\psi\ ,$ (22)

where $\sigma^{0}\equiv\mathbf{1}$ and $\sigma^{i}$ are e.g. the Pauli
matrices (we are following here the notation of [50, 51]). The above
Lagrangian leads to the equations of motion

$\sigma^{\mu}\partial_{\mu}\psi=0\ .$ (23)

A general solution to these equations can be expressed as

$\displaystyle\psi(\bm{x},t)$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}E_{p}}\
\left\\{a_{\bm{p}}(t)u(\bm{p})\ e^{i\bm{p}\bm{x}}+b_{\bm{p}}^{*}(t)u(\bm{p})\
e^{-i\bm{p}\bm{x}}\right\\}\ ,$ (24)

where the time evolution of the coefficients $a_{\bm{p}}$ and $b_{\bm{p}}^{*}$
is given by

$a_{\bm{p}}(t)=a_{\bm{p},0}\ e^{-iE_{p}t}\ ,\
b_{\bm{p}}(t)^{*}=b_{\bm{p},0}^{*}\ e^{iE_{p}t}$ (25)

with $E_{p}\equiv|\bm{p}|$, and where $u(\bm{p})$ are eigenvectors of the
matrix $\sigma^{j}p_{j}$ with eigenvalues $+E_{p}$444Note that $u(-\bm{p})$ is
then an eigenvector with eigenvalue $-E_{p}$. This is why only a single family
of functions $u(\bm{p})$ appears in the expansion of Equation 24. ,

$\sigma^{j}p_{j}\cdot u(\bm{p})=+E_{p}\ u(\bm{p})\ .$ (26)

Note that in the following we will keep the time dependence of $a_{\bm{p}}$
and $b_{\bm{p}}^{*}$ implicit in our notation. The reason is that this
dependence will deviate from Equation 25 once we consider overlapping degrees
of freedom.

Normalising the $u(\bm{p})$ such that

$u(\bm{p})^{\dagger}\cdot u(\bm{p})=E_{p}$ (27)

ensures that they are orthogonal wrt. the Lorentz invariant momentum space
measure

$\mathrm{d}\tilde{p}:=\frac{\mathrm{d}^{3}p}{(2\pi)^{3}\ E_{p}}\ .$ (28)

Up to an irrelevant phase factor we can e.g. choose $u(\bm{p})$ as [51]

$u(\bm{p})=\sqrt{E_{p}}\pmatrix{e^{-i\phi}\sin\frac{\theta}{2}\cr\cos\frac{\theta}{2}}\
,$ (29)

where $\bm{p}=(p\sin\theta\cos\phi,p\sin\theta\sin\phi,p\cos\theta)^{T}$ .

In the quantum version of the above field theory we consider the operator
valued field

$\displaystyle\hat{\psi}(\bm{x},t)$ $\displaystyle=\int\mathrm{d}\tilde{p}\
\left\\{\hat{a}_{\bm{p}}(t)u(\bm{p})\
e^{i\bm{p}\bm{x}}+\hat{b}_{\bm{p}}(t)^{\dagger}u(\bm{p})\
e^{-i\bm{p}\bm{x}}\right\\}\ ,$ (30)

where the operator $\hat{b}_{\bm{p}}^{\dagger}$ can be thought of as creating
an anti-spinor of momentum $\bm{p}$ while $\hat{a}_{\bm{p}}$ is destroying a
spinor with momentum $\bm{p}$. At equal times these operators satisfy the
anti-commutation relations

$\displaystyle
0=\\{\hat{a}_{\bm{p}},\hat{a}_{\bm{q}}\\}=\\{\hat{b}_{\bm{p}},\hat{b}_{\bm{q}}\\}=\\{\hat{a}_{\bm{p}},\hat{b}_{\bm{q}}\\}=\\{\hat{a}_{\bm{p}},\hat{b}_{\bm{q}}^{\dagger}\\}$
(31)
$\displaystyle\\{\hat{a}_{\bm{p}},\hat{a}_{\bm{q}}^{\dagger}\\}=\\{\hat{b}_{\bm{p}},\hat{b}_{\bm{q}}^{\dagger}\\}=(2\pi)^{3}E_{p}\
\delta_{D}(\bm{p}-\bm{q})\ .$ (32)

This ensures that the field operators satisfy the equal-time anti-commutation
relation

$\displaystyle\\{\hat{\psi}(\bm{x}),i\hat{\psi}(\bm{y})^{\dagger}\\}$ (33)
$\displaystyle\ \ \ =i\int\mathrm{d}\tilde{p}\mathrm{d}\tilde{q}\
\left[\\{\hat{a}_{\bm{p}},a_{\bm{q}}^{\dagger}\\}\
u(\bm{p})u(\bm{q})^{\dagger}+\\{\hat{b}_{-\bm{p}}^{\dagger},b_{-\bm{q}}\\}\
u(-\bm{p})u(-\bm{q})^{\dagger}\right]e^{i\bm{p}\bm{x}-i\bm{q}\bm{y}}$
$\displaystyle\ \ \ =i\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}E_{p}}\
\left[u(\bm{p})u(\bm{p})^{\dagger}+u(-\bm{p})u(-\bm{p})^{\dagger}\right]e^{i\bm{p}(\bm{x}-\bm{y})}$
$\displaystyle\ \ \ =i\ \mathbf{1}_{\mathrm{2D}}\ \delta_{D}(\bm{x}-\bm{y})\
,$ (34)

as is needed because $i\hat{\psi}^{\dagger}$ is the conjugate momentum of the
field $\hat{\psi}$ .

###  2.2 Decomposition into qubits

To make it explicit that the above field can be considered as a collection of
qubits, let us constrict $\psi(\bm{x})$ to a box of finite size $L$ . This
means that we have to perform the substitutions

$\mathrm{d}^{2}p\rightarrow\frac{(2\pi)^{3}}{L^{3}}\ ,\
\delta_{D}(\bm{p}-\bm{q})\rightarrow\frac{L^{3}}{(2\pi)^{3}}\delta_{\bm{p},\bm{q}}\
,$ (35)

and that integrals over momenta will be replaced by sums over the discrete
grid $\bm{p}\in\\{\ 2\pi/L(n_{1},n_{2},n_{3})\ |\ n_{i}\in\mathbb{Z}\ \\}$ .
For convenience, we will also consider redefined mode operators

$\hat{c}_{\bm{p}}=\frac{\hat{a}_{\bm{p}}}{(E_{p}V)^{\frac{1}{2}}}\ ,\
\hat{d}_{\bm{p}}=\frac{\hat{b}_{\bm{p}}}{(E_{p}V)^{\frac{1}{2}}}\ ,$ (36)

such that the new operators satisfy the anti-commutation relations

$\displaystyle
0=\\{\hat{c}_{\bm{p}},\hat{c}_{\bm{q}}\\}=\\{\hat{d}_{\bm{p}},\hat{d}_{\bm{q}}\\}=\\{\hat{c}_{\bm{p}},\hat{d}_{\bm{q}}\\}=\\{\hat{c}_{\bm{p}},\hat{d}_{\bm{q}}^{\dagger}\\}$
(37)
$\displaystyle\\{\hat{c}_{\bm{p}},\hat{c}_{\bm{q}}^{\dagger}\\}=\\{\hat{d}_{\bm{p}},\hat{d}_{\bm{q}}^{\dagger}\\}=\delta_{\bm{p},\bm{q}}\
.$ (38)

Our field can now be decomposed in terms of these operators as

$\hat{\psi}(\bm{x},t)=\sum_{\bm{p}}\frac{1}{(E_{p}V)^{\frac{1}{2}}}\left\\{\hat{c}_{\bm{p}}(t)\
u(\bm{p})\ e^{i\bm{p}\bm{x}}+\hat{d}_{\bm{p}}(t)^{\dagger}\ u(\bm{p})\
e^{-i\bm{p}\bm{x}}\right\\}\ .$ (39)

Usually, each of the grid points $\bm{p}$ in the above sum represents a
4-dimensional Hilbert space factor

$\displaystyle\mathcal{H}_{\bm{p}}=\mathcal{H}_{\bm{p}}^{c}\otimes^{\mathrm{JW}}\mathcal{H}_{\bm{p}}^{d}$
(40)

and the total Hilbert space is the tensor product over all these factors,

$\displaystyle\mathcal{H}=\bigotimes_{\bm{p}}^{\mathrm{JW}}\mathcal{H}_{\bm{p}}\
,$ (41)

where the superscript JW again indicates that operators in the individual
Hilbert spaces need to be embedded into the product space via Jordan-Wigner-
factors (cf. D) in order for them to anti-commute (as opposed to commute). The
$\hat{c}_{\bm{p}}$, $\hat{d}_{\bm{p}}$ and their Hermitian conjugates act non-
trivially only on the factors $\mathcal{H}_{\bm{p}}^{c}$ and
$\mathcal{H}_{\bm{p}}^{d}$ respectively. On the factor
$\mathcal{H}_{\bm{p}}^{c}$ (and similarly for $\mathcal{H}_{\bm{p}}^{d}$) we
can define

$\displaystyle\hat{\sigma}_{x,\bm{p}}^{c}$
$\displaystyle=\hat{c}_{\bm{p}}+\hat{c}_{\bm{p}}^{\dagger}$ (42)
$\displaystyle\hat{\sigma}_{y,\bm{p}}^{c}$
$\displaystyle=i\left(\hat{c}_{\bm{p}}-\hat{c}_{\bm{p}}^{\dagger}\right)$ (43)
$\displaystyle\hat{\sigma}_{z,\bm{p}}^{c}$
$\displaystyle=-i\hat{\sigma}_{x,\bm{p}}^{c}\hat{\sigma}_{y,\bm{p}}^{c}=2\hat{c}_{\bm{p}}^{\dagger}\hat{c}_{\bm{p}}-1\
.$ (44)

These operators constitute a Pauli algebra on the qubit Hilbert space
$\mathcal{H}_{\bm{p}}^{c}$ . Note however, that the labels $x,y,z$ are simply
notation, and not meant to indicate directions in physical space. The
Hamiltonian of the field can be expressed in terms of these operators as

$\displaystyle\hat{H}$
$\displaystyle=\sum_{\bm{p}}E_{p}\left\\{\left(\hat{c}_{\bm{p}}^{\dagger}\hat{c}_{\bm{p}}-\frac{1}{2}\right)+\left(\hat{d}_{\bm{p}}^{\dagger}\hat{d}_{\bm{p}}-\frac{1}{2}\right)\right\\}$
(45)
$\displaystyle=\sum_{\bm{p}}\frac{E_{p}}{2}\left\\{\hat{\sigma}_{z,\bm{p}}^{c}+\hat{\sigma}_{z,\bm{p}}^{d}\right\\}\
.$

So our field behaves like a set of non-interacting spins in a
$\bm{p}$-dependent magnetic field.

##  3 Fermion field with overlapping qubits

###  3.1 Chao-Reichert-Sutherland-Vidick embedding

We will now use the scheme of [24] for embedding $N$ qubits into an Hilbert
space of dimension $2^{n}<2^{N}$ (Chao-Reichert-Sutherland-Vidick embedding,
or CRSV-embedding for the rest of this paper) to create a version of the Weyl
field whose effective degrees of freedom approximately follow an area law
scaling. To describe our approach we will focus on the particle part of the
field (the anti-particle part can be treated analogously). We start by
covering the entirety of Fourier space with adjacent but non-overlapping
shells $\\{s\\}$ (cf. our sketch in Figure 2), and we introduce the following
notation.

  * •

$k_{s}\,,\ \Delta_{s}$ :

the central radius and width of Fourier space shell $s$.

  * •

$N_{s}$ :

number of qubits inside Fourier space shell $s$ (i.e. the number of momenta
$\bm{p}\in s$).

  * •

$n_{s}$ :

logarithm (to the base $2$) of the dimension into which the $N_{s}$ will be
embedded.

For our regular grid in Fourier space we expect

$\displaystyle N_{s}\ \approx\ 4\pi k_{s}^{2}\Delta_{s}\ \bigg{/}\
\frac{(2\pi)^{3}}{L^{3}}\ ,$ (46)

where $k_{s}$ is the central radius of shell $s$. The scaling $N_{s}\propto
k_{s}^{2}\Delta_{s}$ indicates a volume scaling of the independent degrees of
freedom, but we will squeeze the qubits of shell $s$ into a Hilbert space that
is too small for them to be independent. In particular we would like to obtain
an area-scaling of the effective degrees of freedom, such that [12]

$\displaystyle\frac{n_{s}}{N_{s}}\propto\frac{k_{s}\Delta_{s}}{k_{s}^{2}\Delta_{s}}=\frac{1}{k_{s}}\
.$ (47)

This will be achieved if

$\displaystyle n_{s}=B\cdot 2\pi k_{s}\Delta_{s}\ \bigg{/}\
\frac{(2\pi)^{2}}{L^{2}}\ ,$ (48)

where $B$ is some constant, and where the factors of $L/2\pi$ are just for
later convenience. Along the lines of [24] we will now proceed as follows:

  * A)

For each shell $s$ we choose a $2^{n_{s}}$-dimensional matrix representation
of the Clifford algebra with generators $\bm{C}_{1}$, …, $\bm{C}_{2n_{s}}$ .

  * B)

In the space $\mathbb{R}^{2n_{s}}$ , we draw $2N_{s}$ random vectors from a
standard normal distribution.

  * C)

We collect these vectors into $N_{s}$ pairs (corresponding to the $N_{s}$
momenta $\bm{p}$ in the Fourier space shell $s$), and choose orthonormal bases
$\\{\ket{v_{\bm{p}}},\ket{w_{\bm{p}}}\\}$ for the $N_{s}$ 2-dimensional
subspaces spanned by these vector pairs.

  * D)

For some orthonormal basis $\\{\ket{e_{j}}\\}$ of $\mathbb{R}^{2n_{s}}$, we
define

$\displaystyle\bm{\sigma}_{x,\bm{p}}^{c}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{v_{\bm{p}}}\bm{C}_{j}\
\ ,\ \
\bm{\sigma}_{y,\bm{p}}^{c}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{w_{\bm{p}}}\bm{C}_{j}\
\ ,\ \ \bm{\sigma}_{z,\bm{p}}^{c}$
$\displaystyle=-i\bm{\sigma}_{x,\bm{p}}^{c}\bm{\sigma}_{y,\bm{p}}^{c}\ ,$

and we set

$\displaystyle\bm{c}_{\bm{p}}=\frac{1}{2}\left(\bm{\sigma}_{x,\bm{p}}^{c}+i\bm{\sigma}_{y,\bm{p}}^{c}\right)\
.$ (50)

(Note that this is slightly different from how [24] had originally defined
their embedding. The reason for this is that we want the creation and
annihilation operators for different modes $\bm{p}_{1},\,\bm{p}_{2}$ to
approximately anti-commute, while [24] wanted operators on different qubits to
approximately commute.)

  * E)

The matrices $\bm{c}_{\bm{p}}^{\dagger}$ and $\bm{c}_{\bm{p}}$ can serve as
creation and annihilation operators in the Hilbert space $\mathcal{H}_{s}^{c}$
corresponding to shell $s$. In principle we now only have to stitch together
different shells to obtain the creation and annihilation operators
$\hat{c}_{\bm{p}}^{\dagger}$ and $\hat{c}_{\bm{p}}$ in the whole Hilbert space

$\mathcal{H}^{c}=\bigotimes_{s}\mathcal{H}_{s}^{c}\ .$ (51)

A naive attempt to do so would be to define
$\hat{c}_{\bm{p}}=\left(\bigotimes_{\bm{p}\notin
s}\mathbf{1}_{s}\right)\otimes\bm{c}_{\bm{p}}$ , i.e. to consider a tensor
product of the operator $\bm{c}_{\bm{p}}$ (which is defined on the Hilbert
space of the shell that contains $\bm{p}$) with the unit operators in all
shells that do now contain $\bm{p}$. This naive approach would not work,
because we need operators $\hat{c}_{\bm{p}}$ and $\hat{c}_{\bm{q}}$ that live
on different shells to be anti-commuting (instead of commuting).

To achieve anti-commutation between different shells, we need to insert the
appropriate Jordan-Wigner strings into the above, naive construction. We
sketch in D how this can be done in practice. But the details of this do not
matter in the following, since they do not impact any anti-commutators that
are relevant to our calculations.

  * F)

Finally, we duplicate the above construction for the $d$-particles and build
the full Hilbert space as the tensor product of the spaces of particles and
anti-particles (again, using an appropriate Jordan-Wigner construction to
ensure anti-commutation between $c$\- and $d$-operators).

Within the above construction the anti-commutator between the operators
$\hat{c}_{\bm{p}}^{\dagger}$ and $\hat{c}_{\bm{q}}$, where $\bm{p}$ and
$\bm{q}$ are in the same Fourier space shell $s$, becomes

$\displaystyle\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}$
$\displaystyle=\frac{1}{4}\sum_{jl}\braket{e_{j}}{v_{\bm{p}}-iw_{\bm{p}}}\braket{e_{l}}{v_{\bm{q}}+iw_{\bm{q}}}\\{\bm{C}_{j},\bm{C}_{l}\\}$
(52)
$\displaystyle=\frac{\braket{v_{\bm{p}}}{v_{\bm{q}}}+\braket{w_{\bm{p}}}{w_{\bm{q}}}+i\braket{v_{\bm{p}}}{w_{\bm{q}}}-i\braket{w_{\bm{p}}}{v_{\bm{q}}}}{2}$
$\displaystyle\equiv\frac{\braket{z_{\bm{p}}}{z_{\bm{q}}}}{2}\ ,$

where in the last line we have introduced the complex vectors
$\ket{z_{\bm{p}}}=\ket{v_{\bm{p}}}+i\ket{w_{\bm{p}}}$ . Note that in general
$\braket{z_{\bm{p}}}{z_{\bm{q}}}\neq 0$ for $\bm{p}\neq\bm{q}$, because the
vectors $\ket{v_{\bm{p}}},\ket{w_{\bm{p}}}$ are more numerous than their
dimension (and because we chose these vector pairs randomly). At the same
time, it is easy to check that $\braket{z_{\bm{p}}}{z_{\bm{p}}}=2$ and hence
$\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{p}}\\}=1$.

The vectors $\ket{v_{\bm{p}}},\ket{w_{\bm{p}}}$ that are used in the above
embedding procedure are chosen at random (cf. steps B and C). As a
consequence, the anti-commutators
$\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}$ for $\bm{p}\neq\bm{q}$
become classical random variables. We demonstrate in B that, as a consequence
of the Johnson-Lindenstrauss theorem (JL-theorem, [25, 52, 53, 24, 54]) the
vectors $\\{\ket{v_{\bm{p}}},\ket{w_{\bm{p}}}\\}$ can be drawn such that with
a probability $>1-\delta$ _all_ anti-commutators in a shell $s$ simultaneously
satisfy

$\displaystyle|\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}|$
$\displaystyle<\sqrt{\frac{16}{n_{s}}\ln\left(\frac{2N_{s}}{\sqrt{\delta}}\right)}$
(53)
$\displaystyle=\sqrt{\frac{32\pi\log\left(k_{s}^{2}\Delta_{s}L^{3}/\left(\pi^{2}\sqrt{\delta}\right)\right)}{k_{s}BL^{2}\Delta_{s}}}\
.$

To asses whether or not this is small, we need to specify values for $L$,
$\Delta_{s}$ and $B$. Our fiducial choice for these parameters is explained in
the following subsection.

###  3.2 Fiducial parameters & satisfying the Bekenstein bound

Our procedure to construct overlapping creation and annihilation operators for
the Weyl field depends on the values of a number of parameters:

  * •

$\Delta_{s}$: the width of the Fourier space shells $\\{s\\}$;

  * •

$B$: the parameter determining the dimension of the Hilbert space factor, into
which the qubits in one Fourier space shell will be embedded;

  * •

$L$: the IR scale that induces the discreteness in Fourier space;

  * •

$\Lambda_{\mathrm{UV}}$: a UV-cutoff.

In the following, we will take the UV scale $\Lambda_{\mathrm{UV}}$ to be a
free parameter of our construction. As we will see later, in order to explain
the existence of a number of observed cosmic neutrino fluxes this scale needs
to be significantly lower than the Planck energy scale
$\Lambda_{\mathrm{Planck}}$. Generally, having a lower $\Lambda_{\mathrm{UV}}$
will decrease effects of holography (or rather: of the mode overlaps present
in our construction) because is allows us to choose a higher value of $B$ (and
thus have smaller mode overlaps) while still satisfying the Bekenstein bound
on the total number of degrees of freedom present in our field. We will
interpret $\Lambda_{\mathrm{UV}}<\Lambda_{\mathrm{Planck}}$ as an indication
that effective field theories of elementary Fermions need to break down
already at sub-Planckian energies.

To fix $L$, we adopt the approach of [13] who chose their IR scale as the
future co-moving particle horizon of a $\Lambda$CDM universe similar to our
own Universe555We assume parameters similar to those found by [55] when
analysing 2-point statistics of temperature fluctuations in the cosmic micro
wave background. This scale is finite in a $\Lambda$-dominated universe, and
it corresponds to the largest co-moving sub-volume of the Universe about which
we can ever collect any information.

For concreteness we will assume a constant relative shell width, i.e.
$\Delta_{s}=\alpha k_{s}$ for some $\alpha\ll 1$ . This is a somewhat
arbitrary modelling choice, and we will quote many of the results in later
parts of this paper in terms of general shells widths $\Delta_{s}$ (or in
terms of general values for $n_{s}$ and $N_{s}$) in order to keep these
results accessible to generalisation. Generally, having wider shells leads to
a stronger suppression of holographic effects. The reason for this can be
found in the Johnson-Lindenstrauss theorem itself, which states that the qubit
overlap $\epsilon$ can be chosen such that $\epsilon^{2}\sim(\ln
N_{s})/n_{s}\sim(\ln\Delta_{s})/\Delta_{s}$ , which decreases as $\Delta_{s}$
is increased. Demanding $\alpha\ll 1$ seems natural because is leads to an
approximately smooth radial mode density function in Fourier space, and in our
fiducial construction we will set $\alpha=0.01\,$.

Finally, we can motivate a choice for $B$ as follows. The overall number of
effective qubits in our construction is given by

$\displaystyle N_{\mathrm{qubit}}\ =\ 2\sum_{\mathrm{shell}\ s}n_{s}\ =\
\frac{BL^{2}}{\pi}\sum_{\mathrm{shell}\ s}k_{s}\Delta_{s}\ $
$\displaystyle\approx\
\frac{BL^{2}}{\pi}\int_{0}^{\Lambda_{\mathrm{UV}}}\mathrm{d}k\ k$ (54)
$\displaystyle=\ \frac{B(L\Lambda_{\mathrm{UV}})^{2}}{2\pi}\ ,$

where the factor of $2$ in the first line takes into account that there are
both particles and anti-particles. At the same time the IR scale $L$ will
correspond to a real-space boundary area $A_{\mathrm{boundary}}\sim 4\pi
L^{2}\,$. The Bekenstein bound on the maximum entropy attainable within such a
boundary should then lead to

$\displaystyle\frac{B(L\Lambda_{\mathrm{UV}})^{2}}{2\pi}$
$\displaystyle\lesssim\pi(L\Lambda_{\mathrm{Planck}})^{2}$ (55)
$\displaystyle\Rightarrow B$ $\displaystyle\lesssim
2\pi^{2}\left(\frac{\Lambda_{\mathrm{Planck}}}{\Lambda_{\mathrm{UV}}}\right)^{2}\
.$ (56)

Note again that a lower UV-scale $\Lambda_{\mathrm{UV}}$ allows us to choose a
higher value for $B$. Since in turn the qubit overlaps scale as $\epsilon\sim
1/\sqrt{B}\ $ this means that holographic effects will be less pronounced. If
our field $\psi$ constituted the only degrees of freedom in the Universe, then
the above bound would be saturated, such that $B\approx
2\pi^{2}\left(\Lambda_{\mathrm{Planck}}/\Lambda_{\mathrm{UV}}\right)^{2}$ and
$n_{s}\approx\pi
L^{2}k_{s}\Delta_{s}\left(\Lambda_{\mathrm{Planck}}/\Lambda_{\mathrm{UV}}\right)^{2}$.
We will instead take into account the possibility (or rather: certainty) that
there are other quantum fields in the Universe. Assuming that the degrees of
freedom of all types of fields (including spacetime degrees of freedom)
individually satisfy an area law scaling leads to

$\displaystyle
B\,\approx\,2\pi^{2}\,\left(\frac{\Lambda_{\mathrm{Planck}}}{\Lambda_{\mathrm{UV}}}\right)^{2}\,\frac{N_{\mathrm{dof},\psi}}{N_{\mathrm{dof,total}}}\
,$ (57)

where $N_{\mathrm{dof,total}}$ are the total number of degrees of freedom
present in the Universe and $N_{\mathrm{dof},\psi}$ are the degrees of freedom
contributed by the field $\psi$ .

With our above choices for $\Delta_{s}$ and $B$ the bound on qubit overlap
from Equation 53 now becomes

$\displaystyle|\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}|$
$\displaystyle<\,\sqrt{\frac{32\pi\log\left(k_{s}^{2}\Delta_{s}L^{3}/\left(\pi^{2}\sqrt{\delta}\right)\right)}{k_{s}BL^{2}\Delta_{s}}}$
(58)
$\displaystyle=\,\sqrt{\frac{16\left[3\ln\left(k_{s}L\right)+\ln\left(\alpha/\pi^{2}\sqrt{\delta}\right)\right]}{\alpha\pi(k_{s}L)^{2}}}\,\sqrt{\frac{N_{\mathrm{dof,total}}}{N_{\mathrm{dof},\psi}}}\,\frac{\Lambda_{\mathrm{UV}}}{\Lambda_{\mathrm{Planck}}}$
$\displaystyle\approx\,\sqrt{\frac{48\log\left(k_{s}L\right)}{\alpha\pi(k_{s}L)^{2}}}\,\sqrt{\frac{N_{\mathrm{dof,total}}}{N_{\mathrm{dof},\psi}}}\,\frac{\Lambda_{\mathrm{UV}}}{\Lambda_{\mathrm{Planck}}}\
\ \ \ \mathrm{for}\ k_{s}\gg L\ .$

As we already noted in Section 1, this bound is decreasing with increasing
wave number $k$ . This means that at higher energies the overlap between
different modes becomes more and more negligible. In Figure 3 we display the
bound for different values of the ratio $N_{\mathrm{dof},\psi}\ /\
N_{\mathrm{dof,total}}$ , but fixing $\alpha=0.01$ and
$\Lambda_{\mathrm{UV}}=\Lambda_{\mathrm{Planck}}$. At everyday scales
$k\sim\mathrm{cm}^{-1}$ the anti-commutators
$\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}$ have absolute values
$\lesssim 10^{-27}$, and at energies probed by the Large Hadron Collider they
drop to $\lesssim 10^{-43}$, indicating that any pair of Fourier modes in this
energy range behaves as almost perfectly independent degrees of freedom. These
drastic results are a direct consequence of the JL-theorem. At fixed
$\epsilon$ it predicts an exponential scaling
$N_{s}\sim\exp(n_{s}/\epsilon^{2})$, cf. B. But in order to achieve holography
we only need polynomial scaling between $N_{s}$ and $n_{s}$! This allows us to
choosing a running $\epsilon$ instead, that quickly decreases with increasing
energy.

Note that the mode overlap can be decreased even further (while still
satisfying the Bekenstein bound) if the UV cutoff $\Lambda_{\mathrm{UV}}$ of
our theory is lowered to values smaller than $\Lambda_{\mathrm{Planck}}\,$.
Cosmic neutrino emission seems to indicate that this is indeed required, in
order to keep (our implementation of) holography consistent with observational
data (cf. Figure 4 as well as Table 1).

###  3.3 Degrees of freedom of other quantum fields

The standard model has 90 Fermionic and 28 Bosonic field degrees of freedom
[56]. Neglecting Bosonic fields, and assuming that all Fermion degrees of
freedom behave in a way similar to our holographic Weyl field, it seems that
$N_{\mathrm{dof},\psi}\ /\ N_{\mathrm{dof,total}}\ <\ 1/90\,$. Of course,
Bosonic degrees of freedom will account for vastly more effective qubit
degrees of freedom, so a much smaller ratio $N_{\mathrm{dof},\psi}\ /\
N_{\mathrm{dof,total}}$ may seem likely. Technically, Bosons should contribute
infinitely many qubits, but in the light of the holographic principle it has
been argued that this is unphysical [11, 12, 13]. Let us assume that Bosonic
degrees of freedom are approximated in the real Universe with generalised
Pauli operators [21, 13] of dimension $\sim 1000\,$. This number is admittedly
ad hoc, but the number of qubits contributed by such approximate Bosonic
degrees of freedom depends only logarithmically on the dimension of the
generalised Pauli operators. We would then expect $N_{\mathrm{dof},\psi}\ /\
N_{\mathrm{dof,total}}\ <\ 1/(90+28\log_{2}1000)\ \approx\ 0.003\,$.

So, if we attempt to model e.g. a neutrino of the standard model with our
holographic version of the Weyl field, then it seems that we should use
$N_{\mathrm{dof},\psi}\ /\ N_{\mathrm{dof,total}}\lesssim 0.003\,$. This
however assumes that all the different fields in the standard model are
independent and that there is no overlap between those fields. We consider
this unlikely, and thus take the somewhat larger value of

$\displaystyle N_{\mathrm{dof},\psi}\ /\ N_{\mathrm{dof,total}}\approx 0.01$
(59)

to be our fiducial choice in the rest of this paper.

###  3.4 Breaking of Lorentz symmetry

Our construction breaks Lorentz invariance for a number of reasons:

  * A)

we assume a hard UV and IR cutoffs;

  * B)

we have employed a naive equal-time version of holography instead of imposing
area scaling on a light-sheet;

  * C)

we have used non-local mode overlaps to build our holographic Fermion field.

We leave it to future work to address point B) by e.g. quantizing our field
directly on a cosmic light-sheet and then imposing holographic mode overlaps
on that sheet. This may also address our need for an IR-cut, since the light-
sheet naturally ends at the cosmic horizon.

In Section 4.2 we show that it is a consequence of point C) that plane waves
in our field have a finite lifetime. In particular, we show that over time the
power in a wave with wave vector $\bm{p}$ is isotropically scrambled to other
modes $\bm{q}$ of similar absolute wave number $|\bm{q}|\approx|\bm{p}|\,$.
Such a breaking of Lorentz symmetry is likely related to ideas of modified
dispersion relations for relativistic particles, which have been proposed as a
generic feature of quantum gravity before (e.g. [57, 58]). At the same time,
we demonstrate that the plane wave lifetime can be made cosmologically large
by choosing a UV-cut that is far below the Planck scale, but still reasonably
above the energies of current particle physics experiments.

This turns point A) into the main cause of our Lorentz symmetry breaking. It
has been argued that quantum fields can be UV-regularized in a Lorentz
invariant manner [59, 60] via dimensional regularisation. But this
regularisation scheme has also been criticised as being a mere mathematical
trick that lacks concrete implementations [61]. Instead of assuming a sharp
cutoff at $k_{\mathrm{UV}}$ we could have modelled our UV-cut as a transition
from a holographic mode density below $k_{\mathrm{UV}}$ to a super-holographic
mode density above $k_{\mathrm{UV}}\,$. This means that instead of squeezing
the $\sim k_{s}^{2}\Delta_{s}$ modes of Fourier space shell $s$ into $\sim
k_{s}\Delta_{s}$ physical modes one would impose a stronger squeezing when
$k_{s}>k_{\mathrm{UV}}\,$. With a sufficiently strong super-holographic
squeezing, this may still satisfy the cosmic Bekenstein bound with a similar
value for $k_{\mathrm{UV}}\,$. We leave it to future work to investigate the
extent to which such a regularisation scheme together with a light-sheet
quantisation would restore Lorentz symmetry at large energies. At the same
time, we would like to stress that there are reasons to believe that the
breaking of Lorentz symmetry is physical [62, 11, 61]. In particular, if
spacetime and local degrees of freedom emerge as approximations to a more
abstract class of quantum theories [43, 63, 64, 65] then also the symmetries
of the emergent spacetime should be approximate. In Section 7 we argue that
such an emergence process would also give rise to stochastic features like the
one present in our random mode overlaps.

##  4 Dynamics of the holographic Weyl field

Let us define

$\displaystyle\hat{\psi}^{c}(x)$
$\displaystyle=\sum_{\bm{p}}\frac{1}{(|\bm{p}|V)^{\frac{1}{2}}}\
\hat{c}_{\bm{p}}(t)\ u(\bm{p})\ e^{i\bm{p}\bm{x}}\ \ ;$
$\displaystyle\hat{\psi}^{d}(x)$
$\displaystyle=\sum_{\bm{p}}\frac{1}{(|\bm{p}|V)^{\frac{1}{2}}}\
\hat{d}_{\bm{p}}^{\dagger}(t)\ u(\bm{p})\ e^{-i\bm{p}\bm{x}}\ ,$ (60)

where we have expanded the field in terms of _time dependent_ ladder operators
$\hat{c}_{\bm{p}}(t)$ and $\hat{d}_{\bm{p}}^{\dagger}(t)$ instead of writing
explicitly what this time dependence is. This is necessary because our use of
overlapping qubits may change time evolution and we may in general have

$\displaystyle\frac{\mathrm{d}\hat{c}_{\bm{p}}}{\mathrm{d}t}\neq-i|\bm{p}|\hat{c}_{\bm{p}}\
\ ,\ \ \frac{\mathrm{d}\hat{d}_{\bm{p}}^{\dagger}}{\mathrm{d}t}\neq
i|\bm{p}|\hat{d}_{\bm{p}}\ .$ (61)

We will investigate one particular choice for the Hamiltonian operator of our
Fermion, namely we will assume that it remains of the form

$\displaystyle\hat{H}$
$\displaystyle=\sum_{\bm{p}}|\bm{p}|\left\\{\left(\hat{c}_{\bm{p}}^{\dagger}\hat{c}_{\bm{p}}-\frac{1}{2}\right)+\left(\hat{d}_{\bm{p}}^{\dagger}\hat{d}_{\bm{p}}-\frac{1}{2}\right)\right\\}$
(62) $\displaystyle\approx\sum_{s}\frac{k_{s}}{2}\sum_{\bm{p}\in
s}\left\\{\hat{\sigma}_{z,\bm{p}}^{c}+\hat{\sigma}_{z,\bm{p}}^{d}\right\\}$
$\displaystyle\equiv\sum_{s}\hat{H}_{s}\ ,$

where in the second line we have made explicit that the field is a sum over
independent (non-overlapping) Fourier space shells $s$, and where we assumed
that all momenta $\bm{p}$ in one shell $s$ have similar absolute values
$|\bm{p}|\approx k_{s}$ . Because of the overlap between the operators
$\hat{c}_{\bm{p}},\,\hat{d}_{\bm{p}}$ in one shell, the spectrum of this
operator is different from the energy spectrum of a standard Weyl field. Also,
the time evolution induced by this Hamiltonian will differ from the standard,
non-overlapping case. In the following, we study instances of both of these
modifications.

###  4.1 Energy spectrum and re-formulation as non-local Heisenberg model

In Section 3 we defined the operator algebra representing an individual
Fourier mode $\bm{p}$ in a Fourier space shell $s$ of the particle sector of
our field as

$\displaystyle\bm{\sigma}_{x,\bm{p}}^{c}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{v_{\bm{p}}}\bm{C}_{j}\
\ ,\ \
\bm{\sigma}_{y,\bm{p}}^{c}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{w_{\bm{p}}}\bm{C}_{j}\
\ ,\ \ \bm{\sigma}_{z,\bm{p}}^{c}$
$\displaystyle=-i\bm{\sigma}_{x,\bm{p}}^{c}\bm{\sigma}_{y,\bm{p}}^{c}\ .$

From these operators we could then define annihilation operators as

$\displaystyle\bm{c}_{\bm{p}}=\frac{1}{2}\left(\bm{\sigma}_{x,\bm{p}}^{c}+i\bm{\sigma}_{y,\bm{p}}^{c}\right)\
.$

This way we constructed a potentially large number of overlapping qubits to
make up our holographic Weyl field. But since the Hilbert space of Fourier
space shell $s$ of our field has dimension $2^{n_{s}}$, we can also decompose
our construction into a set of non-overlapping qubits. Let us e.g. define the
creation and annihilation operators of these qubits as

$\displaystyle\hat{A}_{j}\equiv\frac{1}{2}\left(\bm{C}_{2j-1}+i\bm{C}_{2j}\right)\
,\
\hat{A}_{j}^{\dagger}\equiv\frac{1}{2}\left(\bm{C}_{2j-1}-i\bm{C}_{2j}\right)$
$\displaystyle\Rightarrow\bm{C}_{2j-1}=\hat{A}_{j}+\hat{A}_{j}^{\dagger}\ ,\
\bm{C}_{2j}=-i(\hat{A}_{j}-\hat{A}_{j}^{\dagger})\ .$

In C we show that with these non-overlapping creation and annihilation
operators the Hamiltonian $\hat{H}_{s}$ in each Fourier space shell can be re-
written as

$\displaystyle\hat{H}_{s}$
$\displaystyle=\frac{k_{s}}{2}\left(\hat{A}_{1}^{\dagger},\ \dots\ ,\
\hat{A}_{n}^{\dagger},\ \hat{A}_{n},\ \dots\ ,\
\hat{A}_{1}\right)\bm{K}\pmatrix{\hat{A}_{1}\cr\dots\cr\hat{A}_{n}\cr\hat{A}_{n}^{\dagger}\cr\dots\cr\hat{A}_{1}^{\dagger}}\
,$ (63)

where $\bm{K}$ is a stochastic coefficient matrix which can be characterised
as a generalised Wigner matrix (with the additional property of anti-
persymmetry, cf. C.3). Equation 63 can be considered as defining a non-local
Heisenberg model with general 2-site interactions. In C.1 we show that the
eigenspectrum $\\{\lambda_{H}\\}$ of this Hamiltonian can be deduced from the
positive half of the eigenvalues $\\{\lambda_{K}\\}$ of that coefficient
matrix via

$\displaystyle\\{\lambda_{H}\\}$
$\displaystyle=\left\\{\left.\frac{k_{s}}{2}\sum_{i=1}^{n_{s}}\lambda_{K,i}\,s_{i}\
\right|\ \\{s_{i}\\}\in\\{-1,+1\\}^{n_{s}}\right\\}\ ,$ (64)

i.e. the eigenvalues of $\hat{H}_{s}$ are proportional to sums over the
positive eigenvalues of $\bm{K}$ with all possible sets of prefactors $\pm 1$.
The reason why only positive eigenvalues of $\bm{K}$ appear in this result is
that all eigenvalues $\lambda$ of $\bm{K}$ come in pairs $\pm\lambda$ (because
of the above mentioned anti-persymmetry, cf. C.3).

Up to one additional symmetry, the coefficient matrix $\bm{K}$ is a
generalised Wigner matrix, and the eigenvalues of such matrix ensembles are
very well understood (see e.g. [53, 66, 44]). In particular, a rigidity
property derived by [44] states that in the limit $n_{s}\rightarrow\infty$ the
$i$th of these eigenvalues will be given with increasing accuracy by the
$i/n_{s}$-quantile of a Wigner semi-circle distribution. In C.1 we use this
result to derive that the vacuum energy in a Fourier space shell $s$ as a
function of $k_{s}$, $N_{s}$ and $n_{s}$ is given by

$\displaystyle E_{\min,s}$
$\displaystyle\approx-\frac{N_{s}k_{s}}{2}\cdot\left(\frac{8}{3\pi}\sqrt{\frac{\displaystyle
n_{s}}{\displaystyle N_{s}}}\right)\ .$ (65)

In Figure 5 we have used simulated realisations of overlapping qubits to show
that this result is already accurate for very low numbers of field modes (cf.
A for a description of those simulations). At energies (and mode numbers)
relevant for high-energy physics, it will be even more accurate because of the
asymptotic nature of random matrix theory results [66]. Equation 65 result
represents a strong suppression wrt. the non-overlapping Weyl field, whose
vacuum energy in each shell would simply be $-N_{s}k_{s}/2\,$. When choosing
$\Lambda_{\mathrm{UV}}=\Lambda_{\mathrm{Planck}}$ the total vacuum energy of
the field is lowered by a factor of $\sim 10^{-30}$ (which is however not
enough to alleviate the cosmological constant problem).

###  4.2 Lifetime of plane waves

Figure 6: Comparing our analytical predictions for the lifetime of plane waves
to results obtained from simulated realisations of low numbers of overlapping
qubits. The upper panel displays the expectation value of the bi-commutator of
the occupation number operator $\hat{N}_{1}$, representing e.g. a Fourier mode
$\bm{k}_{1}$, with the total occupation number operator
$\hat{N}_{\mathrm{total}}$ in the Fourier space shell to which $\bm{k}_{1}$
belongs, as a function of the ratio $N_{s}/n_{s}$ (which is proportional to
the shell radius $k_{s}$). This bi-commutator is proportional to the second
time derivative of the expectation value of $\hat{N}_{1}$, and can hence be
used to define a characteristic lifetime for the plane wave state
$\ket{\bm{k}_{1}}$, as we have done in Section 4.2. The expectation values are
taken wrt. the state $\ket{\bm{k}_{1}}$ and wrt. the CRSV-embedding. Symbols
represent results obtained from our simulations (cf. A) and the dashed lines
are the asymptotic results obtained in C.2. The lower panel shows relative
residuals between the two.

To investigate the impact of holography on the dynamics of our field further,
we study the behaviour of plane wave states. Unfortunately, the vacuum state
of our Hamiltonian is not an eigenstate of the occupation number operators
$\hat{N}_{\bm{p}}$ of any of our overlapping Fourier modes $\bm{p}$. As a
consequence, we cannot create a state in which a single mode is excited by
acting on the vacuum state with a creation operator
$\hat{c}_{\bm{p}}^{\dagger}\,$. Instead, we proceed as follows to define a
state which approximates a plane wave excitation of our field with energy
$\bm{p}$:

  * A)

First we consider the subspace of states $\ket{\phi}$ with
$\hat{N}_{\bm{p}}\ket{\phi}=+\ket{\phi}$. These are the states in which we
would measure the occupation number of $\bm{p}$ to be 1.

  * B)

In this subspace, we find the states that minimize
$\left|\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\bra{\phi}\hat{N}_{\bm{p}}\ket{\phi}\right|$.
These plane wave candidate states will be most stable in time. (Note that
$\frac{\mathrm{d}}{\mathrm{d}t}\bra{\phi}\hat{N}_{\bm{p}}\ket{\phi}\equiv 0$
for all eigenstates of $\hat{N}_{\bm{p}}$.)

  * C)

Condition B) is still satisfied by an entire subspace of states (cf. C.2), and
our final selection step is to choose the candidate state that has the lowest
energy expectation value $\bra{\phi}\hat{H}\ket{\phi}$ .

We denote the state defined through conditions A) - C) with $\ket{\bm{p}}$ and
we consider it as the closest equivalent our theory has to the standard plane
wave state $\hat{c}_{\bm{p}}^{\dagger}\ket{0}$ for the non-overlapping Weyl
field. We then estimate a characteristic time scale for the stability of this
plane wave via the equation

$\displaystyle\frac{1}{T_{\mathrm{scramble}}^{2}}\equiv-\mathbb{E}\left\\{\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\bra{\psi_{\mathbf{p}}}\hat{N}_{\mathbf{p}}\ket{\psi_{\mathbf{p}}}\right\\}\
.$ (66)

Here the expectation value $\mathbb{E}\\{\cdot\\}$ is with respect to the
random vectors $\bm{v}_{\bm{p}}$ and $\bm{w}_{\bm{p}}$ that were used to
define the Pauli algebra of the mode $\bm{p}$ in the JLC-embedding (cf.
Section 3). Note that the different Fourier space shells in our construction
do not overlap. Hence, the total occupation number within each individual
shell is conserved, and the power that is lost in the mode $\bm{p}$ over time
is isotropically scrambled to other modes in the same Fourier space shell $s$.
In C.2 we derive analytically that $T_{\mathrm{scramble}}$ for the mode
$\bm{p}$ is given in terms of the parameters of our construction by

$\displaystyle T_{\mathrm{scramble}}\approx
2\pi^{2}\sqrt{\alpha}\left(\frac{\Lambda_{\mathrm{Planck}}}{\Lambda_{\mathrm{UV}}}\right)^{2}\frac{N_{\mathrm{dof},\psi}}{N_{\mathrm{total}}}\sqrt{\frac{L}{k_{s}}}\
,$ (67)

where the quantities appearing in this equation where introduced in Section 3,
and represent

  * •

$\alpha$: relative width of the Fourier space shell $s$,

  * •

$\Lambda_{\mathrm{UV}}/\Lambda_{\mathrm{Planck}}$: UV-cutoff of our field in
Planck units,

  * •

$N_{\mathrm{dof},\psi}$: number of degrees of freedom (qbits) present in our
field

  * •

$N_{\mathrm{total}}$: total number of degrees of freedom present in the
Universe,

  * •

$L$: IR-cutoff,

  * •

$k_{s}$: central radius of the Fourier space shell $s$.

Our analytical results are approximations that hold in the limit
$n_{s}\rightarrow\infty$, $N_{s}\rightarrow\infty$. In Figure 6 we compare
those analytical predictions to results obtained from low-dimensional
simulations of overlapping qubits (cf. A), and we find that they are already
accurate for quite low numbers of qubits - see e.g. the green triangles in
Figure 6, which were obtained from simulating only $12$ overlapping qubits.

Equation 67 is the relation we have used to derive our results in Table 1 and
Figure 4. In Figure 4 we show $cT_{\mathrm{scramble}}$ for different UV-cuts.
In that figure we also indicate the typical energies and travel distances of
neutrinos emitted by a number of identified and resolved sources. The
existence of solar neutrinos seems to demand that neutrino effective field
theory already breaks down at $k_{\mathrm{UV}}<10^{-3}k_{\mathrm{Planck}}$.
This is even more drastic when considering cosmic neutrino observations: we
summarize the UV-cuts required to allow the existence of different observed
fermion signals with know and located sources in Table 1, and the observation
of a $\sim 290\,$TeV neutrino emitted from the blazar TXS 0506+056 [49]
requires $k_{\mathrm{UV}}<470k_{\mathrm{LHC}}$, i.e. only a few orders of
magnitude above modern particle physics experiments. As we discussed in
Section 3.4, the breakdown of our effective theory beyond that energy should
happen either as a sharp cutoff, or as a transition to super-holographic mode
overlaps for modes with $|\bm{k}|>k_{\mathrm{UV}}\,$. This is because any
modes present beyond $k_{\mathrm{UV}}$ need to be squeezed into a small part
of the field’s Hilbert space in order to still satisfy the cosmic Bekenstein
bound.

Note again, that the above considerations assume a ratio of
$N_{\mathrm{dof},\psi}/N_{\mathrm{total}}\sim 0.01$ between the degrees of
freedom present in a single neutrino field and the total degrees of freedom
present in the Universe. In Section 3.3 we argued that this is a realistic
estimate, assuming that Bosonic fields can be modelled via generalised Pauli
operators (GPU [63, 12, 13]) with similar holographic mode overlaps.

###  4.3 Consistency with the cosmic ray spectrum

Figure 7: Top panel: Cosmic ray flux (more concretely the $E^{2}$-Intensity,
which is defined as
$E^{2}\,\mathrm{d}N/\mathrm{d}E\mathrm{d}A\mathrm{d}t\mathrm{d}\Omega$) as a
function of particle energy; using data and tools compiled by [67] as well as
observations reported by [68]. Lower Panel: Zoom into the neutrino
contributions to the cosmic ray spectrum. The grey vertical band indicates the
neutrino UV-cut we have derived in Section 4.2. The width of the band is an
estimate of the mistake we make by assuming a static Universe. But note that a
number of other simplified assumptions may also significantly impact our
results.

As we discussed in the previous subsection, when we interpret our construction
of the holographic Weyl field as a model for neutrinos then the fact that a
$290\,$TeV neutrino has been observed from the far away blazar TXS 0506+056
indicates that neutrino physics should have a UV-cut of
$k_{\mathrm{UV}}<470\,k_{\mathrm{LHC}}$ (where we again take
$k_{\mathrm{LHC}}=14\,$TeV). Above that threshold we expect a rather sharp
cutoff or at least a super-holographic behaviour of the neutrino fields,
because whatever modes are present above $k_{\mathrm{UV}}$ need to be squeezed
into a very small part of the total Hilbert space in order not to violate the
Bekenstein bound.

Of course, our calculation has a number of caveats and relies on a number of
modelling choices. We list several of these in Section 8, but the two main
shortcomings of our model are that it assumes a static background space time
(whereas the cosmos has significantly expanded in the time it took for
neutrinos from TXS 0506+056 to arrive on earth), and that we implement a
naive, equal-time version of holography instead of quantizing our field on a
light-sheet. The UV-cut $k_{\mathrm{UV}}<470k_{\mathrm{LHC}}\,$ may change
once these shortcomings are corrected. But our current result does at least
seem to be consistent with current intensity limits for the cosmic neutrino
flux.

In Figure 7 we have used tools and data compiled by [67] (which includes data
from the databases [69, 70, 71]) as well as observations reported by [68] to
display current measurements of the intensities observed in different cosmic
ray species as a function of particle energy. To date the IceCube experiment
[72] has only established upper limits for neutrino flux beyond the UV-cut we
derived - cf. the blue, open circles in both panels of Figure 7, which
represent direct neutrino detections and the vertical grey band which
indicates our result from the previous subsection. So far, the event that
comes closest to our limit is an indirect detection of an excitation of the
Glashow resonance which presumably took place in Earth’s atmosphere and which
has produced a shower of secondary particles which was observed in IceCube
(cf. orange square; [68]). This event indicates that electroweak theory is
valid until shortly before the energy where our holographic theory for
neutrinos needs to break down. Again, this comparison hinges on a number of
assumptions, part of which we already know to be rather simplistic. To obtain
an estimate of the impact of assuming a static space time, we re-derive our
previous UV-cut when taking the travel distance from TXS0506+056 to be the
distance between us and the blazar at the time when the observed high energy
neutrino was emitted. This indeed moves the UV-cut to higher energies, and the
width of the grey band in Figure 7 represents that shift. Of course, we could
also directly calculate the travel time of neutrinos from TXS0506+056 and use
that to derive a UV-cut. But that would also be a simplification, because we
still wouldn’t properly quantize our field on an expanding background. So for
now, we stick with the above rough estimate of the impact of cosmic expansion.
Another obstacle for further analysis is the stark Lorentz symmetry breaking
of our theory close the the UV-cut Section 3.4. A better approximation of
Lorentz invariance may be achieved by quantizing the field on light-sheets and
by replacing the sharp UV cutoff with a transition to super-holographic mode
overlaps. We leave this for future work.

In the upper panel of Figure 7 we also display measurements of cosmic ray flux
that are attributed to cosmic protons (using data from the Tibet-III [73],
KASCADE-Grande [74] and Telescope Array [75] experiments; as compiled by
[67]). The measurements extend to much higher energies than the holographic
UV-cut we derived for neutrinos, which seems to indicate that in QCD the
breakdown of holography happens at higher energies. There is a number of
subtleties which at the moment prevent us from drawing more concrete
conclusions from this high energy proton flux. First, protons are composite
particles and we did not investigate how to describe such composite states
within our framework. Quark confinement, and the fact that quarks have
fractional electric charges, will plausibly prevent individual proton
constituents from scrambling in the same way that individual neutrino
excitations do. The energy of a proton also does not directly translate to
similar energies for the proton constituents. Finally, it is a possibility
that holography indeed breaks down at different energies for different parts
of the standard model. A direct derivation of a QCD UV scale from the travel
distances of highly energetic protons (as we did in the previous subsection
for neutrinos) is complicated by the fact that high energy protons scatter
with the cosmic microwave background [76, 77]. This mimics to some extent the
plane wave scrambling we have derived here. Though in our construction that
scrambling would not reduce proton energy, but instead create a background of
diffuse, highly energetic particles. For these reasons, we do not take the
existence of proton cosmic rays above our neutrino cutoff as a significant
problem for our model.

Finally, note that some of the high-energy neutrinos detected by IceCube may
be associated with distant gamma ray bursts (GRBs). E.g. a neutrino with an
energy of $\sim 60\,$TeV may be associated with a GRB that was located to be
at redshift $z=1.38$ [58]. If we assume this association to be correct, then
we could also use this pairing to derive a UV-cut for our holographic model of
neutrinos. Estimating our systematic uncertainty due to assuming a static
Universe in the same way as above would lead to an interval
$390\,k_{\mathrm{LHC}}\lesssim k_{\mathrm{UV}}\lesssim
601\,k_{\mathrm{LHC}}\,$, which encompasses our fiducial result
$k_{\mathrm{UV}}\approx 500\,k_{\mathrm{LHC}}$. The authors of [58] also
report an extreme event of a $\sim 1800\,$TeV neutrino that may be associated
with a GRB at a redshift of $z=3.93\,$. Our way of estimating the impact of
cosmic expansion is likely inaccurate for a source at such a high redshift. If
we nevertheless employ that estimate again, and if we assume that the
association of said neutrino with the high redshift GRB is indeed correct, we
arrive at $128\,k_{\mathrm{LHC}}\lesssim k_{\mathrm{UV}}\lesssim
283\,k_{\mathrm{LHC}}\,$. Such a range would indeed be in tension with the
neutrino spectrum displayed in Figure 7, but we deem our calculations
unreliable in this situation. To adequately apply our framework to high-
redshift sources, we would not just need to reformulate it on an expanding
background, but we would also have to implement the covariant entropy bound
(i.e. area scaling on light-sheets) instead of the equal-time area scaling
present in our current model. As mentioned above, we leave this for future
work.

##  5 Real-space anti-commutator

Figure 8: The overlap of Fourier modes generates a non-zero propagator at
large equal-time distances in real space. We show the average amplitude of
this long-range anti-commutator relative to the local anti-commutator when
smoothing the field by different scales $R$ . Even for cosmological smoothing
scales ($R\,\sim 30$ Mpc) this ratio remains at a level of $\sim 1\%$ (cf.
Section 5).

Let us rewrite ($\hat{c}$-part of) the real-space field as

$\displaystyle\hat{\psi}^{c}(x)$ $\displaystyle\approx\sum_{\mathrm{shell}\
s}\frac{1}{(k_{s}V)^{\frac{1}{2}}}\sum_{\bm{p}\in s}\hat{c}_{\bm{p}}(t)\
u(\bm{p})\ e^{i\bm{p}\bm{x}}\ ,$ (68)

where we again made explicit that the field is a sum over independent (non-
overlapping) Fourier space shells $s$, and where we assumed that all momenta
$\bm{p}$ in one shell $s$ have similar absolute values $|\bm{p}|\approx k_{s}$
.

To asses how much the qubit overlap changes the behaviour of our field wrt.
the non-overlapping case, let us consider the equal-time anti-commutator of
$\psi^{c}$ and the $\hat{c}$-part of the momentum field,
$i\psi_{\alpha}^{c}(\bm{x})^{\dagger}$,

$\displaystyle\\{i\psi_{\alpha}^{c}(\bm{x})^{\dagger},\psi_{\beta}^{c}(\bm{y})\\}$
$\displaystyle=i\sum_{\mathrm{shell}\ s}\frac{1}{k_{s}V}\sum_{\bm{p},\bm{q}\in
s}\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}\
u_{\alpha}(\bm{p})^{\dagger}u_{\beta}(\bm{q})\
e^{-i\bm{p}\bm{x}+i\bm{q}\bm{y}}$ (69) $\displaystyle=i\sum_{\mathrm{shell}\
s}\frac{1}{k_{s}V}\sum_{\bm{p},\bm{q}\in
s,\bm{p}\neq\bm{q}}\\{\hat{c}_{\bm{p}}^{\dagger},\hat{c}_{\bm{q}}\\}\
u_{\alpha}(\bm{p})^{\dagger}u_{\beta}(\bm{q})\
e^{-i\bm{p}\bm{x}+i\bm{q}\bm{y}}\ +$ $\displaystyle\ \
+\frac{i}{V}\sum_{\bm{p}}\
\frac{u_{\alpha}(\bm{p})^{\dagger}u_{\beta}(\bm{p})}{|\bm{p}|}\
e^{-i\bm{p}(\bm{x}-\bm{y})}\ .$

In particular, let us look at the trace of this wrt. the spinor components,

$\displaystyle\sum_{\alpha}\\{i\psi_{\alpha}^{c}(\bm{x})^{\dagger},\psi_{\alpha}^{c}(\bm{y})\\}$
$\displaystyle=\frac{i}{2}\sum_{\mathrm{shell}\
s}\frac{1}{k_{s}V}\sum_{\bm{p},\bm{q}\in
s,\bm{p}\neq\bm{q}}\braket{z_{\bm{p}}}{z_{\bm{q}}}\
u(\bm{p})^{\dagger}u(\bm{q})\ e^{-i\bm{p}\bm{x}+i\bm{q}\bm{y}}$ (70)
$\displaystyle\ \ +i\delta_{D}(\bm{x}-\bm{y})$ $\displaystyle\equiv
iC(\bm{x},\bm{y})+i\delta_{D}(\bm{x}-\bm{y})\ .$

The second term in the last line of Equation 70 would be the standard result
in the case of independent qubits, while the first term is a correction due to
qubit overlap. Because of the stochastic way by which we have embedded the
individual qubit operators $\hat{c}_{\bm{p}},\hat{c}_{\bm{p}}^{\dagger}$ into
the physical Hilbert space, the correction $C(\bm{x},\bm{y})$ is in fact a
(classical) random variable. It can be shown to have vanishing expectation
value, and its variance is given by

$\displaystyle\mathbb{E}\left[|C(\bm{x},\bm{y})|^{2}\right]$
$\displaystyle=\frac{1}{4}\sum_{\mathrm{shell}\
s}\frac{\epsilon_{s}^{2}}{k_{s}^{2}V^{2}}\sum_{\bm{p},\bm{q}\in
s,\bm{p}\neq\bm{q}}|u(\bm{p})^{\dagger}u(\bm{q})|^{2}\ ,$ (71)

where $\epsilon_{s}^{2}=\mathbb{E}(|\langle
z_{\bm{p}}|z_{\bm{q}}\rangle|^{2})$ is the variance of the overlap between two
qubits in the same shell. As we show in B.5, this variance is given by

$\epsilon_{s}^{2}=\frac{2}{n_{s}}=\frac{1}{\pi L^{2}k_{s}\Delta_{s}}\ ,$ (72)

where $n_{s}$ is again the effective number of qubits within the physical
Hilbert space of shell $s$ (cf. Section 3). Recall also, that the number of
actual (overlapping) qubits within shell $s$ is given by

$\displaystyle N_{s}=\frac{4\pi k_{s}^{2}\Delta_{s}V}{(2\pi)^{3}}\ .$ (73)

We can hence re-write the variance of $C(\bm{x},\bm{y})$ as

$\displaystyle\mathbb{E}\left[|C(\bm{x},\bm{y})|^{2}\right]$
$\displaystyle=\frac{1}{4}\sum_{\mathrm{shell}\
s}\frac{\epsilon_{s}^{2}}{V^{2}}N_{s}^{2}\frac{1}{N_{s}^{2}}\sum_{\bm{p},\bm{q}\in
s,\bm{p}\neq\bm{q}}\frac{|u(\bm{p})^{\dagger}u(\bm{q})|^{2}}{k_{s}^{2}}$ (74)
$\displaystyle\approx\frac{1}{4}\sum_{\mathrm{shell}\
s}\frac{\epsilon_{s}^{2}}{V^{2}}\left(\frac{4\pi
k_{s}^{2}\Delta_{s}V}{(2\pi)^{3}}\right)^{2}\int\frac{\mathrm{d}\Omega_{\bm{p}}\mathrm{d}\Omega_{\bm{q}}}{(4\pi)^{2}}\frac{|u(\bm{p})^{\dagger}u(\bm{q})|^{2}}{k_{s}^{2}}$
$\displaystyle\equiv\frac{A}{16\pi^{4}}\sum_{\mathrm{shell}\
s}k_{s}^{4}\epsilon_{s}^{2}\Delta_{s}^{2}\ .$

Here we have introduced

$\displaystyle A$
$\displaystyle\equiv\frac{1}{(4\pi)^{2}}\int\mathrm{d}\Omega_{\bm{p}}\mathrm{d}\Omega_{\bm{q}}\frac{|u(\bm{p})^{\dagger}u(\bm{q})|^{2}}{k_{s}^{2}}$
(75)
$\displaystyle=\frac{1}{(4\pi)^{2}}\int\mathrm{d}\theta_{1}\sin\theta_{1}\mathrm{d}\phi_{1}\int\mathrm{d}\theta_{2}\sin\theta_{2}\mathrm{d}\phi_{2}\
\left|\cos\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}+e^{i(\phi_{2}-\phi_{1})}\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}\right|^{2}$
$\displaystyle=\frac{1}{2}\ .$

We can hence conclude that

$\displaystyle\mathbb{E}\left[|C(\bm{x},\bm{y})|^{2}\right]$
$\displaystyle=\frac{1}{32\pi^{5}L^{2}}\sum_{\mathrm{shell}\
s}k_{s}^{3}\Delta_{s}$ (76)
$\displaystyle\approx\frac{1}{32\pi^{5}L^{2}}\int_{0}^{\Lambda_{\mathrm{UV}}}\mathrm{d}k\
k^{3}=\frac{\Lambda_{\mathrm{UV}}^{4}}{128\pi^{5}L^{2}}$
$\displaystyle\Rightarrow\sqrt{\mathbb{E}\left[|C(\bm{x},\bm{y})|^{2}\right]}$
$\displaystyle=\frac{1}{\sqrt{128\pi^{5}}}\frac{\Lambda_{\mathrm{UV}}^{2}}{L}\
.$ (77)

Note that - while the propagator $C$ does depend in $\bm{x}$ and $\bm{y}$ \-
it’s average amplitude between two points is completely independent of those
points and their separation. So our construction induces a long-range
correlation in the spinor field $\psi$ .

To asses the severity of this effect let us introduce a finite smoothing scale
to the real space field, i.e. we consider

$\displaystyle\hat{\psi}_{R}^{c}(x)$ $\displaystyle=\sum_{\mathrm{shell}\
s}\sum_{\bm{p}\in s}\ \frac{e^{-\frac{1}{2}(|\bm{p}|\cdot
R)^{2}}}{(|\bm{p}|V)^{\frac{1}{2}}}\ \hat{c}_{\bm{p}}(t)\ u(\bm{p})\
e^{i\bm{p}\bm{x}}\ ,$ (78)

and similarly for the $\hat{d}$-part of the field. This will change the trace
of the anti-commutator to

$\displaystyle\sum_{\alpha}\\{i\psi_{R,\alpha}^{c}(\bm{x})^{\dagger},\psi_{R,\alpha}^{c}(\bm{y})\\}\
$ $\displaystyle=\ iC_{R}(\bm{x},\bm{y})+\frac{i}{\sqrt{(4\pi)^{3}R^{6}}}\
e^{-\frac{1}{4R^{2}}|\bm{x}-\bm{y}|^{2}}$ (79) $\displaystyle\equiv\
iD_{R}(\bm{x},\bm{y})\ ,$

where the variance of $C_{R}$ is now given by

$\displaystyle\mathbb{E}\left[|C_{R}(\bm{x},\bm{y})|^{2}\right]$
$\displaystyle\approx\frac{1}{32\pi^{5}L^{2}}\int_{0}^{\Lambda_{\mathrm{UV}}}\mathrm{d}k\
k^{3}\ e^{-k^{2}R^{2}}\approx\frac{1}{16\pi^{5}L^{2}R^{4}}$ (80)

as long as $R\gg 1/\Lambda_{\mathrm{UV}}$ . We can then compare the long-range
anti-commutator $C_{R}(\bm{x},\bm{y})$ to the local anti-commutators
$D_{R}(\bm{x},\bm{x})$ and $D_{R}(\bm{y},\bm{y})$ by considering the ratio

$\displaystyle\frac{\mathbb{E}\left[|C_{R}(\bm{x},\bm{y})|^{2}\right]}{\mathbb{E}\left[D_{R}(\bm{x},\bm{x})\right]\
\mathbb{E}\left[D_{R}(\bm{y},\bm{y})\right]}\ =\ \left(\frac{2R}{\pi
L}\right)^{2}\ .$ (81)

So the qubit overlap only significantly impacts the anti-commutation behaviour
of the field on scales $R$ that are comparable to the IR scale $L$ . The above
expression still assumes that our field constitutes the only degrees of
freedom in the Universe. Assuming that the degrees of freedom of all types of
fields (including spacetime degrees of freedom) individually satisfy an area
law scaling, the more general result would be

$\displaystyle\frac{\mathbb{E}\left[|C_{R}(\bm{x},\bm{y})|^{2}\right]}{\mathbb{E}[D_{R}(\bm{x},\bm{x})]\
\mathbb{E}[D_{R}(\bm{y},\bm{y})]}\ =\
\frac{N_{\mathrm{dof,total}}}{N_{\mathrm{dof},\psi}}\left(\frac{2R}{\pi
L}\right)^{2}\ ,$ (82)

where $N_{\mathrm{dof,total}}$ are the total number of degrees of freedom
present in the Universe and $N_{\mathrm{dof},\psi}$ are the degrees of freedom
contributed by the field $\psi$ . Choosing $L$ again as the co-moving size of
our future particle horizon (which is finite in a dark energy dominated
Universe), one would expect a $\sim 1\%$ effect on scales $R\sim 25\
\sqrt{N_{\mathrm{dof},\psi}/N_{\mathrm{dof,total}}}$ Mpc. As before, the
reason why the presence of other degrees of freedom enhances the effects of
holography is that the $N_{s}$ qubits in each shell $s$ then need to be
squeezed into an even smaller number of effective qubits $n_{s}$ in order to
still satisfy the Bekenstein bound.

##  6 Alternative modelling choices

Figure 9: Sketch comparing our fiducial embedding scheme (”CRSV-embedding”) to
an alternative scheme investigated in Section 6. See main text of Section 6.1
for details. Figure 10: Same as Figure 6, but comparing our fiducial model to
alternative analysis choices. See main text of Section 6 for details. Note
that due to numerical limitations we needed to reduce the amount of
simulations run for our alternative Hamiltonian operator, such that for large
ratios $N_{s}/n_{s}$ the results shown here are only averaged over $64$
(instead of $640$) realisations. This is why some of the green triangles
display larger error bars than the simulation results obtained for the other
modelling choices. Figure 11: Same as Figure 5, but comparing our fiducial
model to alternative analysis choices. See main text of Section 6 for details.

###  6.1 non-isotropic overlaps

Our fiducial method of embedding $N$ qubits into a Hilbert space of dimension
$2^{n},\ n<N$, relies on choosing orthonormal pairs of vectors
$\bm{v}_{i},\bm{w}_{i}\ (i=1,\dots,N)$ randomly in $\mathbb{R}^{2n}$ and
defining the Pauli algebra of the $i$th qubit from these vector pairs via
Equations D). The upper panel of Figure 9 displays the $2n\times 2N$ matrix
whose columns consist of such such random vector pairs (in that instance,
$n=4$ and $N=32$). In our fiducial construction each pair
$\bm{v}_{i},\bm{w}_{i}$ represents one Fourier mode of our quantum field in a
thin Fourier space shell, and the scalar products between different vector
pairs determine how much the corresponding Fourier modes overlap.

The above way of constructing operators for the different field modes leads to
mode overlaps that are independent of how far away two wave vectors $\bm{k}$
and $\bm{p}$ are from each other in Fourier space. We want to investigate the
impact of such a non-local overlapping structure on some of our results. To do
so, we study an alternative embedding scheme in which the vectors
$\bm{v}_{i},\bm{w}_{i}$ form a matrix with a sparse structure, as is depicted
in the lower panel of Figure 9. Concretely, this structure is obtained by
populating elements above and below the diagonal of the embedding matrix such
that both horizontally and vertically only one quarter of the matrix elements
are non-zero (and we include non-zero elements in the upper right and lower
left corners to obtain cyclic symmetry). Such an embedding could e.g.
represent a local overlapping structure in a one-dimensional string of field
modes. This would in principle not be appropriate in our case since our
overlapping modes live in Fourier space shells which can be thought of as
approximately two-dimensional. But with this alternative embedding we can
nevertheless qualitatively test for differences between the statistically
isotropic overlaps of our fiducial model and a more local scheme of overlaps.

In Figure 10 we show how this new embedding scheme impacts the lifetime of
plane waves. Modulo a multiplicative factor, this figure displays
$\bra{\bm{k}}\mathrm{d}^{2}\hat{N}_{\bm{k}}/\mathrm{d}t^{2}\ket{\bm{k}}$, i.e.
the expectation value of the second time derivative of the occupation number
operator $\hat{N}_{\bm{k}}$ in a single field mode (modulo a multiplicative
factor) as a function of the energy $|\bm{k}|$ of the plane wave state
$\ket{\bm{k}}$ wrt. which the expectation value is taken (also modulo a
multiplicative factor, since $N_{s}/n_{s}\sim k_{s}$). In Section 4.2 we had
used this second derivative as a definition of $1/T_{\mathrm{scramble}}^{2}$,
where we took $T_{\mathrm{scramble}}$ to represent a characteristic time scale
within which the plane wave will be completely scrambled within its Fourier
space shell. The orange circles in Figure 10 represent values for that second
time derivative obtained from averaging $640$ simulated realisations of the
new embedding scheme, while the blue triangles use our fiducial scheme and the
dashed line represents the analytical prediction derived in C.2. In the limit
of large ratios $N_{s}/n_{s}$ the second time derivative of $\hat{N}_{\bm{k}}$
is larger in the alternative embedding scheme compared to the fiducial scheme
by more than a factor of $2$, indicating a plane wave lifetime that is shorter
by about a factor of $1.5$. This seems to indicate that the statistically
isotropic CJL-embedding leads to more stable plane waves than a more local
embedding scheme. It is however possible that this picture would reverse if we
were to consider wave packets instead of plane waves. Many plane waves will be
superposed in such a packet in a way that is local in Fourier space, and it
may be that local mode overlaps cause such a superposition to become more
stable. At this point neither our analytical nor our simulation tools are
powerful enough to test this hypothesis in a meaningful way, and such an
analysis needs to be carried out in follow-up work.

Finally, we also investigate the impact of the new embedding scheme on the
vacuum energy of our field. Figure 11 compares how that energy is suppressed
in the different embedding schemes, and surprisingly we find that that
suppression is almost identical in both approaches. For the CJL-embedding we
had found in C.1 that the vacuum energy suppression only depends on the ratio
$N_{s}/n_{s}$. Since our alternative embedding cuts down the non-zero elements
of the embedding matrix by the same factor both horizontally and vertically
(cf. our sketch in Figure 9), we speculate that the more local overlap
structure retains the same effective ratio of $N_{s}/n_{s}$, and hence retains
the same amount of vacuum energy suppression. We however do not repeat our
calculations of C.1 for the new embedding scheme, and a more thorough
investigation is again postponed to follow-up work.

###  6.2 alternative Hamiltonian

We also want to consider an alternative, more general form for the Hamiltonian
of our field. In this section we thus take the Hamiltonian in one Fourier
space shell $s$ to be given by

$\displaystyle\hat{H}_{s}$ $\displaystyle=\sum_{\bm{q},\bm{k}\in
s}A_{\bm{q}\bm{k}}\left\\{\left(\hat{c}_{\bm{q}}^{\dagger}\hat{c}_{\bm{k}}-\frac{1}{2}\right)+\left(\hat{d}_{\bm{q}}^{\dagger}\hat{d}_{\bm{k}}-\frac{1}{2}\right)\right\\}\
,$ (83)

where we choose the ansatz
$A_{\bm{q}\bm{k}}=\mathcal{N}_{s}\braket{z_{\bm{k}}}{z_{\bm{q}}}$, such that
the coupling of any two Fourier modes in the Hamiltonian is proportional to
the overlap of these modes within the CRSV-embedding. The time evolution of
$\hat{c}_{\bm{k}}$ with this modified Hamiltonian is given by

$\displaystyle\frac{\mathrm{d}\hat{c}_{\bm{p}}}{\mathrm{d}t}$
$\displaystyle=i\left[\hat{H}_{s},\hat{c}_{\bm{p}}\right]$ (84)
$\displaystyle=\frac{i\mathcal{N}_{s}}{2}\sum_{\bm{q},\bm{k}\in
s}\braket{z_{\bm{k}}}{z_{\bm{q}}}\left\\{\braket{z_{\bm{k}}^{*}}{z_{\bm{p}}}\hat{c}_{\bm{q}}^{\dagger}-\braket{z_{\bm{q}}}{z_{\bm{p}}}\hat{c}_{\bm{k}}\right\\}\
.$

Now remember that we had represented the annihilation operators
$\hat{c}_{\bm{q}}$ by matrices

$\hat{c}_{\bm{q}}=\frac{1}{2}\sum_{j=1}^{2n}\braket{e_{j}}{z_{\bm{q}}}\bm{C}_{j}\
,$ (85)

where $\bm{C}_{j}$ are generators of the Clifford algebra in $2n$ dimensions
and $\ket{e_{j}}$ are an orthonormal basis in $\mathbb{R}^{2n}$. As we had
already discussed in Section 3, Equation 85 constitutes a stochastic
embedding, because the vectors $\ket{z_{\bm{q}}}$ are chosen stochastically in
the CRSV-algorithm. But at least on average we would like the right-hand side
of Equation 84 to equal $-i|\bm{p}|\hat{c}_{\bm{p}}$, which would be the time
derivative of the standard, non-overlapping annihilation operators. This can
be achieved by properly choosing the normalisation factor $\mathcal{N}_{s}\,$.
First, let us calculate the average operator that would appear on the right-
hand side of Equation 84. To do so, we will keep the embedding of the operator
$\hat{c}_{\bm{p}}$ fixed, i.e. we fix the coefficients
$\braket{e_{j}}{z_{\bm{p}}}$, and we will average over the embedding of the
other operators, i.e. over the coefficients $\braket{e_{j}}{z_{\bm{q}}}$,
$\braket{e_{j}}{z_{\bm{k}}}$ with $\bm{q}\neq\bm{p}\neq\bm{k}$. It can then be
shown that

$\displaystyle\mathbb{E}\left[\frac{\mathrm{d}\hat{c}_{\bm{p}}}{\mathrm{d}t}\right]_{\braket{e_{j}}{z_{\bm{p}}}\
\mathrm{fixed}}$ $\displaystyle=-\frac{i\mathcal{N}_{s}}{2}\ \hat{c}_{\bm{p}}\
\sum_{\bm{q}\in s}\mathbb{E}\left[|\langle
z_{\bm{q}}|z_{\bm{p}}\rangle|^{2}\right]_{\braket{e_{j}}{z_{\bm{p}}}\
\mathrm{fixed}}$ (86) $\displaystyle=-2i\mathcal{N}_{s}\hat{c}_{\bm{p}}\
\left\\{\frac{(N_{s}-1)}{n_{s}}+1\right\\}$
$\displaystyle\approx-2i\mathcal{N}_{s}\ \hat{c}_{\bm{p}}\
\left\\{1+\frac{k_{s}L}{2\pi^{3}}\right\\}\ .$

So in order for the average time evolution to equal that of the standard Weyl
field, we need to choose

$\displaystyle\mathcal{N}_{s}=\frac{k_{s}}{2\left(\frac{(N_{s}-1)}{n_{s}}+1\right)}\
.$ (87)

Such a normalisation factor was not necessary in our fiducial Hamiltonian,
because there
$\mathbb{E}[\mathrm{d}\hat{c}_{\bm{p}}/\mathrm{d}t]=-i|\bm{p}|\hat{c}_{\bm{p}}$
was already satisfied to begin with (we do not explicitly show this here, but
it is straight forward to demonstrate).

The green triangles in Figure 10 show
$\bra{\bm{k}}\mathrm{d}^{2}N_{\bm{k}}/\mathrm{d}t^{2}\ket{k}$ as a function of
$N_{s}/n_{s}$ in simulated realisations of overlapping qubits for the
alternative Hamiltonian discussed above. For large ratios $N_{s}/n_{s}$ this
second time derivative seems to approach the same asymptotic behaviour as
plane waves with our fiducial Hamiltonian (blue triangles, and the dashed line
which represents our analytical result from C.2). One may speculate that this
is a result of our choice for the normalisation constant $\mathcal{N}_{s}$
which ensures that the average dynamics of the field modes is the same for our
fiducial and alternative Hamiltonian. We however do not extend our analytical
calculations from C.2 to the more general Hamiltonian and cannot confirm this
speculation at this point. The fact that the lifetime of plane waves doesn’t
seems to be significantly altered between our fiducial Hamiltonian and a more
general one nevertheless lends a degree of robustness to one of our key
results. Of course, we have only considered a rather restricted generalisation
and a much more general class of Hamiltonians should be investigated in
follow-up work. Note in particular that (as we discuss in Section 7) the
ansatz of our paper needs to be more thoroughly motivated by concepts of
quantum mereology, in which a given Hamiltonian dictates which degrees of
freedom are good candidates for semi-classical field degrees of freedom and
thus how the Hamiltonian itself looks like in terms of these degrees of
freedom.

Finally, while plane waves behave quite similarly with our alternative
Hamiltonian, the energy eigenspectrum seems to be significantly altered. The
green triangles in Figure 11 show the suppression of vacuum energy with our
alternative Hamiltonian, and it is much less severe than that present with our
fiducial Hamiltonian. Note also, that the scatter of vacuum energy (wrt.
different realisations of the CRSV-embedding) is much reduced for the
alternative Hamiltonian. Again, we do not attempt to extend our calculations
of C.1 to the more general Hamiltonian and postpone a more thorough analysis
to follow-up work. In particular, at this point it is not clear which form of
the Hamiltonian would be motivated by a more thorough integration of our work
into a quantum mereology framework.

##  7 Why these random overlaps? Context within quantum mereology

In this section, we would like to embed our construction into the context of
_quantum mereology_ [65]. We will argue, that our investigation is in a sense
mereology done the wrong way around. This will also shed light on a peculiar
feature of our model of a holographic Weyl field: the fact that it contains a
stochastic element in the form of randomly chosen embeddings of the field
modes into the smaller, physical Hilbert space.

Let us again consider a thin shell $s$ in Fourier space with radius $k_{s}$
and width $\Delta_{s}\ll k_{s}$. The particle-part of our field (and
similarly, the anti-particle-part) consists of

$\displaystyle N_{s}\,\approx\,\frac{4\pi
k_{s}^{2}\Delta_{s}}{(2\pi/L)^{3}}\,\propto\,k_{s}^{2}\Delta_{s}$ (88)

Fourier modes $\bm{p}\in s$, where $L$ is the IR-cut of our theory. In the
usual Weyl field, each of these modes corresponds to one independent qubit in
the overall Hilbert space. This would however lead to a volume scaling of the
field’s number of degrees of freedom. To achieve an (approximate) area
scaling, we instead sought to represent the field’s Fourier space shell $s$ in
a Hilbert space of dimension $2^{n_{s}}<2^{N_{s}}$ with

$\displaystyle n_{s}\propto k_{s}\Delta_{s}\ .$ (89)

We did so by choosing $2n_{s}$ Clifford generators $\bm{C}_{j}$ in the Hilbert
space of dimension $2^{n_{s}}$ and then representing the Pauli algebra of each
field mode $\bm{p}$ (for the particle-, or $c$-part of the field) as

$\displaystyle\bm{\sigma}_{x,\bm{p}}^{c}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{v_{\bm{p}}}\bm{C}_{j}\
\ ,\ \
\bm{\sigma}_{y,\bm{p}}^{c}\equiv\sum_{j=1}^{2n}\braket{e_{j}}{w_{\bm{p}}}\bm{C}_{j}\
\ ,\ \ \bm{\sigma}_{z,\bm{p}}^{c}$
$\displaystyle=-i\bm{\sigma}_{x,\bm{p}}^{c}\bm{\sigma}_{y,\bm{p}}^{c}\ ,$

where $\braket{e_{j}}{v_{\bm{p}}}$ and $\braket{e_{j}}{w_{\bm{p}}}$ are the
$j$th components of two orthonormal vectors $\bm{v}_{\bm{p}}$ and
$\bm{w}_{\bm{p}}$ that are randomly chosen in $\mathbb{R}^{2n_{s}}$. The
Johnson-Lindenstrauss theorem (cf. our B.3) together with the results of [24]
(cf. our Section 3.1) then guarantees that the field modes for two different
$\bm{p}\neq\bm{q}$ in the same shell $s$ almost anti-commute (up to a small
$\epsilon$, cf. our Equation 53). We have referred to the deviations from
perfect anti-commutation as the overlap between the modes $\bm{p}$ and
$\bm{q}$. That overlap is a stochastic quantity itself, since it is
proportional to the scalar products between the random vectors
$\bm{v}_{\bm{p}},\bm{w}_{\bm{p}}$ and $\bm{v}_{\bm{q}},\bm{w}_{\bm{q}}$ .

Figure 12: Putting our work into the context of the quantum mereology /
quantum first program [63, 65, 43, 64, 21]. The latter conjectures, that
classical physics and in particular classical, local degrees of freedom emerge
from an abstract quantum theory (specified through a Hamiltonian operator in
some Hilbert space) by decomposing the Hilbert space of that theory into
factors that satisfy certain classicality conditions (upper, orange part of
the sketch). Finding and implementing such a mereology algorithm is
challenging [65], and our work (lower, black part of the sketch) does not
quite tackle this task (cf. our discussion in Section 7). But our insights and
the tools we developed can assist future implementations of the quantum
mereology program. And as an additional product, we managed to derive a
concrete phenomenology of the holographic principle.

Why these random overlaps? Why should the true, low-energy EFT of quantum
gravity be described by a model with stochastic ingredients, and what is the
mechanism to choose the random vectors $\bm{v}_{\bm{p}},\bm{w}_{\bm{p}}$ ? We
conjecture that the answer to these questions can be found in the concept of
quantum mereology. In [65] the authors consider general quantum theories,
which are defined by an arbitrary Hilbert space $\mathcal{H}$ as well as a
Hamilton operator $\hat{H}$ on that Hilbert space. The ensemble
$\\{\mathcal{H},\hat{H}\\}$ can be considered as the landscape of all possible
quantum theories. An arbitrarily picked point in that landscape is not
equipped with any pre-defined notion of degrees of freedom, i.e. the Hilbert
space $\mathcal{H}$ does not have any a priori decomposition
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\dots$ into factors
that represent individual degrees of freedom. The authors of [65] also assume
that there are no pre-defined observables other then the Hamiltonian
$\hat{H}$. Instead, they argue that such operators as well as a preferred
Hilbert space factorisation should _emerge_ , by demanding that - under the
evolution dictated by $\hat{H}$ \- the emergent factors and operators should
behave in a quasi-classical way. More concretely, they propose and implement
the following algorithm for quantum mereology:

  * 1.

Choose a candidate factorisation
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\dots$ .

  * 2.

Among all observables of the form
$\hat{O}=\hat{O}_{1}\otimes\hat{O}_{2}\otimes\dots$ , find the one that is
most compatible with the Hamiltonian $\hat{H}$ (i.e. whose commutator with
$\hat{H}$ has the smallest operator norm).

  * 3.

Consider a set of product states
$\\{\ket{\psi}=\ket{\psi}_{1}\otimes\ket{\psi}_{2}\otimes\dots\\}$ that
correspond to peaked configurations in the observables $\hat{O}_{i}$ , i.e.
the $\ket{\psi}_{i}$ are superpositions of the eigenstates of $\hat{O}_{i}$
that are concentrated around one of the eigenvalues of $\hat{O}_{i}$.

  * 4.

Evaluate two measures of classicality for this candidate factorisation: A) how
slowly does entanglement entropy grow when starting in the states
$\\{\ket{\psi}\\}$, and B) how slowly do the peaked states $\\{\ket{\psi}\\}$
disperse (when viewed as wave packets in the eigenspectrum of the operators
$\hat{O}_{i}$). These criteria have been interpreted by [65] as follows: A)
measures the robustness of the observables $\hat{O}_{i}$ towards decoherence,
and B) measures predictability in the sense of asking how long it takes for
peaked states in the observables $\hat{O}_{i}$ to disperse.

  * 5.

Find the factorisation
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\dots$ that maximises
some joint measure of criteria A) and B). This is the emergent, semi-classical
factorisation of $\mathcal{H}$. And the operators $\hat{O}_{i}$ measure
observable degrees of freedom living on the factors $\mathcal{H}_{i}$ .

Originally, [65] implemented their mereology algorithm for the situation where
$\mathcal{H}$ is only split into two factors (a _system_ and an
_environment_), but we want to consider their ideas here in the context of
many degrees of freedom. In particular, we would like to interpret the
$\hat{O}_{i}$ as local degrees of freedom in some effective field theory. To
do that, we would need to complement the original algorithm by some locality
criterion. But such a generalised version of mereology may be difficult to
realise, because it is not guaranteed that an arbitrary Hamiltonian admits a
factorisation in which dynamics are local. The authors of [78] have shown that
such a factorisation rarely exists, and that it is essentially unique if it
exists. Though the situation may be not as dire as these results indicate,
because according to the findings of [79] factorisations in which a given
Hamiltonian appears approximately local do exist for quite general ensembles
of Hamiltonians.

One way to increase the success probability of finding degrees of freedom with
quasi-local dynamics when given an arbitrary Hamiltonian is to make the
mereology algorithm more flexible: instead of insisting that the Hilbert space
$\mathcal{H}$ be split into an exact factorisation and that the observables
$\hat{O}_{i}$ have exactly vanishing (anti-)commutators, we may instead task
mereology with finding operators $\hat{O}_{i}$ that are only quasi-commuting -
similar to the mode operators $\hat{c}_{\bm{p}},\hat{d}_{\bm{p}}$ of our
holographic Weyl field. (Technically, $\hat{c}_{\bm{p}},\hat{d}_{\bm{p}}$ are
of course not local because they are the Fourier space degrees of freedom. But
also the real space degrees of our field suffer random overlaps, as is
demonstrated by our investigation of the real space anti-commutator in Section
5.) In the following, we will assume that the mereology algorithm is given
this additional leeway and that, given a Hamiltonian $\hat{H}$, it identifies
EFT degrees of freedom $\hat{O}_{i}$ that have local dynamics but that can
have small non-zero overlaps.

Now note that we have started the mereology algorithm by randomly picking a
point $(\mathcal{H},\hat{H})$ in the landscape of quantum theories. Even the
points on that landscape with the same Hilbert space $\mathcal{H}$ will still
vary in their Hamiltonian $\hat{H}$, i.e. these points will result in
different EFT degrees of freedom $\hat{O}_{i}$ . So the mereology algorithm we
conjectured above will map an ensemble $\\{\hat{H}\\}$ of Hamiltonian
operators to an ensemble
$\\{\\{\hat{O}_{1}\,,\,\hat{O}_{2}\,,\,\dots\,\\}_{\hat{H}}\\}$ of sets of
quasi-classical degrees of freedom, with each set
$\\{\hat{O}_{1}\,,\,\hat{O}_{2}\,,\,\dots\,\\}_{\hat{H}}$ corresponding to a
different quasi-local EFT. In C, we find evidence that our model of the Weyl
field with overlapping degrees of freedom indeed fits into such a perspective.
We show there that the Hamiltonian $\hat{H}_{s}$ in one Fourier space shell
$s$ of our field can be written as

$\displaystyle\hat{H}_{s}$
$\displaystyle\propto\sum_{j,k=1}^{n}\left(M_{jk}\hat{A}_{j}\hat{A}_{k}+X_{jk}\hat{A}_{j}\hat{A}_{k}^{\dagger}-X_{jk}^{*}\hat{A}_{j}^{\dagger}\hat{A}_{k}-M_{jk}^{*}\hat{A}_{j}^{\dagger}\hat{A}_{k}^{\dagger}\right)\
,$ (90)

where the $\hat{A}_{j},\hat{A}_{j}^{\dagger}$ are non-overlapping creation and
annihilation operators and $M_{jk},X_{ij}$ are random coefficients that depend
on the vectors $\bm{v}_{\bm{p}},\bm{w}_{\bm{p}}$ from above. In particular,
these coefficients have zero mean and identical variances (see C for more
details). So the Hamiltonian above is that of a non-local Heisenberg model
with 2-site interactions and randomly drawn interaction coefficients. This is
a very general ensemble of theories! And nevertheless, with the help of
overlapping degrees of freedom, this can be reformulated into an ensemble of
theories that approximates the standard Weyl field. If we interpret this as an
instance of quantum mereology, then the randomness of the embedding vectors
$\bm{v}_{\bm{p}},\bm{w}_{\bm{p}}$ simply appears to be a result of sampling
from a more general landscape of quantum theories.

There are a number of caveats to this admittedly enthusiastic interpretation
of our results, which we list in the following.

  * •

We have only shown in a quite limited way, that our model indeed approximates
the Weyl field. We demonstrated that plane waves in our field can have cosmic
life times (cf. Section 4), that anti-commutators between Fourier modes are
tiny (cf. Section 3.2), and that notable differences to the real-space
propagator only appear at cosmological scales (cf. Section 5). We have not
studied the full dynamics of our field, and have only limited reason to claim
that the locality of that dynamics is preserved.

  * •

We present a link between a somewhat general ensemble of quantum theories and
an ensemble of theories approximating a quasi-local EFT, but we do not
demonstrate that this link is made via any sort of mereology algorithm.

  * •

While we did reduce our construction to a quite general ensemble of Heisenberg
models, this is still far from sampling the space of all possible
Hamiltonians. In particular, our results are restricted to Heisenberg models
with only 2-site interactions, and we still assume different Fourier space
shells to be uncoupled. (Though note that [79] showed that Hamiltonians with
2-site interactions are approximately unitary equivalent to the ensemble of
all possible Hamiltonians in $\mathbb{C}^{2^{n}}$ with exponentially small
error in the limit $n\rightarrow\infty\,$.)

Part of these caveats are summarised by our sketch in Figure 12. Ideally, the
program of quantum mereology would be implemented by sampling from the
landscape of all possible, abstract Hamiltonians (right hand-side of the
sketch) and then applying a mereology algorithm to re-interpret the latter as
an ensemble of theories with quasi-classical degrees of freedom and dynamics
(left hand-side of the sketch). If we view the process of mereology as
bridging a river between the landscape of abstract quantum theories and the
landscape of potential quasi-classical physics, then the approach of this
paper has rather been the following: we float down the river and throw ropes
to either side of it, in the hopes that they stick somewhere (lower part of
the sketch; and creating a concrete implementation of holography in the
process). As a result we sample neither on the right nor on the left hand-side
of the sketch quite the ensembles of theories we would like to sample.
Nevertheless, we consider our findings as strong evidence, that the mereology
program can work in principle. And the framework of overlapping degrees of
freedom we have developed may assist in future implementations of that
program.

##  8 Discussion

Since Section 1 included a detailed overview of our results, we will summarise
them here only very briefly, and instead focus on open questions. The
essential results are:

  * A)

We have introduced a modified version of the Weyl field whose effective
degrees of freedom obey an approximate area scaling (exact area scaling in
Fourier space), and that satisfies the Bekenstein bound for the total degrees
of freedom inside the cosmic horizon.

  * B)

We find that each pair of Fourier modes of that field still behaves like a
pair of almost independent, non-overlapping degrees of freedom
($|\\{\hat{c}_{\bm{p}},\hat{c}_{\bm{q}}\\}|\sim 10^{-43}$ at LHC energies).
The real space anti-commutator obtains a stochastic, non-local correction
which however becomes significant only on cosmological scales.

  * C)

We define plane wave states in our holographic field and find that they have a
finite lifetime. In particular, the power in a wave of energy $\bm{p}$ is
scrambling to modes of similar absolute value $|\bm{p}|$. We find that this is
not in contradiction to observed cosmological sources of highly energetic
neutrino emission as long as our effective theory breaks down at energies
$k_{\mathrm{UV}}<470k_{\mathrm{LHC}}\,$. We argued that this is still
consistent with current bounds on the validity regime of electroweak theory
from observations of the high-energy cosmic neutrinos. But a potential tension
arises if we naively apply our calculations to a highly energetic neutrino
that according to [58] may be associated with a gamma ray burst detected at a
high redshift ($z\approx 3.93$).

  * D)

We have motivated our construction via quantum mereology, i.e. using the idea
that EFT degrees of freedom should emerge from an abstract theory of quantum
gravity by finding quasi-classical Hilbert space decompositions.

  * E)

We also find that vacuum energy is significantly reduced in our construction,
e.g. by a factor $\sim 10^{-30}$ if $k_{\mathrm{UV}}\sim k_{\mathrm{Planck}}$
and by $\sim 10^{-25}$ if $k_{\mathrm{UV}}\sim 470k_{\mathrm{LHC}}$ (which
corresponds to an overall suppression of $\sim 10^{-74}$ compared to a naive
calculation with $k_{\mathrm{UV}}=k_{\mathrm{Planck}}$ and no holography).
Neither of this is enough to be relevant for the cosmological constant
problem.

All of these findings depend on a number of parameters and modelling choices,
and a large number of questions remain unaddressed. We list those in the
following:

  * •

Several of our results assumed a ratio of
$N_{\mathrm{dof},\psi}/N_{\mathrm{dof,total}}\sim 0.01$, i.e. that our
holographic field makes up $\sim 1\%$ of all degrees of freedom in the
Universe. In Section 3.3 we have only given a heuristic justification for that
number. Our argument did take into account the fact that Bosonic fields may
need to be modelled via generalised Pauli operators (as e.g. done by [63, 12,
13]) and that such fields would contribute significantly more degrees of
freedom to the Universe’s total budget than Fermionic fields. But a more
thorough analysis, that e.g. more closely examines potential overlaps between
different matter fields, is needed.

  * •

While we have considered one alternative to our fiducial Hamiltonian and found
that both Hamiltonians lead to similar plane wave lifetimes, we have not
demonstrated that there isn’t any other alternative Hamiltonian for the
holographic Weyl field that more closely preserves the classical, non-
overlapping dynamics.

  * •

As explained in Section 7, the spirit of our analysis is a reverse (and thus
unsuccessful) version of the quantum mereology program, which aims to describe
the emergence of classical degrees of freedom from abstract quantum theories
(cf. Figure 12). At the same time, the technology we have developed for
describing overlapping degrees of freedom may help future analyses to follow
mereology in the right direction.

  * •

We have studied the dynamics of plane waves, but particle propagation would
more realistically be described via wave packets. The latter would be
described by superpositions of plane waves, but the overlap between the
different Fourier modes of our field may cause such overlaps to display non-
trivial behaviour.

  * •

To define the lifetime of plane waves we have only considered averages of the
second time derivative of mode occupation numbers $\hat{N}_{\bm{p}}\,$.
Significant scatter around those averages could lead to further effective
changes to the dispersion relation of plane waves.

  * •

We have investigated the suppression of vacuum energy of our field. But our
results do in principle allow us to calculate the entire energy eigenspectrum
of the field. A more detailed study of that spectrum may reveal even more
observational signatures of holography.

  * •

We have considered only one alternative to our fiducial way of making degrees
of freedom overlap in such a way that an approximate area scaling is reached.
There maybe a multitude of alternatives, and some of those may preserve
classical, non-overlapping dynamics more closely, especially when
investigating those dynamics beyond plane waves.

  * •

We have implemented a naive, equal-time version of holography instead of
imposing a holographic scaling of degrees of freedom on a light-sheet. We have
also quantized our field in a static, non-expanding background. Both of these
simplification impair the applicability of our calculations to high-redshift
sources such as the ones reported by [58].

  * •

We have imposed a sharp UV cutoff in our model. Instead, UV regularisation
could also take place as a transition from holographic to super-holographic
mode overlaps, such that all modes above the regularisation scale
$k_{\mathrm{UV}}$ are squeezed into a small part of the overall Hilbert space.
Such a regularisation may decrease the severity Lorentz symmetry breaking,
especially when paired with a quantisation that implements the covariant
entropy bound on light-sheets instead of our naive equal-time area scaling.

  * •

According to [8, 41, 42] the degrees of freedom underlying effective field
theory should deplete even more quickly with energy than the naive area
scaling $n_{s}\sim k_{s}\Delta_{s}$ we have assumed. Note also that [42]
investigate a number of observational consequences of EFT mode depletion which
we have not considered. At the same time, a modified version of our
construction may serve as a model that implements the ideas put forward by
[41, 42].

The above points constitute a multitude of open questions and potential
problems for our construction. Nevertheless, the tools we have developed here
may serve to open a fruitful new pathway towards studying observational
signatures of the holographic principle.

©2024. We we would like to thank George Efstathiou, Daniel Gruen, Anik Halder,
Luca Marchetti, Daniele Oriti and Rogerio Rosenfeld for helpful comments and
discussions. O.F. was supported by a Fraunhofer-Schwarzschild-Fellowship at
Universitätssternwarte München (LMU observatory) and by DFG’s Excellence
Cluster ORIGINS (EXC-2094 – 390783311). C.C. acknowledges the support by the
Air Force Office of Scientific Research (FA9550-19-1-0360), the National
Science Foundation (PHY-1733907), and the Commonwealth Cyber Initiative. The
Institute for Quantum Information and Matter is an NSF Physics Frontiers
Center. A.S. acknowledges support by the M. J. Murdock Charitable Trust, and
by Whitman College. We are grateful for the invaluable work of the teams of
the public python packages `NumPy` [80], `SciPy` [81], `CuPy` [82] and
`Matplotlib` [83].

## Data availability

`Python` tools to reproduce our results (analytical predictions and simulated
realisations of overlapping qubits) are publicly available at
https://github.com/OliverFHD/GPUniverse . We also provide data measured in a
suite of simulations of overlapping qubits in that repository.

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