December 2021

HU-EP-21/56-RTG

Homological Quantum Mechanics

Christoph Chiaffrino, Olaf Hohm and Allison F. Pinto

Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We provide a formulation of quantum mechanics based on the cohomology of the
> Batalin-Vilkovisky (BV) algebra. Focusing on quantum-mechanical systems
> without gauge symmetry we introduce a homotopy retract from the chain
> complex of the harmonic oscillator to finite-dimensional phase space. This
> induces a homotopy transfer from the BV algebra to the algebra of functions
> on phase space. Quantum expectation values for a given operator or
> functional are computed by the function whose pullback gives a functional in
> the same cohomology class. This statement is proved in perturbation theory
> by relating the perturbation lemma to Wick’s theorem. We test this method by
> computing two-point functions for the harmonic oscillator for position
> eigenstates and coherent states. Finally, we derive the Unruh effect,
> illustrating that these methods are applicable to quantum field theory.

###### Contents

  1. 1 Introduction
  2. 2 General Approach
    1. 2.1 BV for finite-dimensional Toy Model
    2. 2.2 BV for Quantum Mechanics
    3. 2.3 Homotopy Retract for Harmonic Oscillator
  3. 3 Comparison with Standard Formulations
    1. 3.1 Perturbation Lemma
    2. 3.2 Wick Contractions
    3. 3.3 Path Integral
  4. 4 Harmonic Oscillator
    1. 4.1 Homological Computation for Position Eigenstates
    2. 4.2 Homological Computation for Coherent States
    3. 4.3 Comparison with Operator Language
    4. 4.4 General Boundary Conditions
  5. 5 Unruh Effect
    1. 5.1 Generalities and Homotopy Retract
    2. 5.2 Number Expectation Value
  6. 6 Summary and Outlook
  7. A Homological Algebra

##  1 Introduction

There are two fundamental formulations of quantum mechanics (and, by
extension, of quantum field theory). There is the canonical formulation based
on states as vectors of a Hilbert space and physical observables as self-
adjoint operators acting on this space. This formalism determines amplitudes
from which one then computes probabilities for physical observations. Arguably
this is still the definite formulation of quantum mechanics. Importantly,
however, there is also the Feynman path integral formulation in which the
probability amplitude between states $|q_{i};t_{i}\rangle$ and $\langle
q_{f};t_{f}|$ is represented as

$\langle q_{f};t_{f}|q_{i};t_{i}\rangle=\int{\cal
D}q\,\exp\Big{(}\frac{i}{\hbar}S[q(t)]\Big{)}\;.$ (1.1)

Here $S$ is the classical action for the dynamical variable $q(t)$, and the
integral is to be taken over the space of all trajectories (with fixed initial
value $q_{i}$ and final value $q_{f}$). The path integral formulation is
advantageous in many respects: First, starting from the classical theory
encoded in the action $S$, the path integral directly provides the objects of
physical interest, the probability amplitudes, without having to deal with
Hilbert spaces, states, operators, etc. Second, any symmetries realized as
invariances of the action $S$, such as spacetime Lorentz invariance, are
manifestly realized in the quantum theory (in the absence of anomalies). In
fact, much of modern quantum field theory would be unthinkable without
Feynman’s path integral formulation. The trouble is that it has turned out to
be very difficult, if not impossible, to give a mathematically rigorous
definition of the integral over the space of all kinematically allowed
trajectories. One may discretize the physical problem at hand, in which case
the path integral reduces, after Wick rotation, to well-defined finite-
dimensional Gaussian integrals. This has been applied with great success in
lattice gauge theory, but a general and non-perturbative definition of
Feynman’s path integral remains out of reach.

In this paper we introduce an alternative algebraic formulation of quantum
mechanics based on the (co-)homology of the Batalin-Vilkovisky (BV) algebra
[1, 2]. In this we pick up on a point that to the best of our knowledge was
first made by Gwilliam [3] and by Gwilliam and Johnson-Freyd [4] for finite-
dimensional toy models. We generalize their formulation to be applicable to
genuine physical theories. Like the path integral formulation, this approach
has the advantage of providing a direct way to pass from the classical theory
to the physical quantities of the quantum theory, without having to invoke
Hilbert spaces and the like. However, unlike Feynman’s path integral, whose
rigorous definition would require an infinite-dimensional generalization of
calculus, the formulation presented here is algebraic, employing in particular
methods from algebraic topology. It should be emphasized from the outset that
we do not claim to provide a full-fledged alternative formulation of quantum
mechanics. Rather, at the present moment, this formulation is restricted to
the computation of normalized quantum expectation values with respect to a
certain class of states.

In order to begin explaining this homological approach let us consider a one-
dimensional dynamical system with dynamical variable $q(t)$. The space of all
kinematically allowed trajectories is then the infinite-dimensional vector
space $C^{\infty}(\mathbb{R})$ of smooth functions of one real variable. This
space being infinite-dimensional does not cause any trouble in classical
physics, where the equations of motion effectively make the problem finite-
dimensional: a solution is uniquely determined after picking initial or
boundary conditions, say by fixing the two values $q(t_{i})$ and
$\dot{q}(t_{i})$ at some initial time $t_{i}$. As such, in classical physics
the real space of interest is just $\mathbb{R}^{2}$, which can be viewed as
the phase space of this system. In contrast, in quantum mechanics it is not
sufficient to go ‘on-shell’: the path integral should be taken over all
functions $q(t)$ in $C^{\infty}(\mathbb{R})$, with the ones solving the
classical equations of motion being only the dominant contribution in the
limit of small $\hbar$.

At this stage concepts from topology turn out to be helpful, notably the
notion of homotopy. One considers two shapes or spaces to be homotopy
equivalent if one can be smoothly transformed into the other, as for instance
a closed curve which, without tearing it, can be contracted to a point. Thus,
a one-dimensional curve may be homotopy equivalent to a zero-dimensional
point. Similarly, we will show that passing from the infinite-dimensional
space $C^{\infty}(\mathbb{R})$ of kinematically allowed trajectories to the
finite-dimensional phase space $\mathbb{R}^{2}$ can be viewed as what is
called a homotopy retract. While under homotopy one of course looses some
information, the important fact is that the (co-)homology is homotopy
invariant, this being one of the core techniques in algebraic topology. Since,
as will be argued here, the cohomology encodes the objects of physical
interest in quantum mechanics this technique allows us to circumvent the hard
problem of making sense of the path integral over the infinite-dimensional
space of all trajectories and to give a more direct homological definition of
quantum expectation values.

In order to discuss our main technical results in some detail, we begin with
the algebraic structures encoding the data of physical theories. A
perturbative classical field theory is most directly encoded in a Lie-infinity
or $L_{\infty}$ algebra. This is a differential graded generalization of a Lie
algebra, i.e., a graded vector space $V$ equipped with a differential
$\partial$ obeying $\partial^{2}=0$ and a potentially infinite number of
higher brackets or maps obeying generalized Jacobi identities [7], see [8, 9,
10, 11, 12] for reviews. The subspace $V^{0}$ of degree zero is the ‘space of
fields’ and the subspace $V^{1}$ of degree one is the space in which the
equations of motion live or, in BV language, the space of anti-fields (in the
example above both spaces are $C^{\infty}(\mathbb{R})$). Spaces of higher and
lower degrees then encode gauge symmetries, gauge for gauge symmetries,
Noether identities, etc. [12]. (See also [13, 14, 15] for the $L_{\infty}$
formulation of effective field theory in terms of homotopy transfer.) Given
that $\partial^{2}=0$ we can consider the cohomology of $V$: the space of
$\partial$-closed vectors modulo $\partial$-exact vectors. The cohomology
encodes the on-shell data, i.e. the (perturbative) solutions of the classical
equations of motion modulo gauge transformations.

We can now explain the notion of a homotopy retract [16] (see also [17] for a
pedagogical introduction to the closely related notion of homotopy transfer).
One defines a projector from the infinite-dimensional space of fields to the
finite-dimensional phase space, for instance by projecting a field $\phi$ to
its initial and final values:

$p:V^{0}\rightarrow\mathbb{R}^{2}\;,\qquad
p(\phi)=(\phi(t_{i}),\phi(t_{f}))\;.$ (1.2)

There is also an inclusion map $i:\mathbb{R}^{2}\rightarrow V^{0}$ that
reconstructs the solution from the initial and final values of $\phi$. We have
a homotopy retract if there is a homotopy map $h:V^{1}\rightarrow V^{0}$ from
anti-fields (or equations of motion) to fields so that

$\begin{split}p\circ i&={\rm id}\;,\\\ i\circ p&={\rm
id}-h\circ\partial\;.\end{split}$ (1.3)

The first relation holds by definition of $p$ and $i$, and we will show that
the second relation holds upon defining $h$ in terms of the Green’s function.
These relations tell us that while $p$ has a ‘right-inverse’ it does not have
a ‘left-inverse’; rather, $i$ is only a left-inverse ‘up to homotopy’. This is
as it should be since $p$ is a genuine projector onto the much smaller space
$\mathbb{R}^{2}$. This space in fact equals the cohomology of $V$, which is
homotopy invariant.

In order to use this homotopy retract for quantum mechanics we then pass over
to the closely related BV algebra which is defined on the ‘dual’ space of
smooth functions on $V$ (or rather functionals since $V$ is typically
infinite-dimensional), which we denote by ${\cal F}(V)$.111There is also a
notion of quantum or loop $L_{\infty}$-algebra that is dual to the BV algebra
[7, 18], but in this paper we work with the BV algebra. The algebra structure
is given by the graded commutative and associative product of functions
together with a map $\Delta$, called the BV Laplacian, which obeys
$\Delta^{2}=0$. Importantly, $\Delta$ is not a derivation of the product but
rather is a differential of ‘second order’. This implies that the failure of
$\Delta$ to act as a derivation on the product defines a new structure: the
anti-bracket $\\{\cdot,\cdot\\}$ on functions, which is a graded Lie bracket.
In the conventional BV formalism one defines, given an action $S$, the
homological vector field $Q=\\{S,\;\cdot\;\\}$, which acts as a derivation on
functions and obeys $Q^{2}=0$. From this one defines the BV differential

$\delta:=-i\hbar\Delta+Q\;,$ (1.4)

which satisfies $\delta^{2}=0$, provided the action satisfies the Maurer-
Cartan or master equation $-i\hbar\Delta S+\frac{1}{2}\\{S,S\\}=0$. One also
introduces an odd symplectic form from which the anti-bracket is derived like
the Poisson bracket in standard symplectic geometry. The homological vector
field $Q$ is then the Hamiltonian vector field for the ‘function’ $S$, and the
symplectic form is $Q$-invariant. The BV algebra employed in this paper
deviates from the standard construction in the following subtle way: The BV
differential is defined by (1.4), but $Q$ is not the Hamiltonian vector field
for $S$ but rather equals $\\{S,\;\cdot\;\\}$ only up to boundary terms. The
symplectic form would then not be invariant under $Q$, but in our construction
this is immaterial since the symplectic form makes no appearance. Such
generalizations of the BV formalism were introduced by Cattaneo, Mnev and
Reshetikhin in [19, 20].

The homotopy retract from $V$ to $\mathbb{R}^{2}$ gives rise to a homotopy
retract from the BV algebra on ${\cal F}(V)$ to the space of functions ${\cal
F}(\mathbb{R}^{2})$. The BV algebra is thus transferred to the ordinary
algebra of functions on $\mathbb{R}^{2}$ concentrated in degree zero, with no
non-trivial differential left, but this still encodes the complete cohomology
of $\delta$. Our core technical claim is now that the functions on
$\mathbb{R}^{2}$ in the cohomology space compute quantum expectation values as
follows: Given a functional $F$ of fields whose quantum expectation value we
want to compute (for instance, for a 2-point function one considers
$F=\phi(t)\phi(s)$ for fixed times $t,s$) one determines the functional
$F^{\prime}$ that is equal to $F$ in cohomology (so that $F^{\prime}-F=\delta
G$ for a suitable $G$) and that is just the pullback of a function $f$ on
$\mathbb{R}^{2}$, i.e., $F^{\prime}=f\circ p$, where $p$ is the projector in
(1.2). This function $f$ computes the following normalized quantum expectation
value:

$f(x,y)=\frac{\langle
y;t_{f}|T(F)|x;t_{i}\rangle}{\braket{y;t_{f}}{x;t_{i}}}\,,$ (1.5)

where $T$ denotes the time ordering operator, and $|x;t\rangle$ is the state
satisfying $\hat{\phi}(t)|x;t\rangle=x|x;t\rangle$, and similarly for $y$.
More generally, we introduce a homological procedure to compute such
normalized expectation values with respect to states that are any linear
combination of position and momentum eigenstates.

In perturbation theory, one can give an effective procedure to determine the
function $f$ using the so-called perturbation lemma of homological algebra. In
this case one can prove the above claim by showing that the computations are
equivalent to the familiar techniques based on Wick’s theorem in quantum
perturbation theory. Our formulation is also related to other approaches in
the literature on the homotopy algebra formulation of perturbation theory, see
[21, 22, 23, 24, 25]. While some technical details are similar, and in
particular ref. [22] has been very useful for us, the general homological
formulation presented in this paper turns out to be quite different, as we
will discuss in more detail in the summary section. We will also give a formal
path integral argument supporting the above claim. Most intriguingly, the
homological formulation of quantum mechanics is not restricted to perturbation
theory, yet potentially is mathematically well-defined.

The rest of this paper is organized as follows. In sec. 2 we introduce the
needed background material on BV algebras, and present the main statement of
the homological formulation of quantum mechanics. We also set the stage for
our subsequent applications by giving the homotopy retract for the harmonic
oscillator. In sec. 3 we compare the homological formulation with standard
quantum mechanics in perturbation theory and using Feynman’s path integral. We
then illustrate and apply the homological approach in sec. 4 by computing
expectation values for the harmonic oscillator with respect to position
eigenstates and coherent states, and we verify that the results agree with
those of conventional quantum mechanics. In sec. 5 we show that this approach
is applicable to quantum field theory by re-deriving the Unruh effect, which
to the best of our knowledge provides a new derivation of this effect. We
close in sec. 6 with a summary and outlook. In order to keep the paper self-
contained we include an appendix summarizing the key concepts from homological
algebra.

##  2 General Approach

The goal of this section is to give the homological formulation of quantum
mechanics outlined above. In the first subsection we define the BV algebra and
review the finite-dimensional case following Gwilliam and Johnson-Freyd [3,
4]. We then turn in the second subsection to genuine quantum-mechanical
systems and state the main claim about the homological computation of quantum
expectations values. In the final subsection we introduce a homotopy retract
for the harmonic oscillator, as preparation for the applications in later
sections.

###  2.1 BV for finite-dimensional Toy Model

#### BV algebra

We begin by reviewing the BV algebra and the homological formulation for a
finite-dimensional toy model, following sec. 3 in [3]. Thinking of an ‘action’
function $S(x)$ of a finite number of variables $x^{i}$, $i=1,\ldots,N$, the
BV formalism is defined on the larger space of functions $F(x,x^{*})$ of
$x^{i}$ and new anti-commuting variables $x^{*}_{i}$. We typically consider
functions of the form

$F(x,x^{*})=\sum_{k=0}^{N}f^{i_{1}\ldots i_{k}}(x)x^{*}_{i_{1}}\ldots
x^{*}_{i_{k}}\;,$ (2.1)

where the coefficients are smooth functions of $x$ (for instance polynomials)
and antisymmetric in $i_{1}\ldots i_{k}$. This expansion gives a grading to
the vector space of functions according to the number of $x^{*}$: functions
depending only on $x$ are of degree zero, functions linear in $x^{*}$ are of
degree $-1$, etc. A useful mnemonic is to assign a ghost degree of zero to $x$
and a ghost degree of $-1$ to $x^{*}$. In the following we will display
algebraic relations for homogenous functions of fixed degrees, denoting the
degree of $F$ by $|F|$, etc., and we also write $(-1)^{F}\equiv(-1)^{|F|}$.
More general relations then follow by linearity. The multiplication of
functions equips this space with a graded algebra structure with $F\cdot
G=(-1)^{FG}G\cdot F$.

Since the variables $x^{*}_{i}$ anti-commute one has to be careful with the
notion of derivative: there are left- and right-derivatives which are defined
operationally by the first-order variations

$F(x,x^{*}+\delta x^{*})-F(x,x^{*})=\frac{\partial_{r}F}{\partial
x^{*}_{i}}\delta x^{*}_{i}=\delta x^{*}_{i}\frac{\partial_{l}F}{\partial
x^{*}_{i}}\;,$ (2.2)

so that in general $\frac{\partial_{r}F}{\partial x^{*}_{i}}$ and
$\frac{\partial_{l}F}{\partial x^{*}_{i}}$ differ by a sign:
$\frac{\partial_{r}F}{\partial
x^{*}_{i}}=(-1)^{F+1}\frac{\partial_{l}F}{\partial x^{*}_{i}}$. We follow the
convention that, unless explicitly indicated otherwise, all derivatives are
left derivatives. Also note that second-order derivatives w.r.t. $x^{*}$ are
antisymmetric, $\frac{\partial^{2}}{\partial x^{*}_{i}\partial
x^{*}_{j}}=-\frac{\partial^{2}}{\partial x^{*}_{j}\partial x^{*}_{i}}$. These
derivatives also obey graded Leibniz rules w.r.t. the multiplication of
functions:

$\begin{split}\frac{\partial}{\partial x^{*}_{i}}(F\cdot G)&=\frac{\partial
F}{\partial x^{*}_{i}}\cdot G+(-1)^{F}F\cdot\frac{\partial G}{\partial
x^{*}_{i}}\;.\end{split}$ (2.3)

In the next step one equips this space with a graded Lie bracket and a
differential. The Lie bracket is called the anti-bracket and defined like the
Poisson bracket:

$\\{F,G\\}:=\frac{\partial_{r}F}{\partial x^{*}_{i}}\frac{\partial G}{\partial
x^{i}}-\frac{\partial F}{\partial x^{i}}\frac{\partial_{l}G}{\partial
x^{*}_{i}}\;.$ (2.4)

This bracket has intrinsic degree of $+1$, i.e. $|\\{F,G\\}|=|F|+|G|+1$, and
is graded antisymmetric with the degrees shifted by one, i.e.,

$\\{F,G\\}=-(-1)^{(F+1)(G+1)}\\{G,F\\}\;.$ (2.5)

The anti-bracket obeys the graded Jacobi identity

$\\{\\{F,G\\},H\\}+(-1)^{(F+1)(G+H)}\\{\\{G,H\\},F\\}+(-1)^{(H+1)(F+G)}\\{\\{H,F\\},G\\}=0\;,$
(2.6)

and the following compatibility condition with the product of functions:

$\\{F,GH\\}=\\{F,G\\}H+(-1)^{(F+1)G}G\\{F,H\\}\;.$ (2.7)

There is a differential of intrinsic degree $+1$, called the BV Laplacian,
defined as

$\Delta=-\frac{\partial^{2}}{\partial x^{*}_{i}\partial x^{i}}\;.$ (2.8)

The BV Laplacian squares to zero,

$\Delta^{2}=0\;,$ (2.9)

which is a consequence of ordinary derivatives commuting while derivatives
w.r.t. $x^{*}$ anti-commute. Furthermore, the Laplacian acts as a derivation
on the anti-bracket:

$\Delta\\{F,G\\}=\\{\Delta F,G\\}+(-1)^{F+1}\\{F,\Delta G\\}\;.$ (2.10)

In order to prove this it is convenient to first note that $\Delta$, being a
second-order differential operator, does not act as a derivation w.r.t. the
usual multiplication of functions, but rather the anti-bracket encodes the
failure of $\Delta$ to act as a derivation:

$(-1)^{F}\\{F,G\\}=\Delta(FG)-\Delta FG-(-1)^{F}F\Delta G\;.$ (2.11)

This follows quickly from the definitions (2.4) and (2.8). Acting on this
relation with $\Delta$, using $\Delta^{2}=0$ and this relation again, then
establishes the Leibniz relation (2.10). This state of affairs can be
summarized by saying that the anti-bracket $\\{\cdot,\cdot\\}$ together with
the BV Laplacian $\Delta$ form a differential graded Lie algebra (or an
$L_{\infty}$-algebra without higher brackets). More precisely, here the
grading of the differential graded Lie algebra is given by the above grading
shifted by one, explaining the presence of factors like $F+1$ in many
formulas.

Let us pause here to define the notion of a BV algebra. To motivate this
definition we need to establish another relation. Acting with the Laplacian
(2.8) on the product of three functions yields by a straightforward
computation using (2.3)

$\begin{split}\Delta(FGH)=\;&\Delta(F)GH+(-1)^{F}F\Delta(G)H+(-1)^{F+G}FG\Delta(H)\\\
&+(-1)^{F}\\{F,G\\}H+(-1)^{F+G}F\\{G,H\\}+(-1)^{(F+1)G+F}G\\{F,H\\}\;.\end{split}$
(2.12)

Upon using (2.11) this can be rewritten in a form that employs only the
(graded) multiplication of functions and the BV Laplacian:

$\begin{split}\Delta(FGH)=\;&-\Delta(F)GH-(-1)^{F}F\Delta(G)H-(-1)^{F+G}FG\Delta(H)\\\
&+\Delta(FG)H+(-1)^{F}F\Delta(GH)+(-1)^{(F+1)G}G\Delta(FH)\;.\end{split}$
(2.13)

We can now define a BV algebra:
Definition:
A BV algebra is a vector space equipped with a graded commutative and
associative product and a degree-(+1) map $\Delta$ that satisfies
$\Delta^{2}=0$ and is of second order in the sense of (2.13).
Note that the above definition only refers to the product structure and the
differential $\Delta$, which is the only unconventional ingredient since it
does not obey the Leibniz rule with respect to the product but rather the
‘second-order Leibniz rule’ (2.13). The differential graded Lie algebra
structure for the anti-bracket is then a derived notion, with the anti-bracket
being defined by (2.11). All relations of the differential graded Lie algebra
then follow from the axioms of a BV algebra as does the compatibility relation
(2.7), which can be derived from (2.11) and (2.13).

#### Master Equation and Cohomology

Returning to the application of BV algebras in physical theories we note that
for any differential graded Lie algebra one can write the Maurer-Cartan
equation for a vector (function) of degree one (which in the above grading is
degree zero). An example of such a function is the action $S$, and the Maurer-
Cartan equation reads

$\frac{1}{2}\\{S,S\\}-i\hbar\Delta S=0\;,$ (2.14)

where the differential $\Delta$ was rescaled by $i\hbar$ so that this takes
the form of the BV master equation. Given a solution $S$ of the Maurer-Cartan
equation one can define a new differential $\delta$,

$\delta:=\\{S,\,\cdot\,\\}-i\hbar\Delta\;,$ (2.15)

which squares to zero, $\delta^{2}=0$, if and only if the master equation
(2.14) holds. This follows by a quick computation using $\Delta^{2}=0$, the
Jacobi identity (2.6) and the Leibniz rule (2.10).222Using (2.7) one may
verify that $Q\equiv\\{S,\cdot\\}$ acts as a derivation with respect to the
product, so that with (2.11) the anti-bracket can also be defined in terms of
the full BV differential as  $-i\hbar(-1)^{F}\\{F,G\\}=\delta(FG)-\delta
FG-(-1)^{F}F\delta G\;.$ (2.16) In the present context, an $S$ that does not
depend on $x^{*}$ solves the master equation trivially: both terms in (2.14)
vanish separately. Note, finally, that the master equation (2.14) is also
equivalent to

$\Delta\big{(}e^{\frac{i}{\hbar}S}\big{)}=0\;,$ (2.17)

where the exponential of the degree-zero object $S$ is defined in terms of its
Taylor series. This relation can be verified with (2.11).

Since $\delta^{2}=0$ there is a notion of homology spaces $\frac{{\rm
ker}\,\delta}{{\rm im}\,\delta}$: spaces of $\delta$-closed vectors modulo
$\delta$-exact vectors. Our goal is now to compute this homology for a simple
toy model with one real variable $x$ and action

$S(x)=\frac{1}{2}ax^{2}\;.$ (2.18)

We will see that the homology computes the ‘path integral’ including
$e^{\frac{i}{\hbar}S}$, which here reduce to regular integrals.

We begin by writing out the differential $\delta$ defined in (2.15), using
that here only two variables $x$ and $x^{*}$ with $(x^{*})^{2}=0$ enter:

$\begin{split}\delta&=-\frac{\partial S}{\partial x}\frac{\partial}{\partial
x^{*}}+i\hbar\frac{\partial^{2}}{\partial x\partial x^{*}}\\\
&=-ax\frac{\partial}{\partial x^{*}}+i\hbar\frac{\partial^{2}}{\partial
x\partial x^{*}}\;.\end{split}$ (2.19)

First, we have to determine ${\rm ker}\,\delta$, the space of functions that
are $\delta$-closed. A general function can be written as the superfield

$F(x,x^{*})=f(x)+x^{*}g(x)\;,$ (2.20)

where as above we assume that $f$ and $g$ are polynomials. This yields

$\delta F(x,x^{*})=-axg(x)+i\hbar g^{\prime}(x)\;.$ (2.21)

Setting this to zero gives a differential equation that does not have a non-
trivial solution in polynomials (otherwise one has the solution
$e^{ia\frac{x^{2}}{2\hbar}}$). Therefore, the kernel of $\delta$ is given by
functions (2.20) with $g=0$:

${\rm ker}\,\delta=\\{F(x,x^{*})\equiv f(x)\\}\;.$ (2.22)

Next we have to mod out by ${\rm im}\,\delta$, the space of $\delta$-exact
functions. In order to see the significance of this step let us consider
functions of the form (2.20) with $f=0$ and $g=x^{n}$, which yields with
(2.21)

$\delta(x^{*}x^{n})=-ax^{n+1}+i\hbar nx^{n-1}\;.$ (2.23)

When passing over to homology we identify any two functions that differ by a
$\delta$-exact function. Hence the above relation implies that its right-hand
side is zero in homology or, equivalently, that we have the equivalence

$x^{n+1}\sim\frac{i\hbar}{a}nx^{n-1}\;.$ (2.24)

For instance, $x^{2}$ is equivalent (can be reduced to) a constant,
$x^{2}\sim\frac{i\hbar}{a}$. Similarly, $x^{3}$ can be reduced to $x$, since
$x^{3}\sim\frac{i\hbar}{a}2x$, but noting that $x$ is $\delta$-exact,
$x=-\frac{1}{a}\delta(x^{*})$, this is actually equivalent to zero. Using this
iteratively we have

$x^{n}\ \sim\ \left\\{\begin{array}[]{l l}0&\quad\text{for $n$ odd}\\\
\left(\frac{i\hbar}{a}\right)^{\frac{n}{2}}(n-1)(n-3)\cdot\cdots\cdot
1&\quad\text{for $n$ even }\\\ \end{array}\right.$ (2.25)

The reader may recognize the right-hand side as the ‘quantum expectation
value’ $\langle x^{n}\rangle$ of a zero-dimensional QFT with a single variable
(see, e.g., p. 14 in Zee’s text book [27]). More precisely, let us define the
expectation value for a polynomial $f$ in terms of the convergent Gaussian
integral as

$\langle f\rangle:=\frac{1}{{\cal
N}}\int_{-\infty}^{\infty}dx\,f(x)\,e^{-\frac{ax^{2}}{2\hbar}}\Big{|}_{a\rightarrow-
ia}\;,$ (2.26)

with normalization ${\cal
N}:=\int_{-\infty}^{\infty}dxe^{-\frac{ax^{2}}{2\hbar}}$, where the
substitution $a\rightarrow-ia$ is done after performing the integral. Then the
right-hand side of (2.25) equals $\langle x^{n}\rangle$. More generally, any
polynomial $f$ is equivalent in homology to a complex number, and identifying
its homology class $[f]$ with this number we have

$[f]=\langle f\rangle\;.$ (2.27)

Thus, the homology of the differential $\delta$ computes expectation values.

###  2.2 BV for Quantum Mechanics

We now turn to generic dynamical systems and state our main claim about the
homological formulation of the corresponding quantum theories. For
definiteness let us consider a one-dimensional mechanical model with action

$S[\phi]=\int_{t_{i}}^{t_{f}}\text{d}t\,L(\phi(t),\dot{\phi}(t),t)\,.$ (2.28)

The fields here are smooth functions $\phi$ on the interval $[t_{i},t_{f}]$,
i.e. $\phi\in C^{\infty}([t_{i},t_{f}])$. Formally, such theories include
genuine (quantum) field theories, where the dependence of $\phi$ on spatial
coordinates is suppressed, but for definiteness let us think of ordinary
quantum mechanics.

We start from the BV formalism as in the previous section. This means that the
space of dynamical variables is enlarged to include, in addition to $\phi$,
anti-fields $\phi^{*}$, which we also assume to be smooth, $\phi^{*}\in
C^{\infty}([t_{i},t_{f}])$. Fields and anti-fields are combined into a total
space

$V=V^{0}\oplus V^{1}\;,\qquad V^{0}\equiv C^{\infty}([t_{i},t_{f}])\;,\quad
V^{1}\equiv\Pi C^{\infty}([t_{i},t_{f}])\,,$ (2.29)

which we also sometimes denote by $V^{\bullet}$, to indicate that we refer to
the total space of a chain complex. Moreover, here we used, for any vector
space $X$, the common notation $\Pi X$ to indicate that the elements of $\Pi
X$ have reversed parity compared to $X$. This means, for instance, that taking
the elements $\phi\in C^{\infty}([t_{i},t_{f}])$ to be of even degree, the
elements $\phi^{*}\in\Pi C^{\infty}([t_{i},t_{f}])$ have odd degree, although
in the end they are both just smooth functions. The definition of the BV
algebra given in the previous subsection goes through in the present context,
just with functions replaced by functionals and derivatives replaced by
functional derivatives. Concretely, we consider functionals that are
superpositions of the monomials

$F[\phi,\phi^{*}]=\int\text{d}t_{1}\cdots\text{d}t_{k}\text{d}s_{1}\cdots\text{d}s_{l}f(t_{1},...,t_{k},s_{1},...,s_{l})\phi(t_{1})\cdots\phi(t_{k})\phi^{*}(s_{1})\cdots\phi^{*}(s_{l})\,,$
(2.30)

where the coefficient functions $f(t_{1},...,t_{k},s_{1},...,s_{l})$ are
completely symmetric in the $t_{i}$ and completely antisymmetric in the
$s_{i}$. The degree of such a functional is minus the number of anti-fields
appearing in there, i.e., $|F|=-l$. (The relative minus sign compared to the
degree of $+1$ for $\phi^{*}$ encoded in (2.29) is not a typo but rather due
to functionals being dual to (anti-)fields, hence having the opposite degree.)
We denote this space of functionals by $\mathcal{F}(V)$, which inherits a
grading by the number of $\phi^{*}$, so that we can write ${\cal
F}(V)=\cdots\oplus{\cal F}(V)^{-2}\oplus{\cal F}(V)^{-1}\oplus{\cal
F}(V)^{0}$. The BV algebra structure is defined on this space as in the
previous subsection. For instance, the differential $\delta=Q-i\hbar\Delta$,
which still satisfies $\delta^{2}=0$, consists of two parts. There is the
classical piece (surviving the limit $\hbar\rightarrow 0$) given by the
homological vector field

$Q=-\int_{t_{i}}^{t_{f}}\text{d}t\,EL(\phi(t))\frac{\delta}{\delta\phi^{*}(t)}\,,$
(2.31)

where $EL(\phi(t))=0$ are the Euler Lagrange equations. It also satisfies
$Q^{2}=0$. The second piece is the BV Laplacian

$\Delta=-\int_{t_{i}}^{t_{f}}\text{d}t\,\frac{\delta}{\delta\phi(t)\delta\phi^{*}(t)}\;.$
(2.32)

Note that $Q$ and $\delta$ both decrease the number of $\phi^{*}$ by one and
hence have an intrinsic degree of $+1$. (Sometimes we denote by $Q_{p}$ the
restriction of $Q$ to ${\cal F}(V)^{p}$.)

We have to comment on the following deviation from the standard BV formalism:
in general $\frac{\delta S}{\delta\phi(t)}$ is not equal to $EL(\phi(t))$, and
so in general $Q\neq\\{S,-\\}$. The reason is that these two are only equal up
to boundary terms. (Later it will be crucial to allow for variations along
solutions, and these variations do not vanish on the boundary.) Indeed, we
take the space of fields to be the space of smooth functions defined on the
interval $[t_{i},t_{f}]$ rather than the entire real line. This makes the
action well-defined without having to assume any behavior of the fields at
infinity. However, in contrast to standard treatments of the BV formalism, the
symplectic form $\omega$ inducing the bracket $\\{-,-\\}$ is no longer
invariant under the vector field $Q$ due to boundary terms. An extension of BV
formalism, called BV-BFV formalism, accommodates such features and was
developed by Cattaneo, Mnev and Reshetikhin in [19, 20], but in this work we
never use the symplectic form, so these issues are of no concern to us.

Given that $\delta^{2}=0$ we can define the cohomology of $\mathcal{F}(V)$
with respect to $\delta$, and we will show that this cohomology computes
quantum expectation values. Before turning to this it is instructive to
inspect the cohomology $H^{p}(Q)=\frac{\ker Q_{p}}{\text{im}\,Q_{p-1}}$ of
$\mathcal{F}(V)$ with respect to $Q$, recalling $Q^{2}=0$. Specifically, we
will first establish the following
Claim: The cohomology of $Q$ is isomorphic to the space of “on-shell
functionals”.
Here on-shell functionals refer to functionals on solutions to the equation of
motion $EL(\phi(t))=0$. More precisely, this holds under the assumption that
there are no non-trivial gauge symmetries. In the following we will establish
this statement. To this end let us introduce some notation: We denote the
subspace of solutions by ${\cal E}\subseteq V^{0}$,

$\mathcal{E}=\Big{\\{}\phi\in V^{0}\Big{|}EL(\phi(t))=0\Big{\\}}\;.$ (2.33)

Since $\mathcal{E}$ is a subspace of $V^{0}$, any functional $F$ on $V^{0}$
can be restricted to a functional on $\mathcal{E}$. This defines a projection

$\begin{split}r:\mathcal{F}(V^{0})&\longrightarrow\mathcal{F}(\mathcal{E})\,,\qquad
F\longmapsto F|_{\mathcal{E}}\,.\end{split}$ (2.34)

Let us assume that any functional on $\mathcal{E}$ is obtained in this way. In
other words, any functional on ${\cal E}$ extends to a functional on $V^{0}$.
Then the restriction map $r$ is surjective. In general, however, $r$ is not a
bijection since it may have a non-trivial kernel, but upon modding out the
kernel we have an isomorphism: $\mathcal{F}(V^{0})/{\rm
ker}(r)\cong\mathcal{F}(\mathcal{E})$.

We now compute the cohomologies in order to establish the above claim. We
begin at degree zero and note that functionals of degree zero are annihilated
by $Q$, hence $\ker Q_{0}=\mathcal{F}(V^{0})$. We next show that in degree
zero the image of $Q$ consists of functionals of $\phi$ proportional to the
equation of motion. To see this consider a functional of degree $-1$, i.e.,
$G_{-1}=\int_{t_{i}}^{t_{f}}\text{d}s\,\phi^{*}(s)g_{-1}[\phi,s]$, where
$g_{-1}[\phi,s]=\int dt_{1}\cdots
dt_{k}f_{-1}(t_{1},\ldots,t_{k},s)\phi(t_{1})\cdots\phi(t_{k})$. We then have

$Q(G_{-1})=-\int_{t_{i}}^{t_{f}}\text{d}t\,EL(\phi(t))g_{-1}[\phi,t]\,,$
(2.35)

and so this functional vanishes on ${\cal E}$. Therefore, $Q(G_{-1})$ is in
the kernel of $r$, i.e. $\text{im}\,Q_{-1}\subseteq\ker r$. Assuming that any
functional in the kernel of $r$ is of the form (2.35) (c.f. appendix B in [6]
for a discussion on this assumption), for a suitable function $g_{-1}$, one
has in fact $\text{im}\,Q_{-1}=\ker r$. We then find for the cohomology

$H^{0}(Q)=\frac{\ker Q_{0}}{\text{im}\,Q_{-1}}=\frac{\mathcal{F}(V^{0})}{\ker
r}\cong\mathcal{F}(\mathcal{E})\,.$ (2.36)

So the cohomology in degree zero is in fact equal to the space of functionals
on solutions. This establishes the claim for functionals of degree zero.

Let us now turn to the cohomology in degree minus one. The kernel of $Q_{-1}$
consists of functionals $G_{-1}$ of the form given above (2.35) for which
$Q(G_{-1})=0$. There is a straightforward physical interpretation of this
condition: inspecting the explicit form (2.35) one infers that $Q(G_{-1})=0$
iff $\delta\phi(t)\equiv g_{-1}[\phi,t]$ is a (gauge) invariance of the
action. We next investigate which functionals at degree $-1$ are in the image
of $Q_{-2}$. Consider a generic functional at degree $-2$,

$G_{-2}=\int
ds_{1}ds_{2}\phi^{*}(s_{1})\phi^{*}(s_{2})g_{-2}[\phi,s_{1},s_{2}]\;.$ (2.37)

Computing $Q(G_{-2})$ one obtains functionals of the form $G_{-1}$ given above
(2.35) where

$g_{-1}[\phi,t]=\int_{t_{i}}^{t_{f}}\text{d}s\,EL(\phi(s))g_{-2}[\phi,s,t]\,,\qquad
g_{-2}[\phi,s,t]=-g_{-2}[\phi,t,s]\,.$ (2.38)

Under the interpretation of gauge symmetries these are precisely the trivial
gauge symmetries, which always exist. Their action vanishes once we restrict
to solutions. We therefore learned that the trivial symmetries are in the
image of $Q_{-2}$ and hence that $\frac{\ker Q_{-1}}{\text{im}\,Q_{-2}}$
parametrizes non-trivial gauge symmetries. When there are no non-trivial gauge
symmetries, the cohomology in degree $-1$ therefore vanishes. A similar
argument applies in the case of the cohomologies in arbitrary negative degree.
This completes our discussion of the claim that the cohomology encodes “on-
shell functionals”, i.e., functionals on solutions of the equations of motion.

For the following applications it will be important to have an explicit model
for the space of solutions. Suppose that $\phi_{0}$ is a solution to the
equation of motion. Any such solution is uniquely determined by its boundary
values $x_{i}=\phi(t_{i})$ and $x_{f}=\phi(t_{f})$. We therefore have ${\cal
E}\cong\mathbb{R}^{2}$ and so
$\mathcal{F}(\mathcal{E})\cong\mathcal{F}(\mathbb{R}^{2})$. Any functional on
the space of solutions can be viewed as an ordinary function $f(x_{i},x_{f})$
in two variables.

The space $\mathbb{R}^{2}$ of boundary values naturally embeds into
$V^{\bullet}$. Given $(x_{i},x_{f})\in\mathbb{R}^{2}$, let
$\phi_{x_{i},x_{f}}$ be the unique solution with these boundary conditions,
i.e., we demand that

$(\phi_{x_{i},x_{f}}(t_{i}),\phi_{x_{i},x_{f}}(t_{f}))=(x_{i},x_{f})\,.$
(2.39)

This defines the embedding or inclusion map

$\begin{split}i:\mathbb{R}^{2}\longrightarrow
V^{\bullet}\,,\qquad(x_{i},x_{f})\longmapsto(\phi,\phi^{*})=(\phi_{x_{i},x_{f}},0)\,.\end{split}$
(2.40)

There is also an associated projection

$\begin{split}p:V^{\bullet}\longrightarrow\mathbb{R}^{2}\,,\qquad(\phi,\phi^{*})\longmapsto(\phi(t_{i}),\phi(t_{f}))\,,\end{split}$
(2.41)

satisfying $p\circ i=\text{id}_{\mathbb{R}^{2}}$ according to (2.39).

Given the above maps we can define their pullbacks that act (in the opposite
direction) on the dual spaces of functionals and functions. The pullback of
the inclusion is the map

$i^{*}:{\cal F}(V^{\bullet})\longrightarrow{\cal F}(\mathbb{R}^{2})\;,\qquad
i^{*}(F):=F\circ i\;.$ (2.42)

Thus, this map associates to a functional ${F}$ the function on
$\mathbb{R}^{2}$ defined by $i^{*}(F)(x_{i},x_{f})=F[\phi_{x_{i},x_{f}},0]$.
Similarly, the pullback of the projection is the map

$p^{*}:{\cal F}(\mathbb{R}^{2})\longrightarrow{\cal F}(V^{\bullet})\;,\qquad
p^{*}(f)=f\circ p\;.$ (2.43)

Therefore, any function $f$ defines a functional via
$F[\phi,\phi^{*}]=f(p([\phi,\phi^{*}]))=f(\phi(t_{i}),\phi(t_{f}))$. We say
that a functional $F$ restricts to $\mathbb{R}^{2}$, or that $F$ is an “on-
shell functional”, if it is the pullback of a function on $\mathbb{R}^{2}$. In
order to identify $\frac{\ker Q}{\text{im}\,Q}$ with $\mathbb{R}^{2}$, we can
look for representatives $F$ of equivalence classes $[F]\in\frac{\ker
Q}{\text{im}\,Q}$ that are of that form. Explicitly, given a functional
$F[\phi]$ at degree zero, we can look for a $G[\phi,\phi^{*}]$ of degree $-1$,
such that $F^{\prime}=F+Q(G)$ is the pullback of a function $f$ on
$\mathbb{R}^{2}$ and can be hence be written as $F^{\prime}=f\circ p$.

As an aside we note that the pullback maps $i^{*}$ and $p^{*}$ define what are
called chain maps in the homological language. When we think of
$\mathcal{F}(\mathbb{R}^{2})$ as a complex in degree zero with trivial
cohomological vector field $\tilde{Q}=0$, the maps maps $i^{*}$ and $p^{*}$
obey

$i^{*}\circ Q=\tilde{Q}\circ i^{*}=0\;,\qquad Q\circ
p^{*}=p^{*}\circ\tilde{Q}=0\;,$ (2.44)

which means that

$(QF)\circ i=0,\qquad Q(f\circ p)=0\,.$ (2.45)

The first relation follows since the image of $i$ are the solutions on which
$Q$ vanishes, while the the second relation follows since $f\circ p$ is a
functional of degree $0$.

We now turn to our main statement relating to quantum mechanics. Thinking of
$-i\hbar\Delta$ as a small perturbation, we expect that the cohomology of
$\delta=Q-i\hbar\Delta$ is the same as that of $Q$. We can then search for
functionals in the $\delta$ cohomology class of $F$ that restrict to
$\mathbb{R}^{2}$, i.e. we look for a functional $G$, such that
$F^{\prime}=F+\delta(G)$ and $F^{\prime}=p^{*}(f)=f\circ p$ for some function
$f$ on $\mathbb{R}^{2}$. This function $f$ is unique since $p^{*}$ is
invertible in cohomology. We claim that this function computes a certain
normalized expectation value of $F$. More precisely,

$f(x,y)=\frac{\langle
y;t_{f}|T(F)|x;t_{i}\rangle}{\braket{y;t_{f}}{x;t_{i}}}\,.$ (2.46)

On the right-hand side, we use the language of canonical quantization, where
$T$ denotes the time ordering operation. Furthermore, the state $|x;t\rangle$
is the state that satisfies the eigenvalue equation
$\hat{\phi}(t)|x;t\rangle=x|x;t\rangle$, and similarly for $y$, where
$\hat{\phi}(t)$ is the canonically quantized position operator at time $t$
associated to the classical field $\phi(t)$.

Let us point out that while the above statement was made in the context of a
particular projector mapping the infinite-dimensional vector space $V$ to
$\mathbb{R}^{2}$, this relation between cohomology and quantum expectation
values holds more generally. For instance, we may consider the more general
projectors

$\begin{split}p:\mathcal{F}(V)&\longrightarrow\mathbb{R}^{2}\,,\\\
\phi&\longmapsto(a_{i}\phi(t_{i})+b_{i}\dot{\phi}(t_{i}),a_{f}\phi(t_{f})+b_{f}\dot{\phi}(t_{f}))\,.\end{split}$
(2.47)

In this case, the boundary states change. The incoming state $|x,t_{i}\rangle$
is now an eigenstate of the operator
$a_{i}\hat{\phi}(t_{i})+b_{i}\hat{\pi}(t_{i})$ with eigenvalue $x$, where
$\hat{\pi}(t_{i})$ is the momentum operator at $t=t_{i}$. Similarly, the
outgoing state $\langle y,t_{f}|$ is an eigenstate of
$a_{f}\hat{\phi}(t_{f})+b_{f}\hat{\pi}(t_{f})$ with eigenvalue $y$.

###  2.3 Homotopy Retract for Harmonic Oscillator

In the previous subsection we have argued that for a generic dynamical system
the infinite-dimensional function space of dynamical variables is equivalent
in cohomology to the finite-dimensional phase space (or the space of initial
conditions). Here we discuss further the topological interpretation of this
fact by displaying, for the case of the harmonic oscillator, a so-called
homotopy retract that maps between the corresponding chain complexes. This
will be instrumental for the applications to be discussed below.

The equation of motion for the harmonic oscillator is

$EL(\phi(t))=-(\ddot{\phi}+\omega^{2}\phi)=0\,,$ (2.48)

and defines the BV differential $Q$ according to (2.31). Since the cohomology
of $Q$ describes the space of functionals on the solution space of
$\ddot{\phi}+\omega^{2}\phi=0$, which is isomorphic to the space of initial
conditions $(\phi(t_{i}),\dot{\phi}(t_{i}))\in\mathbb{R}^{2}$, we expect that
the cohomology of $Q$ should be isomorphic to functions on $\mathbb{R}^{2}$.
Instead of working on the spaces of functionals, on which $Q$ acts as a vector
field, we can work directly on the field space encoded in the complex

${0}$${V^{0}}$${V^{1}}$${0\;,}$$\scriptstyle{\partial}$ (2.49)

where we recall that $V^{0}$ and $V^{1}$ are two copies of
$C^{\infty}([t_{i},t_{f}])$. The non-trivial differential is
$\partial(\phi)=\ddot{\phi}+\omega^{2}\phi$.333The space of linear functionals
is the dual space to $V^{\bullet}$, and for a linear $F$ we have
$(QF)[\phi,\phi^{*}]=F[0,\partial(\phi)]$, so that $\partial$ is the map dual
to $Q$ (noting that $\partial(\phi^{*})=0$).  The cohomology of this complex
in degree zero is $\ker\partial$, so it consists of the space of solutions.
Since any solution is determined by its initial condition
$(\phi(t_{i}),\dot{\phi}(t_{i}))$, we can identify the cohomology with the
phase space $\mathbb{R}^{2}$. In degree one, the cohomology is zero, because
any element is $\partial$-exact: for any function $f(t)$ there is a $\phi(t)$
so that

$\partial(\phi)=\ddot{\phi}+\omega^{2}\phi=f\,.$ (2.50)

From the theory of ordinary differential equations we know that a solution,
which we denote by $\phi_{f}$, exists under very mild assumptions on $f$.
Smoothness of $f$ ensures that $\phi_{f}$ is also smooth. We can give an
explicit solution to (2.50) by means of a Green’s function:

$\phi_{f}(t)=\int_{t_{i}}^{t_{f}}\text{d}s\,K(t,s)\,f(s)\;.$ (2.51)

Explicitly, the corresponding kernel function reads

$K(t,s):=\theta(t-s)\frac{\sin\omega(t-s)}{\omega}\;,$ (2.52)

with $\theta$ the step function, and it satisfies

$(\partial^{2}_{t}+\omega^{2})K(t,s)=\delta(t-s)\;.$ (2.53)

Of course, $\phi_{f}(t)$ is unique only up to homogeneous solutions, i.e.
solutions to (2.48). We picked a solution such that
$\phi_{f}(t_{i})=\dot{\phi}_{f}(t_{i})=0$.

Our above analysis shows that the cohomology space
$\frac{\ker\partial}{\text{Im}\,\partial}$ is isomorphic to the phase space
$\mathbb{R}^{2}$, which we can think of as a chain complex in degree zero with
trivial differential. We have the following maps called quasi-isomorphisms:

${0}$${V^{0}}$${V^{1}}$${0}$${0}$${\mathbb{R}^{2}}$${0}$${0}$$\scriptstyle{\partial}$$\scriptstyle{p}$$\scriptstyle{0}$$\scriptstyle{0}$
(2.54)

where

$p(\phi)=(\phi(t_{i}),\dot{\phi}(t_{i}))\,.$ (2.55)

There is also a quasi-isomorphism $i:\mathbb{R}^{2}\rightarrow V^{0}$ in the
other direction, defined by

$i(q,p)=q\cos\omega(t-t_{i})+\frac{p}{\omega}\sin\omega(t-t_{i})\,.$ (2.56)

This obeys $p\circ i=1$.

In the following we will construct a so-called homotopy retract of the
original chain complex $V^{\bullet}$ to its cohomology $\mathbb{R}^{2}$. To
this end we interpret the Green’s function (2.51) as a map $h:V^{1}\rightarrow
V^{0}$, i.e., for any function $f\in V^{1}$ we define

$h(f)(t)=\int_{t_{i}}^{t_{f}}\text{d}s\,K(t,s)\,f(s)\;.$ (2.57)

We can extend $h$ to the full complex $V^{\bullet}$ by defining $h(\phi)=0$
for any field $\phi\in V^{0}$. The map $h$ is called a homotopy retract from
the original complex $V^{\bullet}$ to its cohomology $\mathbb{R}^{2}$. In
general, we say that a complex $(Y,\partial_{Y})$ is a homotopy retract of
$(X,\partial_{X})$ if there are chain maps
$i:(Y,\partial_{Y})\rightarrow(X,\partial_{X})$ and
$p:(X,\partial_{X})\rightarrow(Y,\partial_{Y})$, such that $p\circ i=1$ and

$1-i\circ p=\\{\partial,h\\}=\partial\circ h+h\circ\partial\,.$ (2.58)

A direct computation with $p$ and $i$ defined in (2.55) and (2.56) shows that

$\partial h(f)=f\,,\qquad h\partial(\phi)=\phi-ip(\phi)\,.$ (2.59)

More specifically, the first relation is a direct consequence of the property
(2.53), while the second relation requires two integrations by part in order
to use that the Green’s function obeys the analogue of (2.53) with respect to
its second argument. In this computation, one picks up boundary terms that
precisely constitute $-ip(\phi)$. We thus obtain the homotopy relation (2.58)
if we define $p$ and $i$ to be zero on $V^{1}$. This shows that the maps from
$V^{\bullet}$ to $\mathbb{R}^{2}$ define a homotopy retract.

We should emphasize that the above choice of Green’s function and homotopy is
just one of many. The kernel $K$ of the above retarded Green’s function is, in
particular, not symmetric under $t\leftrightarrow s$. We can also choose
Dirichlet boundary conditions at $t_{i}$ and $t_{f}$, for which the kernel
$K_{DD}(t,s)$ is symmetric and given by

$K_{DD}(t,s)=\theta(t-s)\frac{\sin\omega(t-s)}{\omega}-\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})}\frac{\sin\omega(t_{f}-s)}{\omega}\,.$
(2.60)

This yields the homotopy map

$h(f)=\int_{t_{i}}^{t}\text{d}s\,f(s)\frac{\sin\omega(t-s)}{\omega}-\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})}\int_{t_{i}}^{t_{f}}\text{d}s\,f(s)\frac{\sin\omega(t_{f}-s)}{\omega}\,.$
(2.61)

Although in the form (2.60) the symmetry is not manifest, a non-trivial
computation shows $K_{DD}(t,s)=K_{DD}(s,t)$.444To this end one uses that the
step function obeys $\theta(t-s)=1-\theta(s-t)$ in order to create the first
term with $s$ and $t$ interchanged. The remaining terms can then be rewritten
by use of the identity  $\sin(x-y)=\sin x\,\cos y-\cos x\,\sin y\;,$ (2.62)
after which all terms can be recombined to yield $K_{DD}(s,t)$.  The
corresponding chain maps are

$p(\phi)=(\phi(t_{i}),\phi(t_{f}))\,,$ (2.63)

and

$i(x_{i},x_{f})=x_{i}\frac{\sin\omega(t_{f}-t)}{\sin\omega(t_{f}-t_{i})}+x_{f}\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})}\,.$
(2.64)

As a next step we lift the homotopy retract from the underlying spaces
$V^{\bullet}$ and $\mathbb{R}^{2}$ to the BV complex, i.e., to the space of
functionals on $V^{\bullet}$ and the space of functions on $\mathbb{R}^{2}$.
We recall that the differential on this complex is given by the BV
differential, which reads for the harmonic oscillator

$Q=\int_{t_{i}}^{t_{f}}\text{d}t(\ddot{\phi}(t)+\omega^{2}\phi(t))\frac{\delta}{\delta\phi^{*}(t)}\,.$
(2.65)

In order to realize the homotopy retract on the BV complex we have to use
that, as explained above, the maps $i$ and $p$ can be applied to functionals
via pullback to give $I:=p^{*}$ and $P:=i^{*}$. Specifically, for a functional
$F[\phi,\phi^{*}]$ in $\mathcal{F}(V^{\bullet})$ we obtain the function on
$\mathbb{R}^{2}$:

$P(F)(q,p):=i^{*}(F)(q,p)=F[i(q,p),0]\,,$ (2.66)

while for a function $f\in\mathcal{F}(\mathbb{R}^{2})$ we obtain the
functional on $V^{\bullet}$

$I(f)[\phi,\phi^{*}]:=p^{*}(f)[\phi,\phi^{*}]=f(p(\phi))=f(\phi(t_{i}),\phi^{*}(t_{i}))\,.$
(2.67)

We next have to show how to use the homotopy $h$ on $V^{\bullet}$ to define a
homotopy on $\mathcal{F}(V)$.

We first consider the linear functionals $\phi(t)$ and $\phi^{*}(t)$. To
clarify this notation let us point out that the symbol ‘$\phi(t)$’ can be
interpreted in two natural ways. The standard interpretation is that of a
function $\phi$ that takes different values depending on the variable $t$:
here we think of $\phi$ as fixed and of $t$ as a variable. The second
interpretation of ‘$\phi(t)$’ takes $t$ to be fixed but $\phi$ to be variable.
This then defines a functional: a map that assigns a number to each function
$\phi$, namely the number $\phi(t)$ obtained by evaluating the function $\phi$
at the fixed $t$. Similar remarks apply to the interpretation of
‘$\phi^{*}(t)$’ as a function or as a functional. A subtlety of this notation
is that the degree of $\phi^{*}(t)$ depends on its interpretation: as a
function it has degree $+1$, but as a functional it has degree $-1$, c.f. the
discussion after (2.30) above. It will always be clear from the context which
interpretation applies.

Returning to the problem of defining a homotopy map $H$ on functionals we
first define $H$ on the linear functionals $\phi(t)$ and $\phi^{*}(t)$ by

$\begin{split}H(\phi(t))&:=\int_{t_{i}}^{t}\text{d}s\,\frac{\sin\omega(t-s)}{\omega}\phi^{*}(s)\equiv\int_{t_{i}}^{t_{f}}\text{d}s\,K(t,s)\,\phi^{*}(s)\,,\\\
H(\phi^{*}(t))&:=0\;.\end{split}$ (2.68)

Note that $H(\phi(t))$ does not actually depend on $\phi(t)$. The functional
$H(\phi(t))$ maps a given anti-field $\phi^{*}$ to the number that is given by
the integral on the right-hand side. This functional is linear in $\phi^{*}$
and so has degree $-1$, as it should be since $H$ has intrinsic degree $-1$.
It is now straightforward to show that

$\begin{split}QH(\phi(t))&=\phi(t)-p^{*}i^{*}\phi(t)\;,\\\
HQ(\phi^{*}(t))&=\phi^{*}(t)\,.\end{split}$ (2.69)

To obtain a homotopy on all functionals, we observe that any polynomial
functional is a superposition of products of the functionals $\phi(t)$ and
$\phi^{*}(t)$. So it is enough to know how the homotopy distributes over any
product of functionals. We set

$H(FG)=\frac{1}{2}(H(F)G+(-)^{F}p^{*}i^{*}(F)H(G)+(-)^{FG}\big{\\{}H(G)F+(-)^{G}p^{*}i^{*}(G)H(F)\big{\\}}\,.$
(2.70)

Note that we did a graded symmetrization over $F$ and $G$ on the right-hand
side, which is necessary since $H(FG)$ should be graded symmetric in $F$ and
$G$. The definition of $H$ ensures that the homotopy relation

$QH+HQ=1-p^{*}i^{*}\equiv 1-IP$ (2.71)

is satisfied on products $FG$, as long as it holds on $F$ and $G$
individually. The action of $H$ on any functional $F$ can then by successively
reduced to its action on $\phi(t)$ and $\phi^{*}(t)$, where we define it via
(2.68). This shows that $p^{*}i^{*}$ is homotopic to the identity. So
$\mathcal{F}(V^{\bullet})$ is homotopic to $\mathcal{F}(\mathbb{R}^{2})$. We
therefore proved that the space of functionals on $V^{\bullet}$ is quasi-
isomorphic to the space of functions on phase space $\mathbb{R}^{2}$.

##  3 Comparison with Standard Formulations

In this section we verify from various angles the main statement (2.46) that
relates the cohomology of the BV differential to quantum expectation values.
First, we explain the perturbation lemma and show that it relates to Wick
contractions in the familiar formulation of quantum mechanics. Second, we give
a heuristic argument based on the path integral formulation.

###  3.1 Perturbation Lemma

The homological perturbation lemma provides a recipe to compute the cohomology
of the BV differential and in particular the representative $F^{\prime}$
appearing in the main statement around (2.46). The perturbation lemma states
the following. Suppose that we have homotopic complexes $(X,\text{d}_{X})$ and
$(Y,\text{d}_{Y})$. This means that there are quasi-isomorphisms
$p:(X,\text{d}_{X})\rightarrow(Y,\text{d}_{Y})$ and
$i:(Y,\text{d}_{Y})\rightarrow(X,\text{d}_{X})$, together with a homotopy
$h:(X,\text{d}_{X})\rightarrow(X,\text{d}_{X})$, such that

$\text{d}_{X}\circ h+h\circ\text{d}_{X}=1-i\circ p\,.$ (3.1)

Now assume that we perturb the differential of $X$ by some $\eta$, such that
we still have a chain complex, i.e. $\text{d}_{X}^{\prime}:=\text{d}_{X}+\eta$
obeys $(\text{d}_{X}^{\prime})^{2}=0$. Then there is a differential
$\text{d}_{Y}^{\prime}$ on $Y$, so that $(Y,\text{d}_{Y}^{\prime})$ is still
homotopic to $(X,\text{d}_{X}^{\prime})$. This means that there are new quasi-
isomorphisms
$p^{\prime}:(X,\text{d}_{X}^{\prime})\rightarrow(Y,\text{d}_{Y}^{\prime})$ and
$i^{\prime}:(Y,\text{d}^{\prime}_{Y})\rightarrow(X,\text{d}_{X}^{\prime})$
with homotopy
$h^{\prime}:(X,\text{d}_{X}^{\prime})\rightarrow(X,\text{d}_{X}^{\prime})$
satisfying the homotopy relation

$\text{d}_{X}^{\prime}\circ
h^{\prime}+h^{\prime}\circ\text{d}_{X}^{\prime}=1-i^{\prime}\circ
p^{\prime}\,.$ (3.2)

The perturbation lemma provides explicit formulas for the perturbed data:

$\begin{split}p^{\prime}&=p\circ\sum_{n\geq 0}(-\eta
h)^{n}\,,\qquad\;\;\;i^{\prime}=\sum_{n\geq 0}(-h\eta)^{n}\circ i\,,\\\
h^{\prime}&=h\circ\sum_{n\geq 0}(-\eta
h)^{n}\,,\qquad\quad\text{d}_{Y}^{\prime}=\text{d}_{Y}+p\circ\sum_{n\geq
1}(-\eta h)^{n}\circ\eta\circ i\,.\end{split}$ (3.3)

Here the perturbed differential $\text{d}_{Y}^{\prime}$ is such that the
perturbed projection and inclusion are chain maps, which means that

$\text{d}_{Y}^{\prime}p^{\prime}=p^{\prime}\text{d}_{X}^{\prime}\;,\qquad\text{d}_{X}^{\prime}i^{\prime}=i^{\prime}\text{d}_{Y}^{\prime}\;.$
(3.4)

If the complexes have additional structure, we want the perturbations to
preserve these. For instance, the space of functionals forms an algebra under
multiplication, and the cohomological vector field $Q=\\{S,-\\}$ acts as a
derivation. In order to preserve these properties we first have to assume that
the original homotopy data preserves these. So $i$ and $p$ have to be algebra
morphisms, i.e. $i(FG)=i(F)i(G)$ and $p(FG)=p(F)p(G)$. In addition, we have to
assume that $h$ is a _strong deformation retract_ , which means that $h$ is
subject to the side conditions

$p\circ i=1\,,\quad h^{2}=0\,,\quad p\circ h=0\,,\quad h\circ i=0\,.$ (3.5)

If these conditions are met, the perturbed chain maps $p^{\prime}$ and
$i^{\prime}$ will again be algebra morphisms, and $\text{d}_{Y}^{\prime}$ will
be a derivation.

We may first apply the homological perturbation lemma to a classical
mechanical system. Given an action $S$, we split it as $S_{0}+S_{I}$, where
$S_{0}$ is a free theory (quadratic in fields) and $S_{I}$ is the interaction
term. We now think of $(\mathcal{F}(V),Q_{0}=\\{S_{0},-\\})$ as our initial
complex, which is based on the free action (a sum of harmonic oscillators). We
saw that a Green’s function defines a homotopy on the complex
$(V^{\bullet},\partial)$ to phase space, satisfying $\\{\partial,h\\}=1-i\circ
p$, which in turn defines a homotopy $H$ from $\mathcal{F}(V)$ to
$\mathcal{F}(\mathbb{R}^{2})$, c.f. (2.70), satisfying

$QH+HQ=1-IP\,,\quad I=p^{*}\,,\quad P=i^{*}\,.$ (3.6)

Such a homotopy in terms of a Green’s function defines a strong deformation
retract, i.e. the analogue of (3.5) holds for $H$. We then take the
interaction term $Q_{I}=\\{S_{I},-\\}$ as the perturbation. Since the
functions on solution space are concentrated on degree zero, there will be no
induced differential, but the space $(\mathcal{F}(\mathbb{R}^{2}),0)$ is still
a trivial complex. Since $Q_{0}+Q_{I}$ encodes the full interacting equations
of motion $P^{\prime}$ projects functionals to solutions of the full equations
of motion. So this is the role of the projector $P^{\prime}$: it
perturbatively constructs solutions of the interacting theory using the
Green’s function and a given solution of the free theory.

We now turn to our core application for the quantum case. The perturbation
lemma allows us to show that in perturbation theory any functional $F$ of
degree zero has a representative $F^{\prime}$ in the cohomology of
$\delta=\\{S,-\\}-i\hbar\Delta$ that can be written as $F^{\prime}=f\circ
p=I(f)$ for some function $f$ on phase space. We use a Green’s function to
define a homotopy and view $\eta=\\{S_{I},-\\}-i\hbar\Delta$ as a
perturbation. With the perturbation formulas (3.3) we obtain the new chain
maps

$P^{\prime}=P\sum_{n\geq 0}(-\eta H)^{n}\,,\qquad I^{\prime}=I\,,$ (3.7)

where the second equation is true by degree reasons.555Note that $I$ being
unperturbed implies that, in the dual picture, $p$ is unperturbed, which means
that $p$ does not depend on the interactions. We then define

$F^{\prime}:=I^{\prime}P^{\prime}(F)\;,\qquad f:=P^{\prime}(F)\;.$ (3.8)

The claim is that these are the desired functional and function. Indeed,
$F^{\prime}$ is in the same cohomology class as $F$, since

$F-F^{\prime}=(1-I^{\prime}P^{\prime})(F)=\delta(H^{\prime}(F))\,,$ (3.9)

where we used the homotopy relation together with $H(\delta(F))=0$ for degree
reasons. So the objects required by the statement around (2.46) are given by
$F^{\prime}=I(f)=f\circ p$ and $G=-H^{\prime}(F)$.

###  3.2 Wick Contractions

The perturbation lemma discussed above gives a complete perturbative solution
to the problem of determining the functional $F^{\prime}$ that in cohomology
is equivalent to $F$, together with the function $f$. However, the required
action of the homotopy map $H$ given in (2.70) is inconvenient for
computations due to the explicit symmetrization. The action of the homotopy
can be brought to a more manageable form by splitting the field space into
off-shell and on-shell fields. This has the additional advantage of relating
the perturbation lemma to standard computations in quantum field theory using
Wick contractions. More precisely, for the special case that the homotopy is
given by the Feynman propagator, the perturbation lemma amounts to Wick’s
theorem, but the perturbation lemma is more general, as we will see in the
next section.

We begin by decomposing the field space by using the projection $p$ and
inclusion $i$:

$V=ip(V)\oplus(1-ip)(V)=:V_{p}\oplus V_{u}\;,$ (3.10)

where $V_{p}$ is the ‘physical’ subspace and $V_{u}$ is the ‘unphysical’
subspace. Any field can now be written as $\phi=\phi_{p}+\phi_{u}$, where
$\phi_{p}$ is a solution to the equations of motion. Note that $\phi_{u}$
depends on the choice of homotopy and satisfies the same boundary conditions
as the Green’s function used to construct the homotopy. In the dual picture of
functionals on $V$ we have the induced decomposition

$\mathcal{F}(V)=:\mathcal{F}_{p}\oplus\mathcal{F}_{u}\,,$ (3.11)

where $\mathcal{F}_{p}=\mathcal{F}(V_{p})$ is the space of functionals
depending on $\phi_{p}$ only. More precisely, these are the functionals of the
form (2.30) where all $\phi$ are $\phi_{p}$ and there are no $\phi^{*}$.
Correspondingly, $\mathcal{F}_{u}$ is the space of functionals that contain at
least one $\phi_{u}$ or at least one $\phi^{*}$.

Next we consider the above decomposition at the level of the (functional)
derivatives defining the BV algebra. Recall that the space of physical or on-
shell fields is isomorphic to a finite-dimensional space (in the present case
$V_{p}\cong\mathbb{R}^{2}$), so that we can use ordinary coordinates $(x,y)$
of $\mathbb{R}^{2}$ to label a solution $\phi_{p;x,y}$. This allows us to
introduce partial derivatives along solutions:

$(\partial_{x}{F})[\phi,\phi_{*}]:=\left.\frac{\text{d}}{\text{d}\varepsilon}\right|_{\varepsilon=0}{F}[\phi+\phi_{p;x+\varepsilon,y},\phi_{*}]\,,\qquad(\partial_{y}{F})[\phi]:=\left.\frac{\text{d}}{\text{d}\varepsilon}\right|_{\varepsilon=0}{F}[\phi+\phi_{p;x,y+\varepsilon},\phi_{*}]\,.$
(3.12)

We also introduce a functional derivative in the direction of $V_{u}$ via

$\int\text{d}t\,g_{u}(t)\frac{\delta{F}[\phi,\phi^{*}]}{\delta\phi_{u}(t)}:=\left.\frac{\text{d}}{\text{d}\varepsilon}\right|_{\varepsilon=0}{F}[\phi+\varepsilon
g_{u},\phi^{*}]\,,$ (3.13)

where $g_{u}\in V_{u}$. The formula for
$\frac{\delta{F}[\phi,\phi^{*}]}{\delta\phi_{u}(t)}$ is the same as for the
functional derivative $\frac{\delta{F}[\phi,\phi^{*}]}{\delta\phi(t)}$, the
only difference being that the space of functions entering the integral is
restricted. For $\phi^{*}$ we do not introduce a new notation, since the
$\phi^{*}$ are unaffected by the decomposition.

We next give a chain rule relating $\frac{\delta}{\delta\phi(t)}$ to
$\partial_{x},\partial_{y},\frac{\delta}{\delta\phi_{u}(t)}$. Indeed, we can
represent a functional $F[\phi]$ as a functional $F[\phi_{u},x,y]$
(momentarily suppressing the dependence on $\phi^{*}$) by inverting the
relation $\phi=\phi_{u}+\phi_{p}$,

$F[\phi]=F[\phi_{u}[\phi],x[\phi],y[\phi]]\,.$ (3.14)

A straightforward computation along the lines of familiar (finite-dimensional
or functional) calculus establishes the chain rule

$\frac{\delta}{\delta\phi(t)}=\int\text{d}s\,\frac{\delta\phi_{u}(s)}{\delta\phi(t)}\frac{\delta}{\delta\phi_{u}(s)}+\frac{\delta
x}{\delta\phi(t)}\frac{\partial}{\partial x}+\frac{\delta
y}{\delta\phi(t)}\frac{\partial}{\partial y}\,.$ (3.15)

To illustrate this formula, let us look at the projection and inclusion that
correspond to the homotopy with Dirichlet boundary conditions, for which

$\phi(t)=\phi_{u}(t)+x\frac{\sin\omega(t_{f}-t)}{\sin\omega(t_{f}-t_{i})}+y\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})}\,,$
(3.16)

with inverse relation

$\phi_{u}(t)=\phi(t)-\phi(t_{i})\frac{\sin\omega(t_{f}-t)}{\sin\omega(t_{f}-t_{i})}-\phi(t_{f})\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})},\quad
x=\phi(t_{i}),\quad y=\phi(t_{f})\,.$ (3.17)

Hence, (3.15) gives

$\begin{split}\frac{\delta}{\delta\phi(t)}=&\,\frac{\delta}{\delta\phi_{u}(t)}-\delta(t-t_{i})\int\text{d}s\,\frac{\sin\omega(t_{f}-s)}{\sin\omega(t_{f}-t_{i})}\frac{\delta}{\delta\phi_{u}(s)}-\delta(t-t_{f})\int\text{d}s\,\frac{\sin\omega(s-t_{i})}{\sin\omega(t_{f}-t_{i})}\frac{\delta}{\delta\phi_{u}(s)}\\\
&\,+\delta(t-t_{i})\frac{\partial}{\partial
x}+\delta(t-t_{f})\frac{\partial}{\partial y}\,.\end{split}$ (3.18)

We can now return to our goal of finding a more convenient form of the
homotopy lift. As a first step we define

$V_{h}:=\int\text{d}t\,\text{d}s\,\phi^{*}(t)K(t,s)\frac{\delta}{\delta\phi_{u}(s)}\,,$
(3.19)

where $K$ is a Green’s function for the harmonic oscillator. This is a vector
field (on the infinite-dimensional BV manifold) and hence has a simple action
as a derivation. We will see that, up to a rescaling, this implements the
homotopy action on functionals in $\mathcal{F}_{u}$. In fact, on the linear
functionals $\phi=\phi_{u}+\phi_{p}$ and $\phi^{*}(t)$, $V_{h}$ agrees with
the homotopy $H$ defined in (2.68), since

$V_{h}(\phi_{u}(t)+\phi_{p}(t))=V_{h}(\phi_{u}(t))=\int\text{d}sK(t,s)\phi^{*}(s)\,,\qquad
V_{h}(\phi^{*}(t))=0\,.$ (3.20)

In order to discuss the general case of non-linear functionals we recall the
homological vector field $Q_{0}$ corresponding to the free theory,

$Q_{0}=\int\text{d}t\,\big{(}\ddot{\phi}_{u}(t)+\omega^{2}\phi_{u}(t)\big{)}\frac{\delta}{\delta\phi^{*}(t)}\,,$
(3.21)

where we used $\phi=\phi_{u}+\phi_{p}$ and that $\phi_{p}$ satisfies the
equations of motion and thus drops out. The vector fields $V_{h}$ and $Q_{0}$
are both odd and hence their Lie bracket is given by the anticommutator, which
defines a new vector field:

$\\{Q_{0},V_{h}\\}=\int\text{d}t\,\Big{\\{}\phi_{u}(t)\frac{\delta}{\delta\phi_{u}(t)}+\phi^{*}(t)\frac{\delta}{\delta\phi^{*}(t)}\Big{\\}}=:N\;.$
(3.22)

The expression on the right-hand side is found by a computation using (3.19)
and (3.21). Specifically, this result follows quickly when ignoring boundary
terms, but closer inspection shows that also the boundary terms vanish.666This
can be easily checked for specific boundary conditions such as Dirichlet, but
one can also give a general argument valid for arbitrary boundary conditions
as follows. We write the Lie bracket as  $\\{Q_{0},V_{h}\\}=N+B\,,$ (3.23)
where $B$ encodes possible boundary terms. Since $N$ and $\\{Q_{0},V_{h}\\}$
are vector fields, so is $B$. We will show that $B=0$. Since in (3.20) we saw
that on the linear functionals $\phi=\phi_{u}+\phi_{p}$ and $\phi^{*}(t)$,
$V_{h}$ agrees with the homotopy $H$, we have $\\{Q_{0},V_{h}\\}=1-IP$ on
these functionals. Since $IP(\phi_{u}(t))=IP(\phi^{*}(t))=0$, we find
$\\{Q_{0},V_{h}\\}(\phi_{u}(t))=\phi_{u}(t)\,,\quad\\{Q_{0},V_{h}\\}(\phi^{*}(t))=\phi^{*}(t)\,.$
(3.24) Comparing this with (3.23) acting on $\phi_{u}$ and $\phi^{*}$, for
which $N$ acts as the identity, we learn $B(\phi_{u}(t))=0$ and
$B(\phi^{*}(t))=0$, i.e., $B$ is zero on these linear functionals. General
functionals on $\mathcal{F}_{u}$ are superpositions of functionals of the form
$f(x,y)F[\phi_{u},\phi^{*}]$, where $(x,y)$ are coordinates on $V_{p}$ and
$F[\phi_{u},\phi^{*}]$ is at least linear in $\phi_{u}$ and $\phi^{*}$. The
vector fields $V_{h}$ and $Q_{0}$ act directly on $F[\phi_{u},\phi^{*}]$,
ignoring the function $f(x,y)$ in front, and so does $\\{Q_{0},V_{h}\\}$.
Since $B(\phi_{u}(t))=B(\phi^{*}(t))=0$ and $B$ acts as a vector field via the
Leibniz rule we have $B=0$. The resulting vector field has been called $N$,
because it counts the total number of fields $\phi_{u}$ and anti-fields
$\phi^{*}$:

$N(\phi_{u}(t_{1})\cdots\phi_{u}(t_{k})\phi^{*}(t_{k+1})\cdots\phi^{*}(t_{n}))=n\phi_{u}(t_{1})\cdots\phi_{u}(t_{k})\phi^{*}(t_{k+1})\cdots\phi^{*}(t_{n})\,.$
(3.25)

We claim that on the space ${\cal F}_{u}$ of functionals that are at least
linear in $\phi_{u}$ or $\phi^{*}$, on which $N$ is a positive operator, the
homotopy map is implemented by

$H_{u}:={N}^{-1}V_{h}\,,$ (3.26)

where we have identified $N$ with its eigenvalue (which is always positive on
${\cal F}_{u}$, so $N^{-1}$ is well-defined). Indeed, the homotopy relation is
then satisfied:

$\\{Q_{0},H_{u}\\}={N}^{-1}\\{Q_{0},V_{h}\\}={N}^{-1}N=1\,,$ (3.27)

recalling that the subspace ${\cal F}_{u}$ is projected to $0$ (or,
equivalently, the homotopy $H_{u}$ is a strong deformation retract of
$\mathcal{F}_{u}$ to $0$). Finally, we can extend $H_{u}$ to a homotopy $H$ on
the total space $\mathcal{F}(V)=\mathcal{F}_{u}\oplus\mathcal{F}_{p}$ by
declaring $H$ to be zero on $\mathcal{F}_{p}\subseteq\mathcal{F}(V)$. We then
have $\\{Q_{0},H\\}=1-IP$, which defines a strong deformation retract from
$\mathcal{F}(V)$ to $\mathcal{F}_{p}$.

Having defined, using the above decomposition, the lift of the homotopy map
$H$ we can now revisit the application of the perturbation lemma and relate it
to Wick contractions. In the first step we consider the free harmonic
oscillator with differential $Q_{0}=\\{S_{0},-\\}$ and view $-i\hbar\Delta$ as
the perturbation. Under this perturbation there will neither be a induced
differential on $\mathbb{R}^{2}$, nor a perturbation to the inclusion $I$,
while the perturbed projection is

$P_{1}=P\sum_{n\geq 0}(i\hbar\Delta H)^{n}\,,$ (3.28)

where we denote the perturbed projector by $P_{1}$ since below we will
consider a second perturbation to be denoted $P_{2}$. Since $P_{1}$ is of
degree zero it is zero on functionals with at least one anti-field. On
functionals of fields only, $\Delta H$ acts as

$\Delta
H=-\int\text{d}t\,\text{d}s\,K(t,s)\frac{\delta^{2}}{\delta\phi(t)\delta\phi_{u}(s)}\frac{1}{N}\,.$
(3.29)

Recall that the action of $\frac{\delta}{\delta\phi(t)}$ reduces to
$\frac{\delta}{\delta\phi_{u}(t)}$ when it is integrated against a function
satisfying the boundary conditions of $V_{u}$. The kernel $K(t,s)$ is such a
function and so defining

$C:=\int\text{d}t\,\text{d}s\,K(t,s)\frac{\delta^{2}}{\delta\phi_{u}(t)\delta\phi_{u}(s)}\,,$
(3.30)

we have with (3.29) the relation

$\Delta H=-C\frac{1}{N}\;.$ (3.31)

We now consider a functional $F$ for fixed $t_{1},\ldots,t_{m}$ of the form

$F[\phi_{u},x,y]=\phi_{u}(t_{1})\cdots\phi_{u}(t_{m})f(x,y)\,,$ (3.32)

where $f(x,y)$ is any polynomial in $x$ and $y$. Any other functional is a
superposition of these functionals, so the effect of $P_{1}$ on all
functionals can be deduced from its effect on $F$ by linearity. Since
$P(\phi_{u}(t))=0$, the only non-zero contribution to $P_{1}$ in (3.28) comes
from the term where $\sum_{n\geq 0}(i\hbar\Delta H)^{n}$ eliminates all fields
$\phi_{u}$. This can only happen when $m$ is even, i.e., $m=2k$, for which
only the term with $n=k$ contributes. We then find with (3.31)

$\begin{split}P_{1}(F)&=f(x,y)(-i\hbar C)\frac{1}{2}\cdots(-i\hbar
C)\frac{1}{2k-2}(-i\hbar
C)\frac{1}{2k}\;\phi_{u}(t_{1})\cdots\phi_{u}(t_{2k})\\\
&=f(x,y)\frac{1}{k!}\left(-\frac{i\hbar}{2}C\right)^{k}\phi_{u}(t_{1})\cdots\phi_{u}(t_{2k})\,.\end{split}$
(3.33)

This implies that on arbitrary functionals we have

$P_{1}=P\exp\left(-\frac{i\hbar}{2}C\right)\;.$ (3.34)

Note that to apply this formula, we do not need the split $V_{p}\oplus V_{u}$,
since

$C=\int\text{d}t\,\text{d}s\,K(t,s)\frac{\delta^{2}}{\delta\phi_{u}(t)\delta\phi_{u}(s)}=\int\text{d}t\,\text{d}s\,K(t,s)\frac{\delta^{2}}{\delta\phi(t)\delta\phi(s)}\,,$
(3.35)

because $K(t,s)$ is symmetric. (If $K(t,s)$ were not symmetric, it would not
satisfy the boundary conditions of $V_{u}$ in the variable $s$.)

The above results allow us to establish the core relation with the familiar
techniques of perturbative quantum field theory: The operator $C$ (which is
called $2\partial_{P}$ in lemma 3.4.1 in [5]) generates Wick contractions, and
$P_{1}$ implements Wick’s theorem upon interpreting $P$ as a normal ordering
operation. Of course, $P$ depends on the choice of homotopy $H$, and therefore
on the choice of propagator. We will see that, when choosing the Feynman
propagator, $P$ can indeed be interpreted as the usual prescription to move
“annihilation operators to the right”.

So far we have applied the perturbation lemma to the free theory with
differential $Q_{0}$, viewing $-i\hbar\Delta$ as the perturbation, and related
this to familiar Wick contractions. Next we include the non-linear
interactions of the classical theory, thereby combining the perturbations
$Q_{I}=\\{S_{I},-\\}$ and $-i\hbar\Delta$. Thus, we apply the perturbation
lemma by taking $Q_{I}$ to be a perturbation of $Q_{0}-i\hbar\Delta$, for
which one obtains the perturbed map

$P_{2}=P_{1}\circ\sum_{n\geq 0}(-Q_{I}H)^{n}\,.$ (3.36)

The core result following from the perturbation lemma is then that the
function $f$ computing the quantum expectation value for the functional $F$ is
given by

$f=P_{2}(F)\;.$ (3.37)

We will now show that this prescription coincides with conventional quantum
perturbation theory, i.e., that the operator $P_{2}$ computes expectation
values of the full interacting theory. This is established if we can show that
$P_{2}$ is in fact equal to $\tilde{P}_{2}$ defined by

$\tilde{P}_{2}(F)=\frac{P_{1}\left(F\exp\left(\tfrac{i}{\hbar}S_{I}\right)\right)}{Z}\,,$
(3.38)

with normalization

$Z=P_{1}\exp\left(\tfrac{i}{\hbar}S_{I}\right)\,,$ (3.39)

because this is the familiar method: performing Wick contractions of the
operator $F$ weighted by $\exp(\frac{i}{\hbar}S_{I})$ with the interacting
action $S_{I}$.

The proof that (3.38) indeed represents $P_{2}$ follows [22], which we repeat
in our setting. We first show that (3.38) is consistent with the relation
$P_{2}\circ I=1$ that $P_{2}$ obeys, i.e., that

$\tilde{P}_{2}\circ I=1\,.$ (3.40)

First, using $HI=0$, we have $\sum_{n\geq 0}(i\hbar\Delta H)^{n}I(f)=I(f)$ and
therefore with (3.28)

$\begin{split}P_{1}\Big{(}I(f)e^{i\frac{S_{I}}{\hbar}}\Big{)}&=P\Big{(}\sum_{n\geq
0}(i\hbar\Delta
H)^{n}\Big{(}I(f)e^{i\frac{S_{I}}{\hbar}}\Big{)}\Big{)}=P\Big{(}I(f)\sum_{n\geq
0}(i\hbar\Delta H)^{n}e^{i\frac{S_{I}}{\hbar}}\Big{)}\\\
&=PI(f)P\Big{(}\sum_{n\geq 0}(i\hbar\Delta
H)^{n}e^{i\frac{S_{I}}{\hbar}}\Big{)}=PI(f)Z=fZ\,.\end{split}$ (3.41)

Here we used that $PI=1$ and the fact that $P$ is an algebra morphism, i.e.
$P(FG)=P(F)P(G)$. We next use that the perturbed data with projection $P_{2}$
still obey the homotopy relation:

$1-I\circ P_{2}=H_{2}\circ\delta+\delta\circ H_{2}\,,$ (3.42)

where $\delta=Q_{0}+Q_{I}-i\hbar\Delta$ and $H_{2}=H\sum_{n\geq
0}(i\hbar\Delta-Q_{I})^{n}$. We apply $\tilde{P}_{2}$ to both sides and use
that $\tilde{P}_{2}I=1$, c.f. (3.40). Then,

$\tilde{P}_{2}-P_{2}=\tilde{P}_{2}H_{2}\delta+\tilde{P}_{2}\delta H_{2}\,.$
(3.43)

Since $\tilde{P}_{2}$ is non-zero only in degree zero, we find that
$\tilde{P}_{2}H_{2}\delta=0$, so (3.43) reduces to

$\tilde{P}_{2}-P_{2}=\tilde{P}_{2}\delta H_{2}\,.$ (3.44)

We now show that $Z\tilde{P}_{2}\delta=0$, which then proves that
$\tilde{P}_{2}=P_{2}$. To this end we decompose the BV differential as
$\delta=\delta_{0}+Q_{I}$. We assume that $S_{I}$ does not contain
derivatives. We can then write $Q_{I}=\\{S_{I},\cdot\\}$ due to the absence of
boundary terms. We compute, for a generic functional $F$, by use of (2.16)

$\begin{split}\delta_{0}\big{(}e^{\frac{i}{\hbar}S_{I}}F\big{)}&=\delta_{0}\big{(}e^{\frac{i}{\hbar}S_{I}}\big{)}F+e^{\frac{i}{\hbar}S_{I}}\delta_{0}F-i\hbar\big{\\{}e^{\frac{i}{\hbar}S_{I}},F\big{\\}}\\\
&=e^{\frac{i}{\hbar}S_{I}}\delta_{0}F+e^{\frac{i}{\hbar}S_{I}}\big{\\{}S_{I},F\big{\\}}\\\
&=e^{\frac{i}{\hbar}S_{I}}\delta F\;.\end{split}$ (3.45)

Here we used $\delta_{0}\big{(}e^{\frac{i}{\hbar}S_{I}}\big{)}=0$, which
follows because both $S_{0}$ and $S_{I}$ contain no anti-fields, and
$\\{e^{X},F\\}=e^{X}\\{X,F\\}$ for any degree-zero object $X$, which follows
from (2.7). We then have

$Z\tilde{P}_{2}(\delta F)=P_{1}\big{(}\delta
Fe^{\frac{i}{\hbar}S_{I}}\big{)}=P_{1}\delta_{0}\big{(}e^{\frac{i}{\hbar}S_{I}}F\big{)}=0\,,$
(3.46)

where we used that $P_{1}\delta_{0}=0$, i.e. that $P_{1}$ is a chain map with
respect to $Q_{0}-i\hbar\Delta$, as implied by the perturbation lemma. This
concludes the proof of $P_{2}=\tilde{P}_{2}$.

We end this part by summarizing what we have found. First of all, we showed
that in perturbation theory, any functional $F$ has a representative of the
form $F^{\prime}=f\circ p$. We showed that the perturbative computation of the
$Q_{0}-i\hbar\Delta$ cohomology does Wick contractions, just like we would
expect in a free theory. Also, we showed that for an interacting theory we can
think of the homological perturbation lemma computing expectation values with
respect to the free theory, but with functionals $F$ weighted by the
interacting part $e^{\frac{i}{\hbar}S_{I}}$. This is the usual way in which
Feynman diagrams are computed.

###  3.3 Path Integral

In the previous subsection we saw that Wick’s theorem arises as a consequence
of the perturbation lemma, proving that the latter entails in particular the
usual treatment of quantum theories at the perturbative level. Nevertheless,
we want to shed more light on the homological approach by comparing it to the
path integral derivation.

In the path integral formulation, we formally compute expectation values of
functionals by writing

$\langle
y;t_{f}|T(F[\phi])|x;t_{i}\rangle=\int_{\phi(t_{i})=x}^{\phi(t_{f})=y}{\cal
D}\phi\,F[\phi]\,e^{\frac{i}{\hbar}S[\phi]}\;.$ (3.47)

The left-hand side represents how this object is computed in the operator
language. Here, $|x;t\rangle$ is an eigenstate of the field operator
$\hat{\phi}(t)$ with eigenvalue $x$, i.e.
$\hat{\phi}(t)|x;t\rangle=x|x;t\rangle$. Operators and states can be evolved
in time using the unitary time evolution operator, i.e.

$e^{-\frac{i}{\hbar}Hs}|x;t\rangle=|x;t+s\rangle\,,\quad
e^{\frac{i}{\hbar}Hs}\hat{\phi}(t)e^{-\frac{i}{\hbar}Hs}=\hat{\phi}(t+s)\,.$
(3.48)

When working in the operator formalism, we have to include time ordering
indicated by the letter T. However, we will often not write time ordering
explicitly unless we want to stress its presence.

The integral on the right-hand side of (3.47) is thought of as being performed
over all paths with $\phi(t_{i})=x$ and $\phi(t_{f})=y$. We take the
perturbative route and write $S=S_{0}+S_{I}$, where
$S_{0}=\int\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\omega^{2}\phi^{2}$. When
boundary conditions are fixed, the operator $\partial^{2}+\omega^{2}$ in
$S_{0}$ is invertible, and the integral can be given a meaning in perturbation
theory.

One way to proceed is to pick reference boundary conditions, e.g.
$\phi(t_{i})=\phi(t_{f})=0$. We then write $\phi=\phi_{u}+\phi_{p}$, such that
$\phi_{p}$ is the unique classical solution with the generic boundary
conditions $\phi_{p}(t_{i})=x,\phi_{p}(t_{f})=y$, while $\phi_{u}$ satisfies
$\phi_{u}(t_{i})=\phi_{u}(t_{f})=0$. Explicitly, $\phi_{p}$ is given by
$\phi_{p}=i(x(t_{i}),y(t_{f}))$ and can be viewed as a constant in the context
of the path integral. We then substitute $\phi=\phi_{u}+\phi_{p}$ and assume
that the integral measure is invariant under constant shifts so that ${\cal
D}\phi={\cal D}\phi_{u}$. One computes

$\langle
y;t_{f}|F[\phi]|x;t_{i}\rangle=\int_{\phi_{u}(t_{i})=0}^{\phi_{u}(t_{f})=0}{\cal
D}\phi_{u}\,F[\phi_{u}+\phi_{p}]\,e^{\frac{i}{\hbar}S_{I}[\phi_{u}+\phi_{p}]}e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\frac{1}{2}(\dot{\phi}_{u}^{2}-\omega\phi_{u}^{2})}e^{i\partial
S(\phi_{p})}\;,$ (3.49)

where

$\partial
S=\frac{1}{2}\phi_{p}(t_{f})\dot{\phi}_{p}(t_{f})-\frac{1}{2}\phi_{p}(t_{i})\dot{\phi}_{p}(t_{i})\;.$
(3.50)

Importantly, the boundary action $\partial S$ does not depend on $\phi_{u}$,
which is due to our choice of reference boundary conditions
$\phi_{u}(t_{i})=\phi_{u}(t_{f})=0$. Thus, the phase $e^{i\partial
S(\phi_{p})}$ can be scaled out of the path integral:

$\langle y;t_{f}|F[\phi]|x;t_{i}\rangle=e^{i\partial
S(\phi_{p})}\int_{\phi_{u}(t_{i})=0}^{\phi_{u}(t_{f})=0}{\cal
D}\phi_{u}\,F[\phi_{u}+\phi_{p}]\,e^{\frac{i}{\hbar}S_{I}[\phi_{u}+\phi_{p}]}e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\frac{1}{2}(\dot{\phi}_{u}^{2}-\omega\phi_{u}^{2})}\;.$
(3.51)

Picking $F=1$ we have

$\braket{y;t_{f}}{x;t_{i}}=e^{i\partial
S(\phi_{p})}\int_{\phi_{u}(t_{i})=0}^{\phi_{u}(t_{f})=0}{\cal
D}\phi_{u}\,e^{\frac{i}{\hbar}S_{I}[\phi_{u}+\phi_{p}]}e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\frac{1}{2}(\dot{\phi}_{u}^{2}-\omega\phi_{u}^{2})}\,.$
(3.52)

Our next goal is to define a map
$P_{1}:\mathcal{F}(V^{0})\rightarrow\mathcal{F}(\mathbb{R}^{2})$ in terms of
the path integral that equals the map in (3.34) that implemented Wick’s
theorem. We claim that this map can be written, for any functional
$\tilde{F}$, as

$P_{1}(\tilde{F})=\int_{\phi_{u}(t_{i})=0}^{\phi_{u}(t_{f})=0}{\cal
D}\phi_{u}\,\tilde{F}[\phi_{u}+\phi_{p}]\,e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\frac{1}{2}(\dot{\phi}_{u}^{2}-\omega\phi_{u}^{2})}\,.$
(3.53)

Indeed, once we accept that the path integral is computed by doing Wick
contractions using the propagator with the boundary conditions specified on
the right hand side of (3.53) it is clear that this is equal to the map
(3.34). Note that $P_{1}$ is a function on phase space $\mathbb{R}^{2}$
parametrized by the boundary conditions put on $\phi_{p}$.

We can finally give a path integral expression for the normalized quantum
expectation value of $F$ that agrees with our above result from the
perturbation lemma. In terms of the map (3.53) we find with (3.51) and (3.52)
that

$\frac{\langle
y;t_{f}|F[\phi]|x;t_{i}\rangle}{\braket{y;t_{f}}{x;t_{i}}}=\frac{P_{1}(Fe^{\frac{i}{\hbar}S_{I}})}{P_{1}(e^{\frac{i}{\hbar}S_{I}})}\equiv
P_{2}(F)\,,$ (3.54)

using that the phase $e^{i\partial S(\phi_{p})}$ cancels. This is indeed the
same as the map (3.38) defined by means of the perturbation lemma. This
confirms that the conventional interpretation of the (otherwise ill-defined)
path integral in perturbation theory agrees with the homological formulation.

##  4 Harmonic Oscillator

In this section we illustrate the homological formulation of quantum mechanics
by applying it to the one-dimensional harmonic oscillator. Specifically, we
compute two-point functions both with respect to positions eigenstates and
with respect to coherent states. Using the familiar formulation of quantum
mechanics, based on Hilbert spaces and operators, we then verify that the
homological approach yields the correct results. We hope to illustrate in this
that for certain computations, as in the case of position eigenstates, the
homological approach is more transparent.

###  4.1 Homological Computation for Position Eigenstates

For a free theory like the harmonic oscillator all expectation values can be
computed in terms of the two-point function via Wick’s theorem, which we also
obtained from the perturbation lemma. For this reason, we apply the
homological formulation to the two-point function of the harmonic oscillator.
The BV differential for the quantum harmonic oscillator reads

$\delta=\int_{t_{i}}^{t_{f}}\text{d}t\,\left[\big{(}\ddot{\phi}(t)+\omega^{2}\phi(t)\big{)}\frac{\delta}{\delta\phi^{*}(t)}+i\hbar\frac{\delta^{2}}{\delta\phi^{*}(t)\delta\phi(t)}\right]\,.$
(4.1)

According to the general formulation of section 2.2, for a given a functional
$F$ the time-ordered correlation function can be computed via a function $f$
on $\mathbb{R}^{2}$ that in cohomology is equivalent to $F$:

$f(x,y)=\frac{\langle y|T(F)|x\rangle}{\braket{y}{x}}\;.$ (4.2)

More precisely, one determines a representative $F^{\prime}$ of $F$ in the
$\delta$ cohomology that can be written as $F^{\prime}=f\circ p$, where
$p:V^{\bullet}\rightarrow\mathbb{R}^{2}$ is given by
$p(\phi,\phi^{*})=(\phi(t_{i}),\phi(t_{f}))$. This means that $F^{\prime}$
should only depend on $\phi(t_{i})$ and $\phi(t_{f})$.

Since we want to compute the two-point function, we consider the functional
that, for fixed $t,s\in\mathbb{R}$, is defined by

$F[\phi,\phi^{*}]=\phi(t)\phi(s)\;.$ (4.3)

In order for $F^{\prime}=f\circ p$ to be in the same cohomology as $F$ there
should be a $G$ such that $F-F^{\prime}=\delta G$, where $G$ has degree minus
one. The perturbation lemma in the form (3.37) immediately gives us the
function $f$ as follows:

$f=P_{1}(F)\;,\qquad P_{1}=i^{*}\exp\left(-\frac{i\hbar}{2}C\right)\;,$ (4.4)

where $C$ is defined in terms of the Green’s function (2.60) with Dirichlet
boundary conditions:

$C=\int\text{d}t\,\text{d}s\,K_{DD}(t,s)\frac{\delta^{2}}{\delta\phi(t)\delta\phi(s)}\,.$
(4.5)

Note that here $P_{2}=P_{1}$ since we are considering the free theory. We thus
have

$f=i^{*}\left(1-\frac{i\hbar}{2}C\right)\phi(t)\phi(s)\;,$ (4.6)

using that the higher-order terms in $\hbar$ vanish when acting on the
functional (4.3) with two $\phi$. For the second term on the right-hand side
we need to use (4.5) to compute

$C\big{(}\phi(t)\phi(s)\big{)}=2\,K_{DD}(t,s)\;,$ (4.7)

for which one uses that $K_{DD}$ is symmetric. To evaluate then $f(x,y)$ the
first term in (4.6) maps $x,y$ via the inclusion map (2.64) to a solution
$\phi_{p}$ with boundary conditions $\phi_{p}(t_{i})=x$ and
$\phi_{p}(t_{f})=y$ and then evaluates the functional on this solution. Thus,
we have

$f(x,y)=\prod_{r=t,s}\left\\{\frac{\sin\omega(r-t_{i})}{\sin\omega(t_{f}-t_{i})}y+\frac{\sin\omega(t_{f}-r)}{\sin\omega(t_{f}-t_{i})}x\right\\}-i\hbar
K_{DD}(t,s)\,.$ (4.8)

Via the dictionary (4.2) this gives the desired two-point function.

###  4.2 Homological Computation for Coherent States

In the previous subsection we used the propagator with Dirichlet boundary
conditions to find a certain representative in cohomology of the functional
$F=\phi(t)\phi(s)$. We want to see what happens when we instead use different
propagators. The standard propagator in quantum field theory is the Feynman
propagator, which we will investigate here.

In $(1+0)$-dimensions the Feynman propagator is given by

$h_{F}(f)(t)=i\int_{t_{i}}^{t}\text{d}sf(s)\frac{e^{-i\omega(t-s)}}{2\omega}+i\int_{t}^{t_{f}}\text{d}sf(s)\frac{e^{i\omega(t-s)}}{2\omega}=:h_{+}(f)(t)+h_{-}(f)(t)\,.$
(4.9)

Note that $h_{F}(f)$ is complex, even when $f$ is real. It is therefore not
sufficient to work with the field space $V$, but rather we should work with
the complexified field space $V\otimes\mathbb{C}$. We now want to derive the
associated inclusion $i$ and projection $p_{F}$ from the homotopy relation
$\\{\partial,h_{F}\\}=1-i_{F}\circ p_{F}$. Since (4.9) is a Green’s function
it satisfies $(\partial^{2}_{t}+\omega^{2})h_{F}(f)(t)=f(t)$. On the other
hand, on equations of motion $\ddot{\phi}+\omega^{2}\phi$ we find

$h_{+}(\ddot{\phi}+\omega^{2}\phi)(t)=i\frac{\dot{\phi}(t)}{2\omega}-i\frac{\dot{\phi}(t_{i})}{2\omega}e^{-i\omega(t-t_{i})}+\frac{\phi(t)}{2}-\frac{\phi(t_{i})}{2}e^{-i\omega(t-t_{i})}\,,$
(4.10)
$h_{-}(\ddot{\phi}+\omega^{2}\phi)(t)=-i\frac{\dot{\phi}(t)}{2\omega}+i\frac{\dot{\phi}(t_{f})}{2\omega}e^{i\omega(t-t_{f})}+\frac{\phi(t)}{2}-\frac{\phi(t_{f})}{2}e^{i\omega(t-t_{f})}\,,$
(4.11)

and so for the sum

$\begin{split}h_{F}(\ddot{\phi}+\omega^{2}\phi)(t)=\phi(t)-\frac{1}{2}\left(\phi(t_{i})-\frac{\dot{f}(t_{i})}{i\omega}\right)e^{-i\omega(t-t_{i})}-\frac{1}{2}\left(f(t_{f})+\frac{\dot{f}(t_{f})}{i\omega}\right)e^{i\omega(t-t_{f})}.\end{split}$
(4.12)

We next define new (complex) functionals $a(t)$ and $a^{\dagger}(t)$ by

$\phi(t)=\sqrt{\frac{\hbar}{2\omega}}(a^{\dagger}(t)+a(t))\,,\qquad\dot{\phi}(t)=i\sqrt{\frac{\hbar\omega}{2}}(a^{\dagger}(t)-a(t))\,.$
(4.13)

These expressions are motivated by the mode expansion of the harmonic
oscillator, but we should emphasize that here these are just regular
functions, not quantum operators. In particular, the function $a^{\dagger}$ is
just the complex conjugate of the function $a$, with the notation just
reminding us of the usual raising and lowering operators. In terms of these we
have

$h_{F}(\ddot{\phi}+\omega^{2}\phi)(t)=\phi(t)-\sqrt{\frac{\hbar}{2\omega}}\left(a(t_{i})e^{-i\omega(t-t_{i})}+a^{\dagger}(t_{f})e^{i\omega(t-t_{f})}\right)\,.$
(4.14)

This suggests that we define a projector
$p_{F}:V\otimes\mathbb{C}\rightarrow\mathbb{C}^{2}$ by

$\phi\mapsto(a(t_{i}),a^{\dagger}(t_{f}))\,,\quad\phi^{*}\mapsto 0\;,$ (4.15)

and the inclusion $i_{F}:\mathbb{C}^{2}\rightarrow V^{0}\otimes\mathbb{C}$ by

$(x,y)\mapsto\sqrt{\frac{\hbar}{2\omega}}\left(xe^{-i\omega(t-t_{i})}+ye^{i\omega(t-t_{f})}\right)\,,$
(4.16)

with zero image in $V^{1}$. With these definitions we have $p_{F}\circ
i_{F}=1$.

For a given a field $\phi$ the projector $p_{F}$ gives the complex values
$a(t_{i})$ and $a^{\dagger}(t_{f})$. Let us compare this with the projector
associated to the propagator with Dirichlet boundary conditions, which gives
$(\phi(t_{i}),\phi(t_{f}))$. When relating to canonical quantization, we found
that the correlator computed with the latter prescription used the states
$|x\rangle$ and $\langle y|$ satisfying $\phi(t_{i})|x\rangle=x|x\rangle$ and
$\langle y|\phi(t_{f})=\langle y|y$. Correspondingly, when finding a
representative $F^{\prime}$ of the cohomology of some functional $F$ with
$F^{\prime}=f\circ p_{F}$, we expect that $f$ computes correlators with the
in-state $|z\rangle$ being an eigenstate of $a(t_{i})$ and the out-state
$\langle w|$ being an eigenstate of $a^{\dagger}(t_{f})$. This will be
confirmed in the following.

The eigenstates $|z\rangle$ of the annihilation operator $a$ are called
coherent states. We use the convention $a|z\rangle=z|z\rangle$ and $\langle
z|a^{\dagger}=\langle z|z$. With this convention, $\langle\bar{z}|$ is the
conjugate of $|z\rangle$, where $\bar{z}$ denotes the complex conjugate of
$z$. We can check whether it is reasonable that $p_{F}$ gives rise to
correlators with coherent states by looking again at the two-point function.
By either applying the perturbation lemma or going through the same steps as
in the previous section, we find that the representative $F^{\prime}$ of the
cohomology of $F=\phi(t)\phi(s)$ satisfying $F^{\prime}=f\circ p_{F}$ is given
by

$F^{\prime}=-i\hbar
K_{F}(t,s)+\frac{\hbar}{2\omega}\big{(}a(t_{i})e^{-i\omega(t-t_{i})}+a^{\dagger}(t_{f})e^{i\omega(t-t_{f})}\big{)}\big{(}a(t_{i})e^{-i\omega(s-t_{i})}+a^{\dagger}(t_{f})e^{i\omega(s-t_{f})}\big{)}\,.$
(4.17)

We therefore claim that

$f(w,z)=\frac{\langle w|T(\phi(t)\phi(s))|z\rangle}{\braket{w}{z}}\,,$ (4.18)

where

$f(w,z)=-i\hbar
K_{F}(t,s)+\frac{\hbar}{2\omega}\big{(}ze^{-i\omega(t-t_{i})}+we^{i\omega(t-t_{f})}\big{)}\big{(}ze^{-i\omega(s-t_{i})}+we^{i\omega(s-t_{f})}\big{)}\,.$
(4.19)

It is straightforward to verify equation (4.18) in the familiar operator
language of quantum mechanics, as we do now. We first recall that Wick’s
theorem implies

$T(\phi(t)\phi(s))=-i\hbar K_{F}(t,s)+N(\phi(t)\phi(s))\,,$ (4.20)

where $N$ is the normal ordering operation. We have the operator relation

$\hat{\phi}(t)=\sqrt{\frac{\hbar}{2\omega}}\big{(}a(t_{i})e^{-i\omega(t-t_{i})}+a^{\dagger}(t_{f})e^{i\omega(t-t_{f})}\big{)}\;,$
(4.21)

where $a$ and $a^{\dagger}$ are now interpreted as the creation and
annihilation operators of the harmonic oscillator, satisfying the familiar
commutation relations. Usually the above expression appears in textbooks for
$t_{i}=t_{f}=0$ and normal ordering is defined with respect to $a:=a(0)$ and
$a^{\dagger}:=a^{\dagger}(0)$. But this is the same as normal ordering
$a(t_{i})$ and $a^{\dagger}(t_{f})$, since they only differ from $a$ and
$a^{\dagger}$ by phases. We can now compute

$\langle w|N(\phi(t)\phi(s))|z\rangle\,$ (4.22)

by evaluating $a(t_{i})$ at ${z}$ and $a^{\dagger}(t_{f})$ at $w$. This
follows because $N$ moves all annihilation operators to the right, where we
can then use $a(t_{i})|z\rangle={z}|z\rangle$. Similarly, when creation
operators are on the right, we can use $\langle w|a^{\dagger}(t_{f})=\langle
w|w$. Therefore,

$\langle
w|N(\phi(t)\phi(s))|z\rangle=\frac{\hbar}{2\omega}\big{(}ze^{-i\omega(t-t_{i})}+we^{i\omega(t-t_{f})}\big{)}\big{(}ze^{-i\omega(s-t_{i})}+we^{i\omega(s-t_{f})}\big{)}\braket{z}{w}\,.$
(4.23)

Combining this with (4.20) then proves (4.18).

In case of the Feynman propagator, this result explains why the perturbation
lemma gives Wick’s theorem via the projector $P^{\prime}$ and $P$ can be
interpreted as normal ordering. Recall that $P=i^{*}$ just evaluates
functionals on-shell, with boundary conditions specified by the inclusion map
$i$. But this is just what we did in (4.23). We evaluated $\phi(t)\phi(s)$ on
the solution with $a(t)=z$ at $t=t_{i}$ and $a^{\dagger}(t)=w$ at $t=t_{f}$.
Of course, there is nothing special about the two-point functions considered
here, and so the perturbation lemma says that $P^{\prime}$ is really Wick’s
theorem squeezed between coherent states.

###  4.3 Comparison with Operator Language

In the previous two sections we applied the homological recipe to compute
correlators with respect to position eigenstates and with respect to coherent
states. The respective projectors were given by

$p(\phi,\phi^{*})=(\phi(t_{i}),\phi(t_{f}))\,,\qquad
p_{F}(\phi,\phi^{*})=(a(t_{i}),a^{\dagger}(t_{f}))\,.$ (4.24)

Using Wick’s theorem, for $p_{F}$ it was straightforward to see that our
approach agrees with the operator language. For $p$, however, it is harder to
verify that $f$ defined via $F^{\prime}=f\circ p$ actually computes the
correlator with respect to position eigenstates, although our formal path
integral manipulations above suggest that this must be so. The general claim
following from the homological approach is

$f(x,y)=\frac{\langle
y;t_{f}|T(\phi(t)\phi(s))|x;t_{i}\rangle}{\braket{y;t_{f}}{x;t_{i}}}\,,$
(4.25)

where $f$ is the function whose pullback $p^{*}(f)$ equals $F=\phi(t)\phi(s)$
in cohomology. Our goal in this subsection is to check this statement using
the standard formalism of quantum mechanics. Since the operator computation is
quite involved, for simplicity we set $x=y=0$. Since the perturbation lemma
immediately gives the full result (4.8) we see that the homological approach
is advantageous in this case. When $x=y=0$, (4.8) equals the Green’s function
$K_{DD}(t,s)$. So we want to establish the identity

$-i\hbar K_{DD}(t,s)=\frac{\langle
y=0;t_{f}|T(\phi(t)\phi(s))|x=0;t_{i}\rangle}{\braket{y=0;t_{f}}{x=0;t_{i}}}=:g(t,s)$
(4.26)

in the operator language.

We first review some more facts about coherent states. As already stated, a
coherent state $|z\rangle$ is an eigenstate of the annihilation operator, i.e.

$\hat{a}|z\rangle=z|z\rangle\ \;\;\text{for all }z\in\mathbb{C}\,.$ (4.27)

We use the convention that the hermitian conjugate of $|z\rangle$ is
$\langle\bar{z}|$, so that
$\langle\bar{z}|\hat{a}^{\dagger}=\langle\bar{z}|\bar{z}$. Given a general
state $|\psi\rangle$, its overlap with a coherent state $\langle z|$ gives a
holomorphic function in $z$,

$\psi(z):=\braket{z}{\psi}.$ (4.28)

The inner product of two such states is

$\braket{\psi_{1}}{\psi_{2}}=\frac{1}{\pi}\int\text{d}^{2}z\,\bar{\psi}_{1}(\bar{z})\psi_{2}(z)e^{-|z|^{2}}\,.$
(4.29)

The Hilbert space equipped with this inner product is called the Segal-
Bargmann space. The identity can be written as

$1=\frac{1}{\pi}\int\text{d}^{2}ze^{-|z|^{2}}|z\rangle\langle\bar{z}|\,.$
(4.30)

The creation operator acts by multiplication since

$\langle
z|\hat{a}^{\dagger}|\psi\rangle=z\psi(z)\qquad\Rightarrow\qquad(\hat{a}^{\dagger}\psi)(z)=z\psi(z)\,.$
(4.31)

We can then deduce from the inner product (4.29) that $\hat{a}$ acts by
differentiation, i.e.

$\langle z|\hat{a}|\psi\rangle=\frac{\partial}{\partial
z}\psi(z)\qquad\Rightarrow\qquad(\hat{a}\psi)(z)=\frac{\partial}{\partial
z}\psi(z)\,.$ (4.32)

As a consistency check we note that $[\hat{a},\hat{a}^{\dagger}]=1$ in this
representation. Since the vacuum state $|0\rangle$ is annihilated by
$\hat{a}$, the vacuum is represented by $\psi_{0}(z)=\braket{z}{0}$ that is in
fact constant (and equal to one if we normalize it). Likewise, the $n$th
excited state is

$\braket{z}{n}=\frac{1}{\sqrt{n!}}z^{n}\,,$ (4.33)

while a coherent state reads

$\braket{z}{w}=e^{zw}\;.$ (4.34)

We now come back to the original goal of this section, i.e. establishing the
identity (4.26) in the operator language. To do so, we use Wick’s theorem

$T(\phi(t)\phi(s))=-i\hbar K_{F}(t,s)+N(\phi(t)\phi(s))\,,$ (4.35)

where $K_{F}$ is the Feynman propagator and $N$ denotes normal ordering. Using
this in (4.26), we find

$g(t,s)=-i\hbar K_{F}(t,s)+\frac{\langle
y=0;t_{f}|N(\phi(t)\phi(s))|x=0;t_{i}\rangle}{\braket{y=0;t_{f}}{x=0;t_{i}}}\,.$
(4.36)

In order to compute the overlap involving the normal ordering, we express it
in terms of coherent states using (4.30). We then need to express
$|x=0\rangle$ in terms of coherent states. For an arbitrary state
$|\psi\rangle$ we have

$\braket{x}{\psi}=\frac{1}{\pi}\int\text{d}^{2}z\,e^{-\bar{z}z}\braket{x}{z}\braket{\bar{z}}{\psi}\,.$
(4.37)

This formula can be reduced to an integral over the reals. For example, one
can show that [29]

$\braket{x}{\psi}=\psi(x)=Ce^{-x^{2}/2}\int\text{d}y\,e^{-y^{2}/2}\braket{x+iy}{\psi}\,,$
(4.38)

where $C$ is some constant and $\langle x+iy|$ is a coherent state, and so

$\langle x|=Ce^{-x^{2}/2}\int\text{d}ye^{-y^{2}/2}\langle x+iy|\,.$ (4.39)

In particular,

$\langle x=0|=C\int\text{d}y\,e^{-y^{2}/2}\langle iy|\,.$ (4.40)

We now use this in (4.36) to compute $g(t,s)$. We first compute the
denominator

$Z:=\braket{y=0;t_{f}}{x=0;t_{i}}=\langle
y=0|e^{i\frac{H}{\hbar}(t_{f}-t_{i})}|x=0\rangle\,,$ (4.41)

where we used the time evolution operator with respect to the Hamiltonian
$H=\hbar\omega(a^{\dagger}a+\frac{1}{2})$ of the harmonic oscillator. Thus,
using (4.40), we will need the time evolution of a coherent state. Defining
$T=t_{f}-t_{i}$, we need to compute $e^{i\frac{H}{\hbar}T}|z\rangle$, which
can be done by inserting a complete set of eigenstates $|n\rangle$ of the
Hamiltonian and using the overlap (4.33). With this, we find
$e^{i\frac{H}{\hbar}T}|z\rangle=e^{i\frac{\omega}{2}T}|e^{-i\omega
T}z\rangle$. Defining $\lambda:=e^{-i\omega T}$ we thus have
$e^{i\frac{H}{\hbar}T}|z\rangle=\lambda^{-\frac{1}{2}}|\lambda z\rangle$.
Using this together with (4.40) we have:

$\begin{split}Z&=C^{2}\lambda^{-\frac{1}{2}}\int\text{d}y_{1}\text{d}y_{2}e^{-(y_{2}^{2}+y_{1}^{2})/2}\braket{iy_{2}}{-i\lambda
y_{1}}\\\
&=C^{2}\lambda^{-\frac{1}{2}}\int\text{d}y_{1}\text{d}y_{2}e^{-(y_{2}^{2}+y_{1}^{2})/2+\lambda
y_{1}y_{2}}\\\ &=\frac{C_{1}}{\sqrt{\lambda-\lambda^{3}}}\,,\end{split}$
(4.42)

where we performed the Gaussian integral, and $C_{1}:=2\pi C^{2}$ is another
constant that will cancel in the end. Next we turn to the numerator of (4.36).
Expanding $\phi(t)$ in terms of ladder operators,

$\phi(t)=\sqrt{\frac{\hbar}{2\omega}}(a^{\dagger}e^{i\omega t}+ae^{-i\omega
t})\,,$ (4.43)

and using this in (4.36), we need to compute expectation values of operators
quadratic in $a$ and $a^{\dagger}$. For example, we find that

$\begin{split}\langle
y=0;t_{f}|a^{\dagger}a|x=0;t_{i}\rangle&=C^{2}\lambda^{-\frac{1}{2}}\int\text{d}y_{1}\text{d}y_{2}\,\lambda
y_{1}y_{2}\,e^{-(y_{2}^{2}+y_{1}^{2})/2+\lambda y_{1}y_{2}}\\\
&=\frac{C_{1}\lambda^{\frac{3}{2}}}{(1-\lambda^{2})^{\frac{3}{2}}}=Z\frac{\lambda^{2}}{1-\lambda^{2}}\,.\end{split}$
(4.44)

Similarly, we have

$\displaystyle\langle y=0;t_{f}|aa|x=0;t_{i}\rangle$
$\displaystyle=-Z\frac{e^{i2\omega t_{i}}}{1-\lambda^{2}}\,,$ (4.45)
$\displaystyle\langle y=0;t_{f}|a^{\dagger}a^{\dagger}|x=0;t_{i}\rangle$
$\displaystyle=-Z\frac{e^{-i2\omega t_{f}}}{1-\lambda^{2}}\,.$ (4.46)

Since the operators are normal ordered we do not need to compute $\langle
y=0;t_{f}|aa^{\dagger}|x=0;t_{i}\rangle$. We can now use the above to compute
the normal ordered correlator in (4.36), for which we find after some algebra

$\begin{split}g(t,s)=&-i\hbar K_{F}(t,s)-\hbar\frac{e^{-i2\omega
t_{f}}}{1-\lambda^{2}}(2\omega)^{-1}e^{i\omega(t+s)}-\hbar\frac{e^{i2\omega
t_{i}}}{1-\lambda^{2}}(2\omega)^{-1}e^{-i\omega(t+s)}\\\
&+\hbar\frac{\lambda^{2}}{1-\lambda^{2}}(2\omega)^{-1}(e^{i\omega(t-s)}+e^{-i\omega(t-s)})\,.\end{split}$
(4.47)

In order to relate this to $K_{DD}$ we rewrite the Feynman propagator,

$\begin{split}-iK_{F}(t,s)&=(2\omega)^{-1}\theta(t-s)e^{-i\omega(t-s)}+(2\omega)^{-1}\theta(s-t)e^{i\omega(t-s)}\\\
&=-iK_{R}(t,s)+(2\omega)^{-1}e^{i\omega(t-s)}\,,\end{split}$ (4.48)

where $K_{R}(t,s)=\theta(t-s)\frac{\sin\omega(t-s)}{2\omega}$ is the retarded
propagator. This yields

$\begin{split}g(t,s)=&-i\hbar K_{R}(t,s)-\hbar\frac{e^{-i2\omega
t_{f}}}{1-\lambda^{2}}(2\omega)^{-1}e^{i\omega(t+s)}-\hbar\frac{e^{i2\omega
t_{i}}}{1-\lambda^{2}}(2\omega)^{-1}e^{-i\omega(t+s)}\\\
&+\hbar\frac{1}{1-\lambda^{2}}(2\omega)^{-1}e^{i\omega(t-s)}+\hbar\frac{\lambda^{2}}{1-\lambda^{2}}(2\omega)^{-1}e^{-i\omega(t-s)}\\\
=&-i\hbar
K_{R}(t,s)+i\hbar\frac{\cos\omega(t+s-t_{i}-t_{f})-\cos\omega(t-s+t_{f}-t_{i})}{\sin\omega(t_{f}-t_{i})}\,,\end{split}$
(4.49)

where we reintroduced $t_{i}$ and $t_{f}$ through
$\lambda=e^{-i\omega(t_{f}-t_{i})}$. We can now make use of the identity

$\cos\omega((t-t_{i})+(t_{f}-s))-\cos\omega((t-t_{i})-(t_{f}-s))=-2\sin\omega(t-t_{i})\sin\omega(t_{f}-s)\;,$
(4.50)

to arrive at

$\begin{split}\frac{\langle
y=0;t_{f}|T(\phi(t)\phi(s))|x=0;t_{i}\rangle}{\braket{y=0;t_{f}}{x=0;t_{i}}}&=-i\hbar
K_{R}(t,s)+i\hbar\frac{\sin\omega(t_{f}-s)}{\omega}\frac{\sin\omega(t-t_{i})}{\sin\omega(t_{f}-t_{i})}\\\
&=-i\hbar K_{DD}(t,s)\,,\end{split}$ (4.51)

which is what we wanted to show.

###  4.4 General Boundary Conditions

In the previous subsection we exemplified our approach using two different
projectors, which where given by $p_{DD}(\phi)=(\phi(t_{i}),\phi(t_{f}))$ and
$p_{F}(\phi)=(a(t_{i}),a^{\dagger}(t_{f}))$. Our computations showed that
these determine different types of correlation functions.

We now want to generalize to arbitrary linear boundary conditions. More
precisely, we look at boundary conditions of the form

$x=a\phi(t_{i})+b\frac{\dot{\phi}(t_{i})}{\omega}\,,\qquad
y=c\phi(t_{f})+d\frac{\dot{\phi}(t_{f})}{\omega}\,,$ (4.52)

where the numbers $(a,b,c,d,x,y)$ can in general be complex. The numbers
$(x,y)$ parametrize solutions to the equations of motion. In this way, we
obtain a projector

$\begin{split}p:C^{\infty}([t_{i},t_{f}])\otimes\mathbb{C}&\longrightarrow\mathbb{C}^{2}\,,\\\
\phi&\longmapsto(a\phi(t_{i})+b\frac{\dot{\phi}(t_{i})}{\omega},c\phi(t_{f})+d\frac{\dot{\phi}(t_{f})}{\omega})\,.\end{split}$
(4.53)

As usual, we extend $p$ to $V^{\bullet}\otimes\mathbb{C}$ by setting
$p|_{V^{1}}=0$. We recover $p_{DD}$ when $(a,b)=(c,d)=(1,0)$, while $p_{F}$ is
given by
$(a,b)=(\bar{c},\bar{d})=(\sqrt{\frac{\omega}{2\hbar}},i\sqrt{\frac{\omega}{2\hbar}})$.

A solution with boundary conditions (4.52) is given by

$\phi_{x,y}(t)=\frac{ya\sin\omega(t-t_{i})-yb\cos\omega(t-t_{i})+xc\sin\omega(t_{f}-t)+xd\cos\omega(t_{f}-t)}{(ad-
bc)\cos\omega(t_{f}-t_{i})+(ac+bd)\sin\omega(t_{f}-t_{i})}\,,$ (4.54)

This solution defines an inclusion

$\begin{split}i:\mathbb{C}^{2}&\longrightarrow V^{0}\otimes\mathbb{C}\,,\\\
(x,y)&\longmapsto\phi_{x,y}\,.\end{split}$ (4.55)

which we extend to $V^{\bullet}$ via the inclusion $V^{0}\hookrightarrow
V^{\bullet}$. To find the homotopy $h$ from the identity to $i\circ p$, we
note that the homotopies $h_{DD}$ and $h_{F}$ satisfy the boundary conditions
(4.52) with $x=y=0$. So our ansatz for the homotopy $h(f)$ is the unique
solution to $\ddot{\phi}+\omega^{2}\phi=f$ satisfying $p\circ h=0$. It is
given by

$h(f)(t)=\int_{t_{i}}^{t}f(s)K_{i}(t,s)+\int_{t}^{t_{f}}K_{f}(t,s)\,,$ (4.56)

where

$K_{i}(t,s)=K_{f}(s,t)=\frac{(a\sin\omega(s-t_{i})-b\cos\omega(s-t_{i}))(c\sin\omega(t-t_{f})-d\cos\omega(t-t_{f}))}{(ad-
bc)\omega\cos\omega(t_{f}-t_{i})+(ac+bd)\omega\sin\omega(t_{f}-t_{i})}\,.$
(4.57)

In kernel notation, we have

$K(t,s)=\theta(t-s)K_{i}(t,s)+\theta(s-t)K_{f}(s,t)\,.$ (4.58)

Note that $K(t,s)$ is manifestly symmetric in its arguments. A lengthy
computation now shows that

$\begin{split}h(\ddot{\phi}+\omega^{2}\phi)(t)=\phi(t)&-\bigg{(}a\phi(t_{i})+b\frac{\dot{\phi}(t_{i})}{\omega}\bigg{)}\frac{c\sin\omega(t_{f}-t)+d\cos\omega(t_{f}-t)}{(ad-
bc)\cos\omega(t_{f}-t_{i})+(ac+bd)\sin\omega(t_{f}-t_{i})}\\\
&-\bigg{(}c\phi(t_{f})+d\frac{\dot{\phi}(t_{f})}{\omega}\bigg{)}\frac{a\sin\omega(t-t_{i})-b\cos\omega(t-t_{i})}{(ad-
bc)\cos\omega(t_{f}-t_{i})+(ac+bd)\sin\omega(t_{f}-t_{i})}\,,\end{split}$
(4.59)

as well as $\ddot{h}(f)+\omega^{2}h(f)=f$. Therefore, the homotopy relation
$\\{\partial,h\\}=1-i\circ p$ is satisfied.

One application using these more general projectors and homotopies would be
the computation of correlators with in- and out states living in different
representations of the Hilbert space. For example, one could choose
$(a,b)=(1,0)$ and $(c,d)=(0,1)$. In this case, the homotopy satisfies
Dirichlet boundary conditions at $t=t_{i}$ and Neumann boundary conditions at
$t=t_{f}$. The associated representative of the cohomology then uses position
eigenstates $|x;t_{i}\rangle$ as in-states and momentum eigenstates $\langle
p;t_{f}|$ as out-states.

##  5 Unruh Effect

In this section we present the first application of the homological
formulation in the realm of genuine field theories. Specifically, we apply our
homological method in the context of quantum field theory on curved spacetime
by providing an alternative derivation of the Unruh effect: the quantum effect
according to which the number of particles detected depends on the observer
[30]. In the vacuum state an inertial observer in Minkowski space sees no
particles, while in the same state a uniformly accelerated observer sees a
thermal bath of particles.

###  5.1 Generalities and Homotopy Retract

Let us begin with a brief review of general features of uniformly accelerated
observers in two-dimensional Minkowski spacetime with metric

$\text{d}s^{2}=\text{d}t^{2}-\text{d}x^{2}\,.$ (5.1)

The trajectory of an observer is then parametrized by
$x^{\mu}(\tau)=\big{(}t(\tau),x(\tau)\big{)}$, where $\tau$ is proper time, so
that the 2-velocity $u^{\mu}(\tau)=dx(\tau)/d\tau$ satisfies the normalization
condition

$\eta_{\mu\nu}u^{\mu}u^{\nu}=1\,.$ (5.2)

The Lorentz-invariant condition for the acceleration being constant is
expressed in terms of $a^{\mu}(\tau)=\dot{u}^{\mu}(\tau)$ as

$\eta_{\mu\nu}a^{\mu}(\tau)a^{\nu}(\tau)=-a^{2}\,,$ (5.3)

where $a$ is a constant. The trajectory of a uniformly accelerated observer
satisfying these two conditions can be written as

$t(\tau)=\frac{1}{a}\sinh a\tau\,,\qquad x(\tau)=\frac{1}{a}\cosh a\tau\,.$
(5.4)

Next, let us relate the inertial frame to a frame that is comoving with the
observer. This means that denoting these coordinates by
$(\tilde{t},\tilde{x})$ the observer’s worldline is a vertical line
$\tilde{x}=0$, so that the observer is indeed at rest in this frame. The
Rindler coordinates having this property are defined by

$\displaystyle t$ $\displaystyle=a^{-1}e^{a\tilde{x}}\sinh a\tilde{t}\,,$
(5.5) $\displaystyle x$ $\displaystyle=a^{-1}e^{a\tilde{x}}\cosh
a\tilde{t}\,,$ (5.6)

and the inverse relation

$\displaystyle\tilde{t}$ $\displaystyle=\frac{1}{2a}\ln\frac{x+t}{x-t}\,,$
(5.7) $\displaystyle\tilde{x}$
$\displaystyle=\frac{1}{2a}\ln\big{[}a^{2}(x^{2}-t^{2})\big{]}\,.$ (5.8)

From these relations one finds the metric in Rindler coordinates,

$\text{d}s^{2}=(\text{d}t)^{2}-(\text{d}x)^{2}=e^{2a\tilde{x}}\big{[}(\text{d}\tilde{t})^{2}-(\text{d}\tilde{x})^{2}\big{]}\,,$
(5.9)

which is thus conformally equivalent to the Minkowski metric.

We now consider the action of a massless scalar field $\phi$ in a $1+1$
dimensional spacetime,

$S[\phi]=\frac{1}{2}\int\text{d}^{2}x\,\sqrt{-g}\,g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi\,,$
(5.10)

where $g_{\mu\nu}$ is the metric and $g$ is its determinant. In the inertial
frame,

$S[\phi]=\frac{1}{2}\int\text{d}t\text{d}x\big{[}(\partial_{t}\phi)^{2}-(\partial_{x}\phi)^{2}\big{]}\,.$
(5.11)

The action in the accelerated frame takes the same form:

$S[\phi]=\frac{1}{2}\int\text{d}\tilde{t}\text{d}\tilde{x}\big{[}(\partial_{\tilde{t}}\phi)^{2}-(\partial_{\tilde{x}}\phi)^{2}\big{]}\,,$
(5.12)

as a consequence of the conformal invariance of the action (5.10) in two
dimensions and the Rindler metric (5.9) being conformally equivalent to the
Minkowski metric. The equations of motion are

$\ddot{\phi}-\partial_{x}^{2}\phi=0\,,$ (5.13)
$\partial_{\tilde{t}}^{2}{\phi}-\partial_{\tilde{x}}^{2}\phi=0\,,$ (5.14)

where the dot denotes the partial derivative with respect to time $t$. Note
that as a scalar we have for the coordinate-transformed field
$\tilde{\phi}(\tilde{t},\tilde{x})=\phi(t,x)$, so that in the second equation
we could replace $\phi$ by $\tilde{\phi}$.

As a preparation for the homotopy retract we have to introduce the Fourier
transform with respect to the spatial coordinate, both in inertial and Rindler
coordinates:

$\phi_{k}(t):=\int_{-\infty}^{+\infty}\frac{\text{d}x}{\sqrt{2\pi}}e^{-ikx}\phi(t,x)\;,\qquad\tilde{\phi}_{l}(\tilde{t}):=\int_{-\infty}^{+\infty}\frac{\text{d}\tilde{x}}{\sqrt{2\pi}}e^{-il\tilde{x}}\tilde{\phi}(\tilde{t},\tilde{x})\;.$
(5.15)

Note that even though in the second integral we could replace
$\tilde{\phi}(\tilde{t},\tilde{x})$ by $\phi(t,x)$, the Fourier mode
$\phi_{k}$ as a function of $k$ of course differs from $\tilde{\phi}_{l}$ as a
function of $l$. The inverse relations are

$\phi(t,x)=\int_{-\infty}^{+\infty}\frac{\text{d}k}{\sqrt{2\pi}}e^{ikx}\phi_{k}(t)\;,\qquad\tilde{\phi}(\tilde{t},\tilde{x})=\int_{-\infty}^{+\infty}\frac{\text{d}l}{\sqrt{2\pi}}\,e^{il\tilde{x}}\tilde{\phi}_{l}(\tilde{t})\,.$
(5.16)

Since the scalar functions on the left-hand sides are equal (more precisely,
we have $\phi(t,x)=\tilde{\phi}(\tilde{t}(t,x),\tilde{x}(t,x))$), we have two
different expansions of the same $\phi$ into Fourier modes:

$\phi(t,x)=\int_{-\infty}^{+\infty}\frac{\text{d}k}{\sqrt{2\pi}}e^{ikx}\phi_{k}(t)\;=\int_{-\infty}^{+\infty}\frac{\text{d}l}{\sqrt{2\pi}}\,e^{il\tilde{x}(t,x)}\tilde{\phi}_{l}\big{(}\tilde{t}(t,x)\big{)}\,.$
(5.17)

We will also use the following change of basis for the Fourier modes and their
time derivatives:

$\displaystyle\phi_{k}$
$\displaystyle=\sqrt{\frac{\hbar}{2\omega_{k}}}\big{(}a^{\dagger}_{-k}+a_{k}\big{)}\,,\qquad\quad\quad\,\;\tilde{\phi}_{l}=\sqrt{\frac{\hbar}{2\Omega_{l}}}\big{(}b^{\dagger}_{-l}+b_{l}\big{)}\,,$
(5.18) $\displaystyle\dot{\phi}_{k}$
$\displaystyle=i\sqrt{\frac{\hbar\omega_{k}}{2}}\big{(}a^{\dagger}_{-k}-a_{k}\big{)}\,,\qquad\quad\;\partial_{\tilde{t}}\tilde{\phi}_{l}=i\sqrt{\frac{\hbar\Omega_{l}}{2}}\big{(}b^{\dagger}_{-l}-b_{l}\big{)}\,,$
(5.19)

where $\omega_{k}\equiv\sqrt{k^{2}}$, $\Omega_{l}\equiv\sqrt{l^{2}}$. The
inverse relations read:

$\displaystyle a_{k}$
$\displaystyle=\sqrt{\frac{\omega_{k}}{2\hbar}}\bigg{(}\phi_{k}+\frac{i}{\omega_{k}}{\dot{\phi}_{k}}\bigg{)}\,,\qquad
a_{-k}^{\dagger}=\sqrt{\frac{\omega_{k}}{2\hbar}}\bigg{(}\phi_{k}-\frac{i}{\omega_{k}}{\dot{\phi}_{k}}\bigg{)}\,,$
(5.20) $\displaystyle b_{l}$
$\displaystyle=\sqrt{\frac{\Omega_{l}}{2\hbar}}\bigg{(}\tilde{\phi}_{l}+\frac{i}{\Omega_{l}}{\partial_{\tilde{t}}\tilde{\phi}_{l}}\bigg{)}\,,\qquad
b_{-l}^{\dagger}=\sqrt{\frac{\Omega_{l}}{2\hbar}}\bigg{(}\tilde{\phi}_{l}-\frac{i}{\Omega_{l}}{\partial_{\tilde{t}}\tilde{\phi}_{l}}\bigg{)}\,.$
(5.21)

As for the harmonic oscillator these relations are motivated by the familiar
definition of creation and annihilation operators, but we emphasize that also
here these are just functions.

We now discuss the homotopy retract, beginning with the chain complex defining
the theory:

${0}$${V^{0}}$${V^{1}}$${0\;.}$$\scriptstyle{\partial}$ (5.22)

Here the space of fields and the space of anti-fields are given by

$V^{0}=C^{\infty}([t_{i},t_{f}]\times\mathbb{R})\;,\qquad V^{1}=\Pi
C^{\infty}([t_{i},t_{f}]\times\mathbb{R})\,.$ (5.23)

The notation indicates that the (anti-)fields depend on $t$, restricted to the
interval $[t_{i},t_{f}]$, and the space coordinate $x$ living on the full real
line $\mathbb{R}$. The differential is

$\partial(\phi)=(\partial_{t}^{2}-\partial_{x}^{2})\phi\,.$ (5.24)

The important new feature in field theory is that the projector
$p\,:V^{\bullet}\rightarrow C^{\infty}(\mathbb{R})\times
C^{\infty}(\mathbb{R})$ no longer maps to a finite-dimensional space like
$\mathbb{R}^{2}$ but to infinite-dimensional functions spaces, however, with
functions of one less coordinate. Specifically, the projector evaluates the
functions $a$ and $a^{\dagger}$ defined in (5.20) at $t_{i}$ and $t_{f}$,
respectively:

$\phi\mapsto\big{(}a_{k}(t_{i}),a_{l}^{\dagger}(t_{f})\big{)}\,,\qquad\phi^{*}\mapsto
0\,.$ (5.25)

Next, we need to define the inclusion map $i\,:C^{\infty}(\mathbb{R})\times
C^{\infty}(\mathbb{R})\rightarrow V^{0}$ that takes two functions in momentum
space, say $c(k)$ and $d(k)$, and produces a field in $V^{0}$ (i.e. in the
present example a scalar field in two-dimensional Minkowski space). The proper
inclusion map satisfying $p\circ i=1$ is given by

$\big{(}c,d\big{)}\mapsto\phi_{(c,d)}(t,x):=\int_{-\infty}^{+\infty}\frac{dk}{\sqrt{2\pi}}\,e^{ik{x}}\sqrt{\frac{\hbar}{2\omega_{k}}}\big{(}d(-k)e^{i\omega_{k}(t-t_{f})}+c(k)e^{-i\omega_{k}(t-t_{i})}\big{)}\,.$
(5.26)

The homotopy map $h:V^{1}\rightarrow V^{0}$ is defined, for any $f\in V^{1}$,
in terms of the Green’s function of the operator
$\partial_{t}^{2}-\partial_{x}^{2}$:

$h(f)(t,x)=\int_{t_{i}}^{t_{f}}\text{d}s\int_{-\infty}^{+\infty}\text{d}y\,K(t-s,x-y)f(s,y)\,,$
(5.27)

where the kernel is explicitly given by

$K(t-s,x-y)=\int_{-\infty}^{+\infty}\frac{\text{d}l}{4\pi\omega_{l}}\,i\big{(}\Theta(t-s)e^{-i\omega_{l}(t-s)+il(x-y)}+\Theta(s-t)e^{i\omega_{l}(t-s)-il(x-y)}\big{)}\,.$
(5.28)

Indeed, one can verify that with the above definitions for projector,
inclusion and homotopy the homotopy relation $\partial h+h\partial=1-ip$ is
satisfied. To this end one needs to assume that $\phi(t,x)$ and
$\partial_{x}{\phi}(t,x)$ vanish at $x=-\infty$ and $x=+\infty$.

For completeness we also display the important operations of the dual space of
functionals on which the BV algebra is defined. The BV complex
$\mathcal{F}(V^{\bullet})$ is equipped with the differential,

$Q=\int_{t_{i}}^{t_{f}}\text{d}t\int_{-\infty}^{\infty}\text{d}x\,\,\big{(}\ddot{\phi}(t,x)+\partial_{x}^{2}\phi(t,x)\big{)}\frac{\delta}{\delta\phi^{*}(t,x)}\,.$
(5.29)

In addition, the BV-differential is defined as

$\delta\equiv
Q-i\hbar\Delta\,,\qquad\Delta\equiv-\int_{t_{i}}^{t_{f}}\text{d}t\,\int_{-\infty}^{\infty}\text{d}x\,\,\frac{\delta}{\delta\phi^{*}(t,x)}\frac{\delta}{\delta\phi(t,x)}\,.$
(5.30)

For a functional $F[\phi,\phi^{*}]$ in $\mathcal{F}(V^{\bullet})$, we obtain
the pull-back functional in $\mathcal{F}[C^{\infty}(\mathbb{R})\times
C^{\infty}(\mathbb{R})]$ defined by

$i^{*}(F)(c,d)=(F\circ i)(c,d)\,.$ (5.31)

Similarly, the pullback of a functional $f$ in
$\mathcal{F}[C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})]$ with
respect to the projection is the functional in $\mathcal{F}(V^{\bullet})$
given by

$p^{*}(f)(\phi,\phi^{*})=(f\circ p)(\phi,\phi^{*})\,.$ (5.32)

###  5.2 Number Expectation Value

To derive the Unruh effect, one assumes that the number of particles measured
by an accelerated observer is given by the expectation value of the number
operator with respect to Rindler space, i.e., with respect to creation and
annihilation operators defined with the Fourier modes in Rindler space. More
precisely, one computes

${\cal N}_{k}:=\langle N_{k}\rangle\equiv\langle
0|\hat{b}^{\dagger}_{k}\hat{b}_{k}|0\rangle\,,$ (5.33)

where $\hat{b}_{k}$ and $\hat{b}^{\dagger}_{k}$ are the Rindler space
annihilation and creation operators defined in analogy to (5.21), and
$|0\rangle$ is the Minkowski vacuum state. This state is defined so that it is
annihilated by the inertial frame operator $\hat{a}_{k}$:

$\hat{a}_{k}|0\rangle=0\,.$ (5.34)

For definiteness we take the Heisenberg picture operators $b$ and
$b^{\dagger}$ to be at time $\tilde{t}=0$ (which is equivalent to $t=0$ for
all $x$). The usual textbook computation involves relating the creation and
annihilation operators of the accelerated and inertial frames through
Bogolyubov transformations. We provide an alternative approach which does not
require finding the Bogolyubov transformations. Instead, our strategy is to
define the functional $F[\phi]$ of the massless scalar field $\phi$ to be
given by $b_{k}^{\dagger}b_{k}$, with $b_{k}^{\dagger}$ and $b_{k}$ being
defined in terms of the classical field $\phi$ via (5.21). Following our
approach for the harmonic oscillator in sec. 4, we then find $f(c,d)$ such
that $F^{\prime}=f\circ p$ is in the same cohomology class as $F[\phi]$. Then
$f(c,d)$ computes the expectation value

$f(c,d)=\lim_{\tilde{t}\rightarrow 0}\frac{\langle
d|T\big{(}\hat{b}_{k}^{\dagger}(\tilde{t})\hat{b}_{k}(0)\big{)}|c\rangle}{\braket{d}{c}}\;,$
(5.35)

where $|c\rangle$ and $|d\rangle$ are coherent states with respect to $a_{k}$,
i.e.,

$a_{k}|c\rangle=c(k)|c\rangle\,,$ (5.36)

and analogously for $|d\rangle$. Here we take the limit $\tilde{t}\rightarrow
0$ after performing the computation, as opposed to setting $\tilde{t}=0$ from
the beginning, since some care is needed in order to deal with the step
functions entering the Green’s function. Note that the result does not depend
on whether one takes the limit from above or from below, which follows from
the symmetry of the Green’s function. Finally, in order to find the
expectation value of the Rindler number operator with respect to the Minkowski
vacuum, we set $c=d=0$, i.e.,

${\cal N}_{k}=f(0,0)\,.$ (5.37)

The choice $c=d=0$ is the analog of the equation (5.34) specifying the
Minkowski vacuum.

We begin by expressing the functional $b^{\dagger}_{k}(\tilde{t})b_{k}(0)$ in
terms of $\phi(t,x)$. By taking the Fourier transform of (5.21), one obtains
$b_{k}$ and $b_{k}^{\dagger}$ in terms of $\phi$ and
$\partial_{\tilde{t}}{\phi}$:

$b_{k}(\tilde{t})\,\,\,=\int\text{d}\tilde{x}\,e^{-ik\tilde{x}}\sqrt{\frac{\Omega_{k}}{4\pi\hbar}}\bigg{(}\phi+\frac{i}{\Omega_{k}}\partial_{\tilde{t}}{\phi}\bigg{)}\,,$
(5.38)
$b^{\dagger}_{k}(\tilde{t})=\int\text{d}\tilde{x}\,e^{ik\tilde{x}}\sqrt{\frac{\Omega_{k}}{4\pi\hbar}}\bigg{(}\phi-\frac{i}{\Omega_{k}}\partial_{\tilde{t}}{\phi}\bigg{)}\,.$
(5.39)

For the second equation we use the chain rule to obtain

$\partial_{\tilde{t}}\phi=\frac{\partial
t}{\partial\tilde{t}}\,\dot{\phi}+\frac{\partial
x}{\partial{\tilde{t}}}\,\partial_{x}\phi=e^{a\tilde{x}}\cosh(a\tilde{t})\dot{\phi}+e^{a\tilde{x}}\sinh(a\tilde{t})\partial_{x}\phi\;.$
(5.40)

Note that this is only valid when $x>|t|$ since the Rindler coordinates only
cover this part of the Minkowski spacetime. With (5.38) – (5.40), we can
explicitly write out the functional
$F[\phi]=b^{\dagger}_{k}(\tilde{t})b_{k}(0)$:

$\begin{split}F[\phi]=\int\text{d}\tilde{x}\int\text{d}\tilde{y}\,&e^{ik(\tilde{x}-\tilde{y})}\frac{\Omega_{k}}{4\pi\hbar}\bigg{(}\phi(t,x)\phi(0,y)+\frac{i}{\Omega_{k}}e^{a\tilde{y}}\phi(t,x)\dot{\phi}(0,y)\\\
&+\frac{1}{\Omega_{k}^{2}}e^{a(\tilde{x}+\tilde{y})}\big{(}\cosh(a\tilde{t})\dot{\phi}(0,y)\dot{\phi}(t,x)+\sinh(a\tilde{t})\dot{\phi}(0,y)\partial_{x}\phi(t,x)\big{)}\\\
&-\frac{i}{\Omega_{k}}e^{a\tilde{x}}\big{(}\cosh(a\tilde{t})\phi(0,y)\dot{\phi}(t,x)+\sinh(a\tilde{t})\phi(0,y)\partial_{x}\phi(t,x)\big{)}\bigg{)}\,,\end{split}$
(5.41)

where of course $t$ and $x$ on the right-hand side must be viewed as functions
of $(\tilde{t},\tilde{x})$.

We apply the perturbation lemma to find $f(c,d)$, by using $P_{1}$ in (3.34),

$P_{1}=i^{*}\exp\left(-\frac{i\hbar}{2}C\right)\;,$ (5.42)

where the functional derivatives in the $C$ operator are now with respect to
$\phi(t,x)$:

$C=\int\text{d}t\,\text{d}x\,\text{d}s\,\text{d}y\,K(t-s,x-y)\,\frac{\delta^{2}}{\delta\phi(t,x)\delta\phi(s,y)}\,,$
(5.43)

and $K(t-s,x-y)$ is given in (5.28). Applying $P_{1}$ on $F[\phi]$,

$\begin{split}P_{1}(F)(c,d)\,=\,&\,\,\,i^{*}F(c,d)\\\
&-{i\hbar}\int\text{d}\tilde{x}\int\text{d}\tilde{y}\,\,e^{ik(\tilde{x}-\tilde{y})}\frac{\Omega_{k}}{4\pi\hbar}\bigg{(}K(t,x-y)-\frac{i}{\Omega_{k}}e^{a\tilde{y}}\partial_{t}K(t,x-y)\\\
&-\frac{1}{\Omega_{k}^{2}}e^{a(\tilde{x}+\tilde{y})}\big{[}\cosh(a\tilde{t})\,\partial_{t}\partial_{t}K(t,x-y)+\sinh(a\tilde{t})\partial_{t}\partial_{x}K(t,x-y)\big{]}\\\
&-\frac{i}{\Omega_{k}}e^{a\tilde{x}}\big{[}\cosh(a\tilde{t})\partial_{t}K(t,x-y)+\sinh(a\tilde{t})\partial_{x}K(t,x-y)\big{]}\bigg{)}\,.\end{split}$
(5.44)

There are no further terms in the expansion of
$\exp\left(-\frac{i\hbar}{2}C\right)$ because $F[\phi]$ only contains two
$\phi$s. Let us start by treating the first term on the right-hand side of
(5.44). Since we set $c=d=0$, the inclusion to the space of fields (5.26) is
$i(0,0)=0$. Therefore, with (5.31), the first term on the right-hand side of
(5.44) vanishes:

$i^{*}F(0,0)=0\,.$ (5.45)

Next, we take the limit $t=\tilde{t}=0$. After inserting the derivatives of
$K(t,s)$, using (5.28), and writing these in terms of Rindler coordinates, we
obtain

$\begin{split}f(0,0)&=P_{1}(F)(0,0)\\\
&=\int\text{d}\tilde{x}\int\text{d}\tilde{y}\int\text{d}l\,\frac{\Omega_{k}}{16\pi^{2}\omega_{l}}e^{ik(\tilde{x}-\tilde{y})}e^{ila^{-1}(e^{a\tilde{x}}-e^{a\tilde{y}})}\bigg{(}1+\frac{1}{\Omega_{k}^{2}}e^{a(\tilde{x}+\tilde{y})}\omega_{l}^{2}-\frac{\omega_{l}}{\Omega_{k}}e^{a\tilde{x}}-\frac{\omega_{l}}{\Omega_{k}}e^{a\tilde{y}}\bigg{)}\,.\end{split}$
(5.46)

We now perform the change of variables:

$u=e^{a\tilde{x}}\,,\quad\frac{1}{au}du=d\tilde{x}\,,$ (5.47)
$v=e^{a\tilde{y}}\,,\quad\frac{1}{av}du=d\tilde{y}\,.$ (5.48)

Then $f(0,0)$ takes the form

$\begin{split}f(0,0)=\int_{0}^{\infty}\text{d}u\int_{0}^{\infty}\text{d}v\int_{-\infty}^{\infty}\text{d}l\,e^{ika^{-1}(\ln
u-\ln
v)}\frac{\Omega_{k}}{16\pi^{2}a^{2}\omega_{l}}e^{ila^{-1}(u-v)}\bigg{(}\frac{1}{uv}+\frac{\omega_{l}^{2}}{\Omega_{k}^{2}}-\frac{\omega_{l}}{\Omega_{k}}\frac{1}{v}-\frac{\omega_{l}}{\Omega_{k}}\frac{1}{u}\bigg{)}\,.\end{split}$
(5.49)

Evaluating the integrals over $u$ and $v$,777 For this computation we used the
integral identities:
$\begin{split}\int_{0}^{\infty}dx\;e^{iA\ln(x)}e^{iBx}x^{-1}\;=\;(-iB)^{-iA}\Gamma(iA)\,,\qquad\int_{0}^{\infty}dx\;e^{iA\ln(x)}e^{iBx}\;=\;(iA)(-iB)^{-1-iA}\Gamma(iA)\,.\end{split}$
(5.50)

$\begin{split}&f(0,0)=\int_{-\infty}^{\infty}\text{d}l\,\,\frac{\Omega_{k}}{4\pi^{2}a^{2}\omega_{l}}\Gamma\bigg{(}-\frac{ik}{a}\bigg{)}\Gamma\bigg{(}\frac{ik}{a}\bigg{)}(-1)^{ik/a}\,.\end{split}$
(5.51)

As a consistency check, one may verify that the integrand in (LABEL:0bb0comp)
coincides with the expression in equation (8.43) of [31]. By using the
identity for Gamma functions,

$|\Gamma(ik/a)|^{2}=\frac{\pi a}{k\sinh(\pi k/a)}=\frac{2\pi
a}{|k|}\frac{e^{\pi|k|/a}}{(e^{2\pi|k|/a}-1)}\,,$ (5.52)

we obtain

$\begin{split}f(0,0)&=(e^{2\pi|k|/a}-1)^{-1}\int_{-\infty}^{\infty}\text{d}l\,\,\frac{1}{2\pi
a\omega_{l}}\,,\end{split}$ (5.53)

as long as we choose $(-1)^{ik/a}=e^{-\pi|k|/a}$. The expectation value of the
number of particles observed by an accelerated observer is a Bose-Einstein
distribution with the Unruh temperature

$T=\frac{\hbar a}{2\pi k_{B}}\;,$ (5.54)

where $k_{B}$ is the Boltzmann constant. The divergent integral in (5.53) is
also present in the conventional derivation of the Unruh effect (see, e.g.,
chapter 8 in [31]) and is interpreted as the infinite volume of the entire
space.

##  6 Summary and Outlook

The main result of this paper is a (partial) reformulation of quantum
mechanics that parallels the path integral in that there is no reference to
Hilbert spaces, states, operators, etc. However, in contrast to the path
integral formulation, the homological approach presented here is algebraic,
based on the cohomology of the BV algebra. In this one employs a homotopy
retract from the infinite-dimensional space of all possible trajectories to
the finite-dimensional phase space, thereby circumventing the problem to make
sense of the path integral over the full infinite-dimensional space. We have
shown with a number of examples that the homological formulation allows one to
compute concrete quantum expectation values that agree with those determined
by standard quantum mechanics. However, so far this reformulation is not
complete: it only provides a prescription to compute certain normalized
quantum expectation values with respect to certain states. It then remains as
the most important outstanding problem to explore whether this homological
formulation could be completed to a full-fledged reformulation of quantum
mechanics (and quantum field theory).

It is instructive to compare the techniques presented here with other
approaches in the literature, see [22, 23, 24, 25]. The idea in these
references is to pass via homotopy transfer from the $L_{\infty}$-algebra of a
given theory to a ‘minimal model’ or ‘on-shell’ $L_{\infty}$-algebra on the
cohomology (i.e. with all differentials being trivial). These $L_{\infty}$
brackets compute (at least tree-level) scattering amplitudes, but one has to
overcome some technical challenges. First, in order for the action and inner
product to be well-defined the space of functions is restricted to Schwartz
functions, but then there is no cohomology, no on-shell fields and hence no
minimal model (for Schwartz functions $\square\phi=0$ implies $\phi=0$). One
attempts to circumvent this problem by adding on-shell states by hand in
degree zero and degree one, so that there is a minimal model. However, a
priori the $L_{\infty}$ brackets are then not well-defined since the product
of two on-shell fields is neither on-shell nor a Schwartz function. In order
to remedy this ref. [23] introduces certain regularizing factors in the
products of fields.

In the homological formulation of this paper these issues do not arise. An
important reason is that we do not consider smooth functions on $\mathbb{R}$
but rather on the finite integral $[t_{i},t_{f}]$. Then it is not necessary to
restrict to Schwartz functions and so there is non-trivial cohomology.888These
remarks apply in the realm of quantum mechanics where dynamical variables
depend only on time. In genuine quantum field theories, the functional
dependence on the spatial coordinates is probably best chosen to be of
Schwartz type, but there is still cohomology due to the time dependence being
more general. However, since in our formulation the cohomology is concentrated
in degree zero there is no non-trivial $L_{\infty}$-algebra on this space;
rather, the quantum expectation values are computed by the functions of the
homotopy retract of the BV algebra, as described in the main text.
Technically, the price to pay for working with general smooth functions on
$[t_{i},t_{f}]$ is that the symplectic form encoded in the anti-bracket is no
longer invariant under the vector field $Q$. However, in our formulation the
symplectic form does not enter, and so this does not cause any problems. (See
also [19, 20] where such a more general BV formalism was developed.)

We close this paper with a brief list of interesting open problems:

  * •

Most intriguingly, the homological formulation is arguably mathematically
well-defined for non-perturbative problems. It would then be important to
apply and illustrate this method for genuinely non-perturbative problems.

  * •

In this paper we have dealt with theories without gauge symmetries, so it
would be interesting to consider gauge field theories such as Yang-Mills
theory. Since the BV formalism was originally introduced in order to deal with
subtle issues of gauge theories it should be straightforward to set up the
corresponding BV algebra. However, it would still be instructive to work out
the homotopy retract and the details of the homological formulation.

  * •

One of the potentially most fruitful applications of the homological
formulation may arise for quantum field theory on curved spacetime, where
traditional flat space techniques to quantization have often been awkward. Our
investigation was in fact motivated by the desire to find a systematic recipe
to obtain in cosmological perturbation theory the quantum correlation
functions directly from the $L_{\infty}$-algebra or the dual BV algebra. In
[28] we gave an interpretation of the passing over to gauge invariant so-
called Bardeen variables of cosmological perturbation theory in terms of
homotopy transfer but it remains to give a similar procedure for the
computation of cosmological correlation functions.

### Acknowledgements

We would like to thank Roberto Bonezzi, Tomas Codina and Felipe Diaz-Jaramillo
for useful discussions.

This work is funded by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No
771862) and by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation), “Rethinking Quantum Field Theory”, Projektnummer
417533893/GRK2575.

##  Appendix A Homological Algebra

In this work we will frequently use the language of homological algebras. In
this appendix we introduce all definitions and facts we will use.