# Kinetic Field Theory:
Higher-Order Perturbation Theory

Lavinia Heisenberg<EMAIL_ADDRESS>Shayan Hemmatyar
<EMAIL_ADDRESS>Stefan Zentarra<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093
Zurich, Switzerland  Institute for Theoretical Physics, Heidelberg University,
Philosophenweg 16, 69120 Heidelberg, Germany

###### Abstract

We give a detailed exposition of the formalism of Kinetic Field Theory (KFT)
with emphasis on the perturbative determination of observables. KFT is a
statistical non-equilibrium classical field theory based on the path integral
formulation of classical mechanics, employing the powerful techniques
developed in the context of quantum field theory to describe classical
systems. Unlike previous work on KFT, we perform the integration over the
probability distribution of initial conditions in the very last step. This
significantly improves the clarity of the perturbative treatment and allows
for physical interpretation of intermediate results. We give an introduction
to the general framework, but focus on the application to interacting $N$-body
systems. Specializing the results to cosmic structure formation, we reproduce
the linear growth of the cosmic density fluctuation power spectrum on all
scales from microscopic, Newtonian particle dynamics alone.

###### keywords:

Kinetic Field Theory (KFT) , Perturbation Theory , N-Body Systems , Cosmic
Structure Formation , Cosmology

††journal: Physical Review D

myformat . [] myformatshort th [],

###### Contents

  1. 1 Introduction
  2. 2 KFT Framework
    1. 2.1 Path Integral Formulation of Classical Mechanics
    2. 2.2 Free Generating Functional
    3. 2.3 Interactions
    4. 2.4 Choice of Free Motion
  3. 3 Interacting N-body Systems
    1. 3.1 Generating Functional
    2. 3.2 Density Correlation Functions
  4. 4 KFT Perturbation Scheme
    1. 4.1 Taylor Expansion
    2. 4.2 Feynman Rules
    3. 4.3 Symmetry Factors and Shot-Noise
    4. 4.4 Particle Trajectories and Interpretation
    5. 4.5 Density Correlation Functions and Power Spectra
  5. 5 Integration over Initial Conditions
    1. 5.1 Toy Model: Gaussian Distribution
    2. 5.2 Toy Model: Shell-Crossing
    3. 5.3 Correlated Initial Conditions
    4. 5.4 Correlation Hierarchy
    5. 5.5 Correlations vs. Interactions
  6. 6 Reproducing Linear Cosmic Structure Growth
    1. 6.1 Free Linear Power Spectrum
    2. 6.2 Higher-Order Linear Power Spectrum
  7. 7 Conclusions and Outlook
  8. Acknowledgements
  9. A Particles on an Expanding Background
  10. B Poisson Equation for the Initial Density Contrast

##  1 Introduction

In a 1988 paper [1], E. Gozzi outlined how the successful path integral
formalism of quantum field theory can be used to describe classical systems as
well. In the following years, Gozzi further developed and studied this
approach in collaboration with M. Reuter and W.D. Thacker [2, 3] and others
[4, 5]. Relatively recently, G.F. Mazenko and S.P. Das [6, 7] developed a
perturbation scheme and applied the formalism to phase transitions in glass.
Based on this, the group of M. Bartelmann began to apply the path integral
approach of classical mechanics to the study of cosmic structure formation
[8]. Since then, the approach has been subject to significant further
development [9, 10, 11, 12, 13] and is now known as _Kinetic Field Theory
(KFT)_ [14].

Conceptually, KFT mimics the approach of numerical simulations to describe the
time evolution of an interacting $N$-particle system. Via use of path
integrals over phase space trajectories, one keeps track of the positions and
momenta of all particles over time. Mathematically, this information is stored
in the generating functional $Z$ of the theory. Collective quantities like,
e.g., the density field can be extracted via functional derivatives of $Z$.
Explicit expressions for the generating functional are obtained either via a
perturbative approach [8] or through the use of suitable approximations [9].

There is one significant difference between the formalism of KFT and numerical
$N$-body simulations. Namely, the latter always have to start out with some
explicit initial conditions, i.e., initial positions and momenta for all
particles. A priori this is also true for KFT. However, in an analytic setting
we can abstractly integrate over a probability distribution of initial
conditions to obtain expectation values. This would be prohibitively expensive
for numerical $N$-body simulations which therefore have to resort to other
methods of extracting macroscopic quantities. The integration over initial
conditions in KFT yields an ensemble average which in many applications is the
observable we are interested in.

In this article we revisit to the initial perturbative approach to KFT in [8].
We give a self-contained and pedagogical introduction to the framework of KFT
in a general setting and then specialize it to interacting $N$-body systems.
Unlike the perturbative treatment in [8], we perform the integration over the
initial conditions not on the level of the generating functional, but on the
level of observables. This allows us to interpret intermediate results
describing observables of a system with explicit initial conditions. Upon
integrating them over a suitable probability distribution, we obtain
statistically averaged expectation values, i.e., macroscopic observables
describing the collective system. While ultimately mathematically equivalent,
this change in procedure makes the perturbative treatment significantly more
transparent and simpler.

Our ultimate aim is to supply a simple and consistent procedure for obtaining
observables in cosmic structure formation, in particular the non-linear
density fluctuation power spectrum. The initial probability distribution for
this particular application is relatively complicated. For the sake of self-
containedness and clarity, we repeat the discussion from appendix $A$ of [8].
In particular, we explain how the integration over initial conditions fits
together with the perturbative treatment. Having done so, we will be able to
obtain analytic parameter-free perturbative expressions for the non-linear
density fluctuation power spectrum. These expressions are expanded in both the
initial correlations and the interactions. More precisely, on the one hand we
expand the density and momentum correlations imprinted in the initial
conditions of the $N$-particle system and on the other hand we treat the
particle interactions perturbatively by expanding the forces relative to the
free particle trajectories.

Surprisingly, we find that the expansion in the initial correlations is
significantly more difficult than the expansion in the interactions. Indeed,
for higher-order correlations our expressions acquire integrals over wave
vectors which are challenging to solve even numerically. Conversely, each
interaction order merely supplies a simple time integral. To conclude this
article, we calculate the density fluctuation power spectrum in first order
(i.e. linear) correlations and up to very high interaction order. Combining
the contributions for linear correlations and arbitrarily high interaction
orders precisely reproduces the linear growth of the power spectrum at all
scales. This reveals that any non-linearities in the power spectrum are solely
due to higher-order correlations in the initial conditions and thus can be
studied completely independently of the linear evolution. Results for higher-
order correlations will be presented in a future article [15].

##  2 KFT Framework

Let us consider a generic classical physical system. Denote its space of
possible states by $\boldsymbol{X}$, such that any point
$\boldsymbol{x}\in\boldsymbol{X}$ uniquely characterizes a state of this
system.111All bold faced symbols take values in $\boldsymbol{X}$, i.e., they
are generally some high-dimensional vectors or tensors. In practice, we
usually take $\boldsymbol{X}$ to be phase space consisting of all
(generalized) coordinates and associated momenta. Any evolution of such a
system can be described as a map
$\boldsymbol{\varphi}\colon\mathbb{R}\rightarrow\boldsymbol{X}$ which
associates to any point in time $t\in\mathbb{R}$ a state of the system
$\boldsymbol{\varphi}(t)\in\boldsymbol{X}$. However, the vast majority of such
maps describe unphysical evolution. We are only interested in the maps which
satisfy the equations of motion which we symbolically write as
$\boldsymbol{E}(\boldsymbol{\varphi}(t))=\boldsymbol{0}$. In almost all cases,
the equations of motion are differential equations which require that the
change of the state is determined only by the state of the system at the same
instance in time. For classical systems, given a state
$\boldsymbol{x}\in\boldsymbol{X}$ at initial time $t_{i}$, there is exactly
one solution of the equations of motion
$\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$ which satisfies
$\boldsymbol{\tilde{\varphi}}(t_{i};\boldsymbol{x})=\boldsymbol{x}$.

In a more general setting, we might not precisely know the initial state
$\boldsymbol{x}\in\boldsymbol{X}$, but instead our knowledge of the system at
initial time $t_{i}$ may be encoded in form of a probability distribution
$P(\boldsymbol{x})$ on the space $\boldsymbol{X}$. In case
$P(\boldsymbol{y})=\delta_{D}(\boldsymbol{y}-\boldsymbol{x})$ is given by a
Dirac $\delta$-function, we recover the deterministic setting above. However,
if the probability distribution is more complicated, the state of the system
is stochastic for all times $t$. This means that for any time $t$ there is a
unique probability distribution $P(\boldsymbol{x};t)$ describing the system.
In fact, it is simple to give an explicit expression for this probability
distribution by using the solution to the equations of motion,

$\displaystyle
P(\boldsymbol{x};t)=P\left(\boldsymbol{\tilde{\varphi}}(-t;\boldsymbol{x})\right)\,,$
(1)

where on the right-hand side we have the initial probability distribution.
Obviously it is $P(\boldsymbol{x};t_{i})=P(\boldsymbol{x})$.

The probability distribution $P(\boldsymbol{x};t)$ allows us to extract
expressions for observables at any point in time. As an observable we regard
in this context any function
$\mathcal{O}\colon\boldsymbol{X}\rightarrow\mathbb{R}$, i.e., a map which
assigns to a state of a system some number (this can easily be generalized to
tensor-valued observables). For such an observable, we define its expectation
value at time $t$ as

$\displaystyle\langle\mathcal{O}\rangle(t)=\int_{\boldsymbol{X}}\mathrm{d}\boldsymbol{x}\,\mathcal{O}(\boldsymbol{x})P(\boldsymbol{x};t)\,.$
(2)

In the deterministic case
$P(\boldsymbol{y})=\delta_{D}(\boldsymbol{y}-\boldsymbol{x})$, we simply have

$\displaystyle\langle\mathcal{O}\rangle(t)=\tilde{\mathcal{O}}(t;\boldsymbol{x})\coloneqq\mathcal{O}\left(\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})\right)\,.$
(3)

Here we defined the notation $\tilde{\mathcal{O}}(t;\boldsymbol{x})$ for the
observable evaluated along the solution of the equations of motion for initial
value $\boldsymbol{x}$ at time $t$.

From the discussion above one may arrive at a seemingly simple recipe to
calculate observables: determine the solution $\boldsymbol{\tilde{\varphi}}$
and use it to express any quantity of interest. However, as is well known,
even for seemingly simple classical systems one can prove that there is no
elementary expression for $\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$. In
many cases one can determine $\boldsymbol{\tilde{\varphi}}$ numerically
instead, but this can become computationally costly for systems with a high-
dimensional space of states $\boldsymbol{X}$. A prime example for this is an
interacting $N$-particle system. The formalism of KFT provides an alternative
approach for calculating observables by analytical methods which we want to
introduce below.

###  2.1 Path Integral Formulation of Classical Mechanics

The formalism of KFT is based on the path integral formulation of classical
mechanics which to the best of our knowledge was first proposed by E. Gozzi
[1]. Readers familiar with the path integral approach to Quantum Field Theory
(QFT) will notice many similarities between KFT and QFT. In fact, these
similarities are one of the main theoretical motivations to use KFT, as one
can use the powerful tools developed in the framework of QFT to describe the
evolution of a classical physical system.

Let us consider a classical physical system which at initial time $t_{i}$ is
in a state $\boldsymbol{x}\in\boldsymbol{X}$. Denote the solution of the
equations of motion $\boldsymbol{E}(\boldsymbol{\varphi})=\boldsymbol{0}$
subject to these initial conditions by
$\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$. Further, let
$\boldsymbol{y}\in\boldsymbol{X}$ be some possible state of the system at a
later time $t>t_{i}$. Then the transition probability
$T^{fi}(\boldsymbol{y},t;\boldsymbol{x},t_{i})$, i.e., the probability of
observing $\boldsymbol{y}$ at $t>t_{i}$ given initial conditions
$\boldsymbol{x}$ at $t=t_{i}$, is given by

$\displaystyle
T^{fi}(\boldsymbol{y},t;\boldsymbol{x},t_{i})=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}^{\boldsymbol{\varphi}(t)=\boldsymbol{y}}\mathcal{D}\boldsymbol{\varphi}\,\delta_{D}\left[\boldsymbol{\varphi}(t)-\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})\right]\,.$
(4)

The interpretation of this quantity is rather simple: While the path integral
adds contributions from all possible paths in state space $\boldsymbol{X}$
from $\boldsymbol{x}$ to $\boldsymbol{y}$, the $\delta_{D}$-functional (a
generalization of the Dirac $\delta$-function) picks out a single one of
these. This is reminiscent of taking the classical limit in the path integral
formulation of quantum mechanics, where the classical path supplies the
dominant contribution as $\hbar\rightarrow 0$.

Clearly, the transition probability
$T^{fi}(\boldsymbol{y},t;\boldsymbol{x},t_{i})$ is one if
$\boldsymbol{y}=\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$ and zero
otherwise. This is not surprising given that the evolution of classical
systems with explicitly specified initial conditions is deterministic. Using
the fact that the path selected by the $\delta_{D}$-functional is the solution
of the equations of motion, we can replace condition
$\boldsymbol{\varphi}(t)=\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$
encoded in its argument by the condition
$\boldsymbol{E}(\boldsymbol{\varphi})=\boldsymbol{0}$ which avoids explicit
reference to the solution $\boldsymbol{\tilde{\varphi}}$. Generally, such a
change in the argument of a $\delta_{D}$-functional changes the normalization
according to

$\displaystyle\delta_{D}\left[\boldsymbol{E}(\boldsymbol{\varphi})\right]=\left\lvert\det\left(\frac{\delta\boldsymbol{E}}{\delta\boldsymbol{\varphi}}\left(\boldsymbol{\tilde{\varphi}}\right)\right)\right\rvert^{-1}\,\delta_{D}\left[\boldsymbol{\varphi}-\boldsymbol{\tilde{\varphi}}\right]\,.$
(5)

This is a generalization of the identity

$\displaystyle\delta_{D}\left(f(x)\right)=\left\lvert\det\left(\frac{\partial
f}{\partial
x}\left(\tilde{x}\right)\right)\right\rvert^{-1}\,\delta_{D}\left(x-\tilde{x}\right)$
(6)

for the usual Dirac $\delta$-function which is valid if the function $f$ has
exactly one root $x=\tilde{x}$. However, as proven in equation $(3.51)$ of
[2], the determinant in equation (5) is constant in the case of Hamiltonian
equations of motion. This may be understood in terms of Liuville’s theorem
which states that the Hamiltonian flow in phase space preserves volumes. The
numerical value of the determinant is irrelevant and can be absorbed into the
normalization of the path integral.

While it is possible to keep the determinant in equation (5), it would
unnecessarily complicate our treatment. Therefore, we shall restrict ourselves
to the case of Hamiltonian dynamics, i.e., the space of states
$\boldsymbol{X}$ actually is phase space and the equations of motion
$\boldsymbol{E}(\boldsymbol{\varphi})=\boldsymbol{0}$ are of Hamiltonian form

$\displaystyle\boldsymbol{E}(\boldsymbol{\varphi})=\partial_{t}\boldsymbol{\varphi}-\boldsymbol{\mathcal{J}}\,\partial_{\boldsymbol{\varphi}}H(\boldsymbol{\varphi})=\boldsymbol{0}\,.$
(7)

Here, $\boldsymbol{\mathcal{J}}$ is the usual symplectic matrix and $H$ is the
Hamiltonian. Using equation (5), the transition probability can be expressed
as

$\displaystyle T^{fi}(\boldsymbol{y},t;\boldsymbol{x},t_{i})$
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}^{\boldsymbol{\varphi}(t)=\boldsymbol{y}}\mathcal{D}\boldsymbol{\varphi}\,\delta_{D}\left[\boldsymbol{E}(\boldsymbol{\varphi})\right]$
(8)
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}^{\boldsymbol{\varphi}(t)=\boldsymbol{y}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\exp\left(i\int_{t_{i}}^{t}\mathrm{d}t^{\prime}\,\boldsymbol{\chi}(t^{\prime})\cdot\boldsymbol{E}(\boldsymbol{\varphi}(t^{\prime}))\right)\,.$
(9)

In the second line, we used a standard functional Fourier transform to write
the $\delta_{D}$-functional as an exponential function. The object
$\boldsymbol{\chi}$ is a purely auxiliary abstract mathematical function and
does not have an obvious physical interpretation. In analogy to the path
integral approach to QFT, we may regard the exponent as the action of the
system under consideration.

Further building on the analogy to QFT, we now define the so-called
_generating functional_ which is the central mathematical object of KFT.
Integrating over the final state $\boldsymbol{y}$ of the system (or,
equivalently, setting
$\boldsymbol{y}\coloneqq\boldsymbol{\tilde{\varphi}}(\boldsymbol{x},t)$) to
get rid of the endpoint of the path integral over $\boldsymbol{\varphi}$ and
introducing a source field $\boldsymbol{J}$ for $\boldsymbol{\varphi}$, it is

$\displaystyle
Z[\boldsymbol{J};\boldsymbol{x}]=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\exp\left(i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{\chi}(t^{\prime})\cdot\boldsymbol{E}(\boldsymbol{\varphi}(t^{\prime}))+i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\varphi}(t^{\prime})\right)\,.$
(10)

The generating functional contains complete information about the physical
system under consideration. Any observable depending on the phase space
position at a single time $t$ can be derived from it by means of functional
derivation with respect to $\boldsymbol{J}(t)$.222In principle, it is also
possible to construct observables depending on the phase space variables on a
time-interval (e.g. a backward lightcone). While this may become relevant in
the future, here we restrict ourselves to the simpler case. Indeed, for a
given observable $\mathcal{O}\colon\boldsymbol{X}\rightarrow\mathbb{R}$ it is

$\displaystyle\tilde{\mathcal{O}}(t;\boldsymbol{x})$
$\displaystyle={\left.\kern-1.2pt\mathcal{O}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t)}\right)\,Z[\boldsymbol{J};\boldsymbol{x}]\vphantom{\big{|}}\,\right|_{\boldsymbol{J}=\boldsymbol{0}}}$
(11)
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\mathcal{O}\left(\boldsymbol{\varphi}(t)\right)\,\exp\left(i\int_{t_{i}}^{t}\mathrm{d}t^{\prime}\,\boldsymbol{\chi}(t^{\prime})\cdot\boldsymbol{E}(\boldsymbol{\varphi}(t^{\prime}))\right)$
(12)
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\,\mathcal{O}\left(\boldsymbol{\varphi}(t)\right)\,\delta_{D}\left[\boldsymbol{E}(\boldsymbol{\varphi})\right]$
(13)
$\displaystyle=\mathcal{O}\left(\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})\right)\,.$
(14)

We used the notation $\tilde{\mathcal{O}}$ introduced in equation (3) to
denote the observable as evaluated along the solution of the equations of
motion $\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$ for initial value
$\boldsymbol{x}$ – which evidently is exactly what we obtain via this
differentiation. In fact, this can also be seen by performing the integration
over $\boldsymbol{\chi}$ directly in the generating functional which yields

$\displaystyle Z[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\,\exp\left(i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\varphi}(t^{\prime})\right)\,\delta_{D}\left[\boldsymbol{E}(\boldsymbol{\varphi})\right]$
(15)
$\displaystyle=\exp\left(i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\tilde{\varphi}}(t^{\prime};\boldsymbol{x})\right)\,.$
(16)

We remark that we implicitly assume that the observable $\mathcal{O}$ is an
analytic function of the state $\boldsymbol{x}\in\boldsymbol{X}$, i.e., it is
possible to write $\mathcal{O}$ as convergent power series in $\boldsymbol{x}$
(to make equation (11) mathematically rigorous).

As a simple example for an observable, take the phase space position
$\boldsymbol{\mathcal{O}}(\boldsymbol{y})=\boldsymbol{y}$ for
$\boldsymbol{y}\in\boldsymbol{X}$. It has values in phase space
$\boldsymbol{X}$ itself (as opposed to being real-valued), but this is no
issue at all. Using the prescription (11), we obtain

$\displaystyle\boldsymbol{\tilde{\mathcal{O}}}(t;\boldsymbol{x})$
$\displaystyle={\left.\kern-1.2pt\mathcal{O}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t)}\right)\,Z[\boldsymbol{J};\boldsymbol{x}]\vphantom{\big{|}}\,\right|_{\boldsymbol{J}=\boldsymbol{0}}}$
(17)
$\displaystyle={\left.\kern-1.2pt\frac{\delta}{i\,\delta\boldsymbol{J}(t)}\exp\left(i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\tilde{\varphi}}(t^{\prime};\boldsymbol{x})\right)\vphantom{\big{|}}\,\right|_{\boldsymbol{J}=\boldsymbol{0}}}$
(18) $\displaystyle=\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x}).$ (19)

Thus, the observable $\boldsymbol{\mathcal{O}}$ precisely extracts the phase
space trajectory which is indeed the phase space position as evaluated along
the solution of the equations of motion.

###  2.2 Free Generating Functional

If we are able to find an explicit analytical expression for the generating
functional, i.e., express $Z[\boldsymbol{J};\boldsymbol{x}]$ as an elementary
functional of $\boldsymbol{J}$, we would know everything about the physical
system under consideration. Indeed, we could simply obtain the phase space
trajectory of the system by functional derivation. Hence, it is expected that
in any non-trivial case the path integrals in the generating functional cannot
be evaluated analytically without explicitly referring to the solution of the
equations of motion. As such, the formalism as developed so far is not all
that useful for obtaining explicit results for observables of a classical
physical system.

To remedy this, we make the crucial assumption that the equations of motion
can be split into a free part and an interaction part, i.e.,

$\displaystyle\boldsymbol{E}(\boldsymbol{\varphi})=\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{E_{I}}(\boldsymbol{\varphi})=\boldsymbol{0}\,,$
(20)

where we demand that the free part $\boldsymbol{E_{0}}(\boldsymbol{\varphi})$
is given by a linear differential operator acting on $\boldsymbol{\varphi}$. A
simple example would be
$\boldsymbol{E_{0}}(\boldsymbol{\varphi})=\partial_{t}\boldsymbol{\varphi}$
which shows that this decomposition is always possible for Hamiltonian
equations of motion. Ultimately, we will perform a perturbative expansion of
the interaction part $\boldsymbol{E_{I}}(\boldsymbol{\varphi})$, such that it
is advantageous to include as many terms as possible into the free part of the
equations of motion.

Any linear differential operator admits a Green’s function (also known as
propagator) which can be used to write down a solution to the associated
differential equation. In the specific case of
$\boldsymbol{E_{0}}(\boldsymbol{\varphi})=\boldsymbol{0}$, it is

$\displaystyle\boldsymbol{\bar{\varphi}}(t;\boldsymbol{x})=\boldsymbol{\mathcal{G}}(t,t_{i})\,\boldsymbol{x}\,,$
(21)

where we denote the free solution by $\boldsymbol{\bar{\varphi}}$ and the
Green’s function by $\boldsymbol{\mathcal{G}}(t,t^{\prime})$. Note that the
free solution usually describing straight paths has been overset by a bar and
should not be confused with the phase space trajectory
$\boldsymbol{\tilde{\varphi}}$ generally describing curved paths and
accordingly featuring a tilde.

Using the decomposition of the equation of motion, we can simplify the
generating functional significantly. Suppressing the time dependence for
compactness of notation, we find

$\displaystyle Z[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\exp\left(i\int\mathrm{d}t^{\prime}\left(\boldsymbol{\chi}\cdot\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{\chi}\cdot\boldsymbol{E_{I}}(\boldsymbol{\varphi})+\boldsymbol{J}\cdot\boldsymbol{\varphi}\right)\right)$
(22)
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\exp\left(i\int\mathrm{d}t^{\prime}\,\boldsymbol{\chi}\cdot\boldsymbol{E_{I}}(\boldsymbol{\varphi})\right){\left.\kern-1.2pt\exp\left(i\int\mathrm{d}t^{\prime}\left(\boldsymbol{\chi}\cdot(\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{K})+\boldsymbol{J}\cdot\boldsymbol{\varphi}\right)\right)\vphantom{\big{|}}\,\right|_{\boldsymbol{K}=\boldsymbol{0}}}$
(23)
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\exp\left(i\int\mathrm{d}t^{\prime}\,\frac{\delta}{i\,\delta\boldsymbol{K}}\cdot\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}}\right)\right){\left.\kern-1.2pt\exp\left(i\int\mathrm{d}t^{\prime}\left(\boldsymbol{\chi}\cdot(\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{K})+\boldsymbol{J}\cdot\boldsymbol{\varphi}\right)\right)\vphantom{\big{|}}\,\right|_{\boldsymbol{K}=\boldsymbol{0}}}$
(24)
$\displaystyle={\left.\kern-1.2pt\exp\left(i\int\mathrm{d}t^{\prime}\,\frac{\delta}{i\,\delta\boldsymbol{K}}\cdot\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}}\right)\right)\,Z_{0}[\boldsymbol{J},\boldsymbol{K};\boldsymbol{x}]\vphantom{\big{|}}\,\right|_{\boldsymbol{K}=\boldsymbol{0}}}\,.$
(25)

We introduced another auxiliary function $\boldsymbol{K}$ which acts as a
source field for $\boldsymbol{\chi}$ in order to move the interaction term in
front of the path integrals. In the last line we isolated the so-called _free
generating functional_

$\displaystyle Z_{0}[\boldsymbol{J},\boldsymbol{K};\boldsymbol{x}]$
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\int\mathcal{D}\boldsymbol{\chi}\,\exp\left(i\int\mathrm{d}t^{\prime}\left(\boldsymbol{\chi}\cdot(\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{K})+\boldsymbol{J}\cdot\boldsymbol{\varphi}\right)\right)$
(26)
$\displaystyle=\int_{\boldsymbol{\varphi}(t_{i})=\boldsymbol{x}}\mathcal{D}\boldsymbol{\varphi}\,\delta_{D}\left[\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{K}\right]\,\exp\left(i\int\mathrm{d}t^{\prime}\,\boldsymbol{J}\cdot\boldsymbol{\varphi}\right)$
(27)
$\displaystyle=\exp\left(i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\bar{\varphi}}(t;\boldsymbol{x})-i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime\prime}\,\boldsymbol{J^{\intercal}}(t^{\prime})\,\boldsymbol{\mathcal{G}}(t^{\prime},t^{\prime\prime})\,\boldsymbol{K}(t^{\prime\prime})\right)\,.$
(28)

The second term in the last line is to be understood as a matrix product and
precisely cancels the additional factor $\boldsymbol{K}$ in the equation
$\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{K}=\boldsymbol{0}$. We
remark that in performing the integration over $\boldsymbol{\varphi}$ we
assumed that this equation has symplectic structure such that the associated
functional determinant is unity. This is generically the case for Hamiltonian
equations of motion.

Using the explicit form obtained for the free generating functional, we can
immediately perform the functional derivatives with respect to
$\boldsymbol{K}$ in equation (25).333Note that the term
$\boldsymbol{J^{\intercal}}(t^{\prime})\,\boldsymbol{\mathcal{G}}(t^{\prime},t^{\prime\prime})$
obtained from the differentiation can be moved past the functional derivative
in
$\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)$.
Indeed, $\boldsymbol{\mathcal{G}}(t^{\prime},t^{\prime\prime})=\boldsymbol{0}$
if $t^{\prime}\leq t^{\prime\prime}$ such that the following procedure is
possible: We write the exponential as a power series and in each term we sort
the products by their time variable $t^{\prime\prime}$ (this is accomplished
by splitting the integral into time-ordered regions and considering each of
them separately). In each product, we move all the functional derivatives with
respect to $\boldsymbol{K}$ to the right and let them act on the free
generating functional. The resulting terms then can be moved back to the
original positions of the corresponding functional derivatives, allowing the
power series to be converted back into an exponential. This yields

$\displaystyle Z[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=\exp\left(-i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime\prime}\,\boldsymbol{J^{\intercal}}(t^{\prime})\,\boldsymbol{\mathcal{G}}(t^{\prime},t^{\prime\prime})\,\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)\right)\,Z_{0}[\boldsymbol{J};\boldsymbol{x}]\qquad\text{with}$
(29) $\displaystyle Z_{0}[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=\exp\left(i\int_{t_{i}}^{\infty}\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\bar{\varphi}}(t^{\prime};\boldsymbol{x})\right)\,.$
(30)

Evidently, there are no path integrals left in these expressions and there is
no explicit reference to the solution of the equations of motion. Instead,
only the free trajectory appears which can be easily calculated using the
Green’s function. The free generating functional takes exactly the same form
as the generating functional (16) with $\boldsymbol{\tilde{\varphi}}$ replaced
by $\boldsymbol{\bar{\varphi}}$.

###  2.3 Interactions

The expression in equations (29) and (30) for the generating functional
strongly suggests a perturbative treatment of the interactions by means of a
Taylor expansion. We pursue this strategy momentarily, but as a preparatory
step we need to investigate the term
$\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)$
in slightly more detail. As usual, the derivative in the argument is to be
understood by the virtue of an analytic power series for the function
$\boldsymbol{E_{I}}$. As such, we have the identity

$\displaystyle\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)\,\exp\left(i\int\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\bar{\varphi}}(t^{\prime};\boldsymbol{x})\right)=\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t^{\prime\prime};\boldsymbol{x})\right)\,\exp\left(i\int\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\bar{\varphi}}(t^{\prime};\boldsymbol{x})\right)\,.$
(31)

This comes as no surprise as this is exactly the purpose for which we
introduced the functional derivative in equation (24) – it extracts the
trajectory from the free generating functional. This identity can be
generalized to the case we will be facing in the perturbative treatment.
There, in addition to the generating functional, the derivative in
$\boldsymbol{E_{I}}$ can also act on single factors of $\boldsymbol{J}$. We
can use

$\displaystyle\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)(i\,\boldsymbol{J}\cdot\boldsymbol{A})\cdots=\left((i\,\boldsymbol{J}\cdot\boldsymbol{A})\,\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)+(\boldsymbol{A}\cdot\boldsymbol{\nabla})\,\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)\right)\cdots\,,$
(32)

where $\boldsymbol{A}$ is a placeholder for terms of the form
$\boldsymbol{\mathcal{G}}(t^{\prime},t^{\prime\prime})\,\boldsymbol{E_{I}}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t^{\prime\prime})}\right)$
and $\cdots$ denotes further terms on which the derivatives may act. The
appearance of two terms on the right-hand side is due to the product rule. The
second term warrants some further remarks as it may seem counter-intuitive
that we obtain a derivative acting on $\boldsymbol{E_{I}}$. However, this can
once again be understood by virtue of power series. Indeed, for a monomial

$\displaystyle(\partial_{x})^{n}(ax)\cdots=\left((ax)(\partial_{x})^{n}+an\,(\partial_{x})^{n-1}\right)\cdots\,,$
(33)

where the factor $n$ in the second term accounts for the choice of which of
the derivatives acts on the term $ax$. Observe that resulting expression has
the form of the derivative of the monomial. The generalization to a power
series is straightforward as the same procedure can be applied term-by-term.

It is possible to proceed in full generality. In higher-orders we obtain
higher derivatives of the interaction term $\boldsymbol{E_{I}}$ and it is
possible to devise a diagrammatic representation of terms order by order to
keep track of how the derivatives (which mathematically are tensors) are
contracted with the remaining terms. We are happy to supply such prescription
upon request, but refrain from detailing it here. Instead we momentarily
specify our physical system to be an interacting $N$-body system in an attempt
to sacrifice generality for clarity. Nonetheless, below we give the explicit
expressions for the generating functional in full generality up to second
order in a Taylor expansion of the exponential in equation (29):

$\displaystyle Z_{0}[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=\exp\left(i\int\mathrm{d}t^{\prime}\,\boldsymbol{J}(t^{\prime})\cdot\boldsymbol{\bar{\varphi}}(t^{\prime};\boldsymbol{x})\right)\,,$
(34) $\displaystyle Z_{1}[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=-i\iint\mathrm{d}t_{1}^{\prime}\,\mathrm{d}t_{1}^{\prime\prime}\,\boldsymbol{J^{\intercal}}(t_{1}^{\prime})\,\boldsymbol{\mathcal{G}}(t_{1}^{\prime},t_{1}^{\prime\prime})\,\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{1}^{\prime\prime};\boldsymbol{x})\right)Z_{0}[\boldsymbol{J};\boldsymbol{x}]\,,$
(35) $\displaystyle Z_{2}[\boldsymbol{J};\boldsymbol{x}]$
$\displaystyle=-\iiiint\mathrm{d}t_{1}^{\prime}\,\mathrm{d}t_{1}^{\prime\prime}\,\mathrm{d}t_{2}^{\prime}\,\mathrm{d}t_{2}^{\prime\prime}\,\boldsymbol{J^{\intercal}}(t_{1}^{\prime})\,\boldsymbol{\mathcal{G}}(t_{1}^{\prime},t_{1}^{\prime\prime})\,\left[\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{1}^{\prime\prime};\boldsymbol{x})\right)\,\boldsymbol{J^{\intercal}}(t_{2}^{\prime})\,\right.$
$\displaystyle\hskip
43.36464pt\left.-\,i\,\delta_{D}(t_{1}^{\prime\prime}-t_{2}^{\prime})\,\boldsymbol{\nabla}\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{1}^{\prime\prime};\boldsymbol{x})\right)\,\right]\boldsymbol{\mathcal{G}}(t_{2}^{\prime},t_{2}^{\prime\prime})\,\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{2}^{\prime\prime};\boldsymbol{x})\right)Z_{0}[\boldsymbol{J};\boldsymbol{x}]\,.$
(36)

Note that we denote with $Z_{1}$ and $Z_{2}$, respectively, the additional
contributions from first and second order. The full second order generating
functional is given by the sum $Z_{0}+Z_{1}+Z_{2}$. Observables may be
obtained via functional derivatives with respect to the source field
$\boldsymbol{J}$. As an example, the contributions to the phase space
trajectory $\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$ up to second order
is obtained by acting with $\frac{\delta}{i\,\delta\boldsymbol{J}(t)}$
(evaluated at $\boldsymbol{J}=\boldsymbol{0}$) on the expressions above. We
obtain

$\displaystyle\boldsymbol{\tilde{\varphi}}_{0}(t;\boldsymbol{x})$
$\displaystyle=\frac{\delta}{i\,\delta\boldsymbol{J}(t)}Z_{0}[\boldsymbol{0};\boldsymbol{x}]=\boldsymbol{\bar{\varphi}}(t;\boldsymbol{x})=\boldsymbol{\mathcal{G}}(t,t_{i})\,\boldsymbol{x}\,$
(37) $\displaystyle\boldsymbol{\tilde{\varphi}}_{1}(t;\boldsymbol{x})$
$\displaystyle=-\int\mathrm{d}t_{1}^{\prime\prime}\,\boldsymbol{\mathcal{G}}(t,t_{1}^{\prime\prime})\,\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{1}^{\prime\prime};\boldsymbol{x})\right)\,$
(38) $\displaystyle\boldsymbol{\tilde{\varphi}}_{2}(t;\boldsymbol{x})$
$\displaystyle=\iint\mathrm{d}t_{1}^{\prime\prime}\,\mathrm{d}t_{2}^{\prime\prime}\,\boldsymbol{\mathcal{G}}(t,t_{1}^{\prime\prime})\,\boldsymbol{\nabla}\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{1}^{\prime\prime};\boldsymbol{x})\right)\,\boldsymbol{\mathcal{G}}(t_{1}^{\prime\prime},t_{2}^{\prime\prime})\,\boldsymbol{E_{I}}\left(\boldsymbol{\bar{\varphi}}(t_{2}^{\prime\prime};\boldsymbol{x})\right)$
(39)

Note that the term containing two factors of $\boldsymbol{J}$ at second order
vanishes for this particular observable because the leftover factor
$\boldsymbol{J}$ is set to zero (for the same reason, the functional
derivative cannot act on $Z_{0}$ at any positive order). The first order phase
space trajectory, given by
$\boldsymbol{\tilde{\varphi}}_{0}(t;\boldsymbol{x})+\boldsymbol{\tilde{\varphi}}_{1}(t;\boldsymbol{x})$,
coincides with the so-called Born approximation which adds the interactions as
evaluated along the free trajectory and is a common ad hoc approximation used,
e.g., in scattering problems.444Note that if we replace the free trajectory
$\boldsymbol{\bar{\varphi}}$ inside $\boldsymbol{E_{I}}$ on the right-hand
side of equation (38) by the actual phase space trajectory
$\boldsymbol{\tilde{\varphi}}$, the sum of the resulting term and the free
trajectory reproduce $\boldsymbol{\tilde{\varphi}}$ exactly. Indeed, we have
the well-known integral equation
$\displaystyle\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})=\boldsymbol{\bar{\varphi}}(t;\boldsymbol{x})-\int_{t_{i}}^{t}\mathrm{d}t_{1}\,\boldsymbol{\mathcal{G}}(t,t_{1})\,\boldsymbol{E_{I}}\left(\boldsymbol{\tilde{\varphi}}(t_{1};\boldsymbol{x})\right)\,.$
The Born approximation is the first step in solving this integral equation
iteratively (this process is sometimes referred to as iterative Born
approximation). We remark that the result of this iteration in second order
(where one inserts the (first order) Born approximation for the trajectory on
the right-hand side) does not coincide with the second order KFT trajectory –
infinitely many terms of second order iterative Born approximation are
assigned to higher-orders in KFT. We elaborate on this in footnote 7 using the
diagrammatic treatment of KFT perturbation theory. Here it is derived as the
first order perturbative approximation in the KFT framework.

###  2.4 Choice of Free Motion

As a last topic for the general treatment, we want to briefly discuss the
splitting of the equations into a free and an interaction part. It should be
stressed that this is a choice which is to a certain degree arbitrary. Indeed,
one can always take the trivial splitting $\boldsymbol{E_{0}}=\boldsymbol{0}$
and $\boldsymbol{E_{I}}=\boldsymbol{E}$ regarding the entirety of the
equations of motion as interactions. This way, the Green’s function is the
identity and the free trajectory is constant
$\boldsymbol{\bar{\varphi}}(t;\boldsymbol{x})=\boldsymbol{x}$. However, this
is a poor choice because the interaction part of the equations of motion is
only treated perturbatively, while the free part is solved exactly. Hence, it
is advantageous to include as many terms of the equations of motion as
possible into the free part $\boldsymbol{E_{0}}$. The ability to do so is of
course limited by the requirement of the existence of a Green’s function for
the free motion. Nonetheless, minimizing the interaction part
$\boldsymbol{E_{I}}$ generally is expected to improve the accuracy and
convergence of the perturbative treatment.

This situation is familiar from many other areas of physics. Fundamentally,
what we consider as free (or background) evolution and what we regard as
interactions (or perturbations) on top of it, is a choice. As an example,
given a density field $\rho(q)$ we may regard it as the sum of a background
component $\bar{\rho}(q)$ and a perturbation $\delta\rho(q)$, i.e.,
$\rho(q)=\bar{\rho}(q)+\delta\rho(q)$. However, there is a large freedom in
performing this split and depending on the situation different choices might
be advantageous. In the example, the most natural split is obtained by
introducing a condition like $\frac{1}{V}\int\mathrm{d}q\,\rho(q)=\bar{\rho}$.
But if we, e.g., describe the density field of gas in a galaxy, one might want
to consider the dark matter halo as part of the background density.

Likewise in our situation, there is a natural choice for the free motion.
Newton’ first law asserts that the reference motion is uniform along straight
lines. His second law can then be used to determine the acceleration relative
to this reference motion. Commonly we would claim that this acceleration is
due to a force caused by interactions. However, in some situations it may be
advantageous to include part of this acceleration into the free motion. For
example, on an expanding background, we usually prefer to use comoving
coordinates. In doing so, the physical trajectories of the particles are no
longer straight lines. While this makes the free motion a bit more
complicated, it has the advantage that the forces between particles are no
longer sourced by the density field, but only by the density excess over the
mean density (cf. A).

There is an even more sophisticated choice. It is possible to use the
Zel’dovich approximation for the free motion and model the interactions
relative to it [16, 17]. In this case it is less clear what the interaction
potential should be and, in fact, it is not even clear whether a simple two-
body interaction can reproduce the forces between the particles. Recent work
suggests, however, that this is indeed possible and demonstrates that a
Yukawa-like interaction potential describes the force relative to Zel’dovich
trajectories well [18]. Here, we limit ourselves to the more conventional case
of using uniform motion along straight paths in comoving coordinates as our
free motion.

We emphasise that the choice of free motion does not have any effect on
observables. The trajectory $\boldsymbol{\tilde{\varphi}}$ and all other
observables are exactly the same for any split of the equations of motion
$\boldsymbol{E}=\boldsymbol{E_{0}}+\boldsymbol{E_{I}}$. The free trajectory
$\boldsymbol{\bar{\varphi}}$ and the interaction part $\boldsymbol{E_{I}}$ can
be altered significantly, though, which can have significant effects on
computational complexity and the validity of approximations. For example, the
perturbative treatment of interactions presented in this work may have very
different convergence properties and speed depending on the choice of free
motion. It might be an interesting task for future works to formalize and
study the symmetry
$\left(\boldsymbol{E_{0}},\boldsymbol{E_{I}}\right)\mapsto\left(\boldsymbol{E_{0}}+\boldsymbol{\xi},\boldsymbol{E_{I}}-\boldsymbol{\xi}\right)$
and find optimal choices for $\boldsymbol{\xi}$.

##  3 Interacting N-body Systems

The discussion above was completely general, but also rather abstract. The
only assumption we made was that the equations of motion are of Hamiltonian
form and split into a free and an interaction part. We needed their symplectic
structure to omit a functional determinant which otherwise would have been a
bit of a nuisance. As such, we have been studying a Hamiltonian system
governed by an equation
$\boldsymbol{E_{0}}(\boldsymbol{\varphi})+\boldsymbol{E_{I}}(\boldsymbol{\varphi})=\boldsymbol{0}$
evolving from initial conditions $\boldsymbol{x}\in\boldsymbol{X}$ along a
phase space trajectory $\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})$.

In this section we specialize our abstract treatment and notation to
interacting $N$-body systems. We stress that it is possible, albeit
notationally laborious, to keep the entire perturbative expansion abstract and
general. In particular, it is possible to define Feynman rules for the general
setting which yield the Feynman rules derived below as special cases. We
refrain from listing them here or even in a dedicated appendix as this would
require to introduce an excessive amount of non-trivial notation or,
alternatively, work with equations featuring a zoo of indices.

###  3.1 Generating Functional

Let us consider a system of $N$ interacting particles in three-dimensional
Euclidean space. The state of this physical system can be described by a point
in phase space $\boldsymbol{x}\in\boldsymbol{X}\cong\mathbb{R}^{6N}$. Let us
take one such point as our initial conditions and define

$\displaystyle\boldsymbol{\tilde{\varphi}}\colon\mathbb{R}\times\boldsymbol{X}\rightarrow\boldsymbol{X}\qquad\text{with}\qquad\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})=\left(\tilde{\varphi}_{1}(t;\boldsymbol{x}),\tilde{\varphi}_{2}(t;\boldsymbol{x}),\ldots,\tilde{\varphi}_{N}(t;\boldsymbol{x})\right)^{\intercal}\in\boldsymbol{X}$
(40)

to be the phase space trajectory of the system. Here,
$\tilde{\varphi}_{j}(t;\boldsymbol{x})$ is a six-dimensional vector describing
the position and momentum of the $j\textsuperscript{th}$ particle at time $t$.
As we have to be able to access the position and momentum parts of this and
other objects frequently below, let us introduce the following notation: For
any point $x$ in single-particle phase space $X\cong\mathbb{R}^{6}$ we write
${x}^{(q)}$ for its position and ${x}^{(p)}$ for its momentum, i.e., it is
$x=\left({x}^{(q)},{x}^{(p)}\right)^{\intercal}\in X$. Similarly, the
$3N$-dimensional vector containing all the positions is given by
$\boldsymbol{{x}^{(q)}}=\left({x}_{1}^{(q)},{x}_{2}^{(q)},\ldots,{x}_{N}^{(q)}\right)^{\intercal}$.

Using this notation and particularizing to an interacting $N$-body system, the
equations of motion for the $j\textsuperscript{th}$ particle are given by

$\displaystyle\partial_{t}\varphi_{j}(t)-\begin{pmatrix}\frac{1}{m}{\varphi}_{j}^{(p)}(t)\\\
-m\,\partial_{{\varphi}_{j}^{(q)}}V\left(t;\boldsymbol{\varphi^{(q)}}\right)\end{pmatrix}=0\,.$
(41)

The potential $V$ is given by

$\displaystyle
V\left(t;\boldsymbol{\varphi^{(q)}}\right)=\frac{1}{2}\sum_{j}\sum\limits_{l\neq
j}v\left({\varphi}_{j}^{(q)}(t)-{\varphi}_{l}^{(q)}(t),t\right)$ (42)

featuring the two-particle interaction potential $v$. We allow for an explicit
time-dependence of this potential for later convenience. It is useful to
perform a Fourier transform resulting in

$\displaystyle
V\left(t;\boldsymbol{\varphi^{(q)}}\right)=\frac{1}{2}\sum_{j}\sum\limits_{l\neq
j}\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\,v(k^{\prime},t)\,\exp\left(ik^{\prime}\left({\varphi}_{j}^{(q)}(t)-{\varphi}_{l}^{(q)}(t)\right)\right)\,,$
(43)

where we use the symbol $v$ for the Fourier transform of the two-particle
interaction potential, too. We abuse notation in this way throughout and
distinguish functions and their Fourier transforms only via their arguments.
For Newtonian gravitational interactions in Newtonian gauge we have
$v(k^{\prime})\propto-\frac{1}{\left\|k^{\prime}\right\|^{2}}$ and the
splitting of the equations of motion is given by

$\displaystyle(E_{0})_{j}$
$\displaystyle=\partial_{t}\varphi_{j}(t)-\begin{pmatrix}\frac{1}{m}{\varphi}_{j}^{(p)}(t)\\\
0\end{pmatrix}\,,$ (44) $\displaystyle(E_{I})_{j}$
$\displaystyle=\begin{pmatrix}0\\\ \sum\limits_{l\neq
j}\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\,i\,m\,k^{\prime}\,v(k^{\prime},t)\,\exp\left(ik^{\prime}\left({\varphi}_{j}^{(q)}(t)-{\varphi}_{l}^{(q)}(t)\right)\right)\end{pmatrix}\,.$
(45)

A Green’s function for the free equations of motion is given by

$\displaystyle\mathcal{G}(t,t^{\prime})=\begin{pmatrix}\theta(t-t^{\prime})\,\mathcal{I}_{3}&g_{qp}(t,t^{\prime})\,\mathcal{I}_{3}\\\
0&g_{pp}(t,t^{\prime})\,\mathcal{I}_{3}\end{pmatrix}$ (46)

with $g_{qp}(t,t^{\prime})=\frac{1}{m}(t-t^{\prime})\,\theta(t-t^{\prime})$
and $g_{qp}(t,t^{\prime})=\theta(t-t^{\prime})$ in Newtonian gauge on a static
background. We keep the propagators $g_{qp}(t,t^{\prime})$ and
$g_{pp}(t,t^{\prime})$ as well as the interaction potential
$v(k^{\prime},t^{\prime})$ general below to allow for gauge changes in the
final results. Moreover, this leaves sufficient freedom to transfer our
results to an expanding background necessary for the cosmological case (cf.
A).

Using the explicit forms for the free and interaction parts of the equations
of motion, we can simplify the expressions in equations (29) and (30) for the
generating functional. Indeed, we obtain

$\displaystyle Z[\boldsymbol{{J}^{(q)}};\boldsymbol{x}]$
$\displaystyle=\exp\left(\sum\limits_{j}\sum\limits_{l\neq
j}\iint\mathrm{d}t^{\prime}\,\mathrm{d}t^{\prime\prime}\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\,m\,g_{qp}(t^{\prime},t^{\prime\prime})\,v(k^{\prime},t^{\prime\prime})\right.$
$\displaystyle\hskip 43.36464pt\left.{J}_{j}^{(q)}(t^{\prime})\cdot
k^{\prime}\,\exp\left(ik^{\prime}\left(\frac{\delta}{i\,\delta{J}_{j}^{(q)}(t^{\prime\prime})}-\frac{\delta}{i\,\delta{J}_{l}^{(q)}(t^{\prime\prime})}\right)\right)\right)\,Z_{0}[\boldsymbol{{J}}^{(q)};\boldsymbol{x}]\,,$
(47) $\displaystyle Z_{0}[\boldsymbol{{J}^{(q)}};\boldsymbol{x}]$
$\displaystyle=\exp\left(i\sum_{j}\int\mathrm{d}t^{\prime}\,{J}_{j}^{(q)}(t^{\prime})\cdot\bar{{\varphi}}{}_{j}^{(q)}(t^{\prime};\boldsymbol{x})\right)\qquad\text{with}$
(48) $\displaystyle\bar{{\varphi}}{}_{j}^{(q)}(t^{\prime};\boldsymbol{x})$
$\displaystyle={x}_{j}^{(q)}+g_{qp}(t^{\prime},t_{i})\,{x}_{j}^{(p)}\,.$ (49)

Note that we have set the momentum part of the source field
$\boldsymbol{{J}^{(p)}}$ to zero as we are exclusively interested in
observables featuring positions of particles. We remark for later reference
that this avoids any appearance of the function $g_{pp}(t,t^{\prime})$ here
and below.

###  3.2 Density Correlation Functions

Aside from the trajectories themselves, the density and its $r$-point
correlation functions are the most interesting observables for an interacting
$N$-body system. In fact, the collective information encoded in the density
correlation functions tends to be much more useful in practice compared to the
microscopic information encoded in the specific positions of the individual
particles. In our applications we integrate the observables over a probability
distribution of initial conditions making the specific realization of the
system irrelevant. Instead, we are interested in statistical quantities like
the density fluctuation power spectrum. In this section we explain how density
$r$-point functions can be obtained from the generating functional.

The particle number density at position $q$, given the positions of particles
$\boldsymbol{{\varphi}^{(q)}}(t)$ at time $t$, is

$\displaystyle\rho\left(q;\boldsymbol{{\varphi}^{(q)}}(t)\right)=\sum_{j}\delta_{D}\left(q-{\varphi}_{j}^{(q)}(t)\right)\,$
(50)

Performing a Fourier transformation which introduces a wave vector $k$
conjugate to $q$, the density takes the form

$\displaystyle\rho\left(k;\boldsymbol{{\varphi}^{(q)}}(t)\right)=\sum_{j}\exp\left(-i\,k\cdot{\varphi}_{j}^{(q)}(t)\right)\,.$
(51)

We remark that these equations are valid both for the comoving and the
Eulerian densities (cf. A) because the normalization of the Dirac
$\delta$-function, resp. the wave vectors, compensate tacitly the factors of
the scale factor. Thus, the equations written down here and below are valid
both for a static and an expanding background as long as we regard the wave
vector as comoving in the expanding case.

One of the advantages of working in Fourier space is that it is very easy to
construct the density correlation functions. Indeed, the (Fourier space)
density $r$-point correlation function simply is

$\displaystyle G_{\underset{r\leavevmode\nobreak\
\text{times}}{\underbrace{\rho\cdots\rho}}}(k_{1},\ldots,k_{r};\boldsymbol{{\varphi}^{(q)}}(t))=\prod_{s=1}^{r}\rho(k_{s};\boldsymbol{\varphi}(t))\,.$
(52)

Written out, this yields

$\displaystyle
G_{\rho\cdots\rho}(k_{1},\ldots,k_{r};\boldsymbol{{\varphi}^{(q)}}(t))=\sum_{\\{j_{1},\ldots,j_{r}\\}}\exp\left(-i\,\sum_{s=1}^{r}k_{s}\cdot{\varphi}_{j_{s}}^{(q)}(t)\right)\,,$
(53)

where the sum in front of the exponential runs over all $r$-tuples of
particles indices $1\leq j_{s}\leq N$.

The observables defined above are all given in terms of the positions of the
ensemble of particles $\boldsymbol{{\varphi}^{(q)}}(t)$. In order to calculate
these observables for the specific system under consideration, we ought to
replace these positions by the solution of the equations of motion
$\boldsymbol{\tilde{\varphi}^{(q)}}(t;\boldsymbol{x})$ for intial values
$\boldsymbol{x}$. However, we do not attempt to find this solution explicitly,
but instead use the identity in equation (11), i.e.,

$\displaystyle\tilde{\mathcal{O}}(t;\boldsymbol{x})=\mathcal{O}\left(\boldsymbol{\tilde{\varphi}}(t;\boldsymbol{x})\right)={\left.\kern-1.2pt\mathcal{O}\left(\frac{\delta}{i\,\delta\boldsymbol{J}(t)}\right)\,Z[\boldsymbol{J};\boldsymbol{x}]\vphantom{\big{|}}\,\right|_{\boldsymbol{J}=\boldsymbol{0}}}\,.$
(54)

Explicitly, we replace the positions $\boldsymbol{{\varphi}^{(q)}}(t)$ by a
functional derivative with respect to $\boldsymbol{{J}^{(q)}}(t)$ in the
expressions for the density and the $r$-point correlation function above. This
yields the operator

$\displaystyle\hat{G}_{\rho\cdots\rho}(k_{1},\ldots,k_{r},t)=\sum_{\\{j_{1},\ldots,j_{r}\\}}\exp\left(-i\,\sum_{s=1}^{r}k_{s}\cdot\frac{\delta}{i\,\delta{J}_{j_{s}}^{(q)}(t)}\right)$
(55)

with which we can act on the generating functional
$Z[\boldsymbol{J};\boldsymbol{x}]$ to obtain the r-point correlation function.
The density operator $\hat{\rho}(k,t)$ is obtained for the special choice
$r=1$.

##  4 KFT Perturbation Scheme

In this section we present a perturbation scheme for obtaining the density
correlation functions for an interacting $N$-body system. We stress again that
the entire procedure can be done for any Hamiltonian system allowing for a
separation of free motion and interactions. For clarity, concreteness and
simplicity of notation we focus on interacting $N$-body systems here which are
also our main application.

Above we already hinted at the possibility of performing a Taylor expansion of
the exponential containing the interaction part of the equations of motion in
the generating functional (29). In doing so, we obtain a polynomial expression
containing in each term factors of the source field $\boldsymbol{J}$ and
functional derivatives with respect to it. To the very right-hand side of the
expression is the free generating functional which also contains
$\boldsymbol{J}$. Evidently, in each term a certain amount of combinatorics is
required to keep track of all the ways the derivatives could act on the
various factors of $\boldsymbol{J}$.

These combinatorics can be easily done algorithmically via successive product
rules, but this is neither enlightening, nor efficient. Instead we take
inspiration from the path integral formulation of QFT which arrives at very
similar expressions for observables. There, the method of diagrammatically
representing mathematical expressions has been applied with great success.
While our expressions are somewhat unusual from a QFT point of view – most of
our functional derivatives come inside Fourier phase factors – it is
nonetheless possible to obtain a diagrammatic representation in our case, too.
Below we derive our Feynman rules and use them to find perturbative
expressions for the observables we are interested in.

###  4.1 Taylor Expansion

Let us consider the generating functional for interacting $N$-body systems
given in equation (3.1). Replacing the exponential by its Taylor expansion we
have the expression

$\displaystyle Z[\boldsymbol{{J}^{(q)}};\boldsymbol{x}]$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\left[\sum\limits_{j}\sum\limits_{l\neq
j}\iint\mathrm{d}t^{\prime}\,\mathrm{d}t^{\prime\prime}\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\,m\,g_{qp}(t^{\prime},t^{\prime\prime})\,v(k^{\prime},t^{\prime\prime})\right.$
$\displaystyle\hskip 43.36464pt\left.\vphantom{\sum\limits_{l\neq
j}}{J}_{j}^{(q)}(t^{\prime})\cdot
k^{\prime}\,\exp\left(i\,k^{\prime}\cdot\left(\frac{\delta}{i\,\delta{J}_{j}^{(q)}(t^{\prime\prime})}-\frac{\delta}{i\,\delta{J}_{l}^{(q)}(t^{\prime\prime})}\right)\right)\right]^{n}\,Z_{0}[\boldsymbol{{J}^{(q)}};\boldsymbol{x}]$
(56)

with the free generating functional as given in equation (48). It is
worthwhile to point out that up to this point the entire treatment has been
exact. Perturbatively, the $n\textsuperscript{th}$ order contribution, denoted
below by $Z_{n}$, is simply given by the $n\textsuperscript{th}$ term in this
sum. The full expression for the generating functional to
$n\textsuperscript{th}$ order is the sum of the first $n$ terms in the sum.
Evidently, this truncation constitutes an approximation which may have a
limited validity. Taylor’s theorem comes with estimates of the truncation
error, but it is not trivial to apply them to our case of a derivative
operator present in the expansion. We intend to investigate this in future
works.

Just like one would do in QFT, we intend to derive Feynman rules to
diagrammatically represent the perturbative expressions for observables. The
class of observables we want to focus on are density correlation functions as
introduced in section 3.2. The usefulness of the Fourier transform becomes
evident now, as the phase factors coming from the interaction are similar to
the ones coming from the observables. This allows for a unified and hence
simplified treatment. We remark that similar Feynman rules can be obtained
for, e.g., the trajectories of particles yielding a general recipe for the
expressions in equations (37) to (39).

Let us write down the expression for the $n\textsuperscript{th}$ order density
$r$-point correlation function. It is obtained by acting with the operator
$\hat{G}_{\rho\cdots\rho}$ given in equation (55) on the
$n\textsuperscript{th}$ term of the sum in equation (4.1). We obtain

$\displaystyle\left(G_{\rho\cdots\rho}\right)_{n}(k_{1},\ldots,k_{r},t;\boldsymbol{x})$
$\displaystyle=\frac{1}{n!}\sum_{\\{m_{1},\ldots,m_{r}\\}}\exp\left(-i\,\sum_{s=1}^{r}k_{s}\cdot\frac{\delta}{i\,\delta{J}_{m_{s}}^{(q)}(t)}\right)$
$\displaystyle\qquad\left[\sum\limits_{j}\sum\limits_{l\neq
j}\iint\mathrm{d}t^{\prime}\,\mathrm{d}t^{\prime\prime}\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\,m\,g_{qp}(t^{\prime},t^{\prime\prime})\,v(k^{\prime},t^{\prime\prime})\,{J}_{j}^{(q)}(t^{\prime})\cdot
k^{\prime}\right.$ $\displaystyle\qquad\left.\vphantom{\sum\limits_{l\neq
j}}\exp\left(i\,k^{\prime}\cdot\left(\frac{\delta}{i\,\delta{J}_{j}^{(q)}(t^{\prime\prime})}-\frac{\delta}{i\,\delta{J}_{l}^{(q)}(t^{\prime\prime})}\right)\right)\right]^{n}\,{\left.\kern-1.2pt\vphantom{\sum\limits_{l\neq
j}}Z_{0}[\boldsymbol{{J}^{(q)}};\boldsymbol{x}]\vphantom{\big{|}}\,\right|_{\boldsymbol{{J}^{(q)}}=\boldsymbol{0}}}\,.$
(57)

Despite the seeming complexity, this expression has actually quite a pleasant
structure. The key mathematical identity to use is

$\displaystyle\exp\left(k\cdot\frac{\delta}{\delta{J}_{m}^{(q)}(t)}\right)\,{J}_{j}^{(q)}(t^{\prime})\cdot
k^{\prime}\,\cdots\,Z_{0}[\boldsymbol{{J}^{(q)}};\boldsymbol{x}]$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\left({J}_{j}^{(q)}(t^{\prime})\cdot
k^{\prime}+\delta_{jm}\,\delta_{D}\left(t-t^{\prime}\right)\,k\cdot
k^{\prime}\right)\,\exp\left(k\cdot\frac{\delta}{\delta{J}_{m}^{(q)}(t)}\right)\,\cdots\,Z_{0}[\boldsymbol{{J}^{(q)}};\boldsymbol{x}],$
(58)

which can be seen as a special case of the discussion around equation (33).
Again the dots $\cdots$ denote further terms on which the derivative may act.
Given that the phase factors are not modified, they ultimately act in this
form on the free generating functional $Z_{0}$. Hence, in the final expression
we can replace all the functional derivatives by the free trajectory
$\boldsymbol{\bar{\varphi}^{(q)}}$. The only difficulty is to keep track of
which of the factors of ${J}_{j}^{(q)}$ is acted upon by which of the phase
factors. Note that this action happens exactly once as any remaining factors
of ${J}_{j}^{(q)}$ would cause the entire term to vanish upon setting the
source field to zero.

###  4.2 Feynman Rules

We want to find a diagrammatic representation of equation (4.1). The first
step in achieving this is to understand the term in the square bracket which
is taken to the $n\textsuperscript{th}$ power. It contains both factors of the
source field $\boldsymbol{{J}^{(q)}}$ as well as derivatives with respect to
it. Hence, the ordering of the individual factors is crucial. Back in equation
(25) the ordering was irrelevant because the two derivatives commute. But as
soon as we performed the functional derivative with respect to
$\boldsymbol{K}$ in equation (29) we had to be careful with this issue. In
fact, in footnote 3, we explained in detail how it is even possible to perform
these derivatives – by time ordering all terms and exploiting the fact that
the Green’s function is causal. This allowed us to move the factors of
$\boldsymbol{J}$ back to the left past some of the functional derivatives.

We can postpone the step in which we move the factors of $\boldsymbol{J}$ back
past the functional derivatives for the moment. Then all these factors – here
in the form of ${J}_{j}^{(q)}$ – are towards the very right of the expression
just next to the free generating functional from which they were obtained via
differentiation. All Fourier phase factors containing the functional
derivatives are left of it. Hence, a priori the situation is such that any
phase factor could act on any ${J}_{j}^{(q)}$.

We choose this as the starting point for formulating our Feynman rules. Let us
use the phase factors as the vertices of our diagrams:

$k_{s}$, $t$
$\displaystyle\coloneqq\sum_{m_{s}}\exp\left(-i\,k_{s}\cdot{\bar{\varphi}}_{m_{s}}^{(q)}(t;\boldsymbol{x})\right)\,,$
(59) $a$ $\displaystyle\coloneqq\sum\limits_{j_{a}}\sum\limits_{l_{a}\neq
j_{a}}\int_{t_{i}}^{\infty}\mathrm{d}t_{a}^{\prime}\int\frac{\mathrm{d}^{3}k_{a}^{\prime}}{(2\pi)^{3}}\,m\,v(k_{a}^{\prime},t_{a}^{\prime})\,\exp\left(i\,k_{a}^{\prime}\cdot\left({\bar{\varphi}}_{j_{a}}^{(q)}(t_{a}^{\prime};\boldsymbol{x})-{\bar{\varphi}}_{l_{a}}^{(q)}(t_{a}^{\prime};\boldsymbol{x})\right)\right)\,.$
(60)

We call the first of them _density vertex_ and the second _interaction
vertex_. Note that we replaced the functional derivatives with the free
trajectory in accordance with our earlier comments. The remaining parts of the
expression in equation (4.1) are encoded in the edges of the diagrams. These
are

$a$$b$ $\displaystyle\coloneqq
g_{qp}(t_{b}^{\prime},t_{a}^{\prime})\,k_{a}^{\prime}\cdot
k_{b}^{\prime}\,\left(\delta_{j_{a}j_{b}}-\delta_{j_{a}l_{b}}\right)\,,$ (61)
$a$$k_{s}$, $t$ $\displaystyle\coloneqq-
g_{qp}(t,t_{a}^{\prime})\,k_{a}^{\prime}\cdot k_{s}\,\delta_{j_{a}m_{s}}\,,$
(62)

where the uncoloured vertices mean that the corresponding vertex factors are
not yet included.

We claim the following:555We use graph-theoretical language here: A tree is a
connected and acyclic graph, i.e., a connected diagram without loops. A forest
is the disjoint union of trees. The expression (4.1) is given by the sum over
all possible forests formed from the vertices and edges defined above. Each
forest consists of $r$ trees of which each has one root given by a density
vertex. All other vertices are interaction vertices. We comment on our notion
of equivalence of diagrams in section 4.3 below. The mathematical expression
encoded by a diagram is the product of all vertices times the product of all
edges (this way all the factors coming from the edges appear under the sums
and integrals as they should). The $n\textsuperscript{th}$ order contribution
to the $r$-point density correlation function is the set of diagrams with
exactly $n$ interaction vertices and $r$ density vertices.

The crucial ingredient for proving this claim is the time ordering due to
causality. Mathematically this is encoded in the Heaviside function
$\theta(t-t^{\prime})$ inside the propagator $g_{qp}(t,t^{\prime})$ which
ensures that functional derivatives may only act on factors ${J}_{j}^{(q)}$
with earlier time argument. This prevents loops and is also the reason why the
density vertices (which come with the highest possible time coordinate $t$)
are the roots of the trees. Upon time ordering the diagram, all interaction
vertices have exactly one outgoing edge due to the fact that each factor
${J}_{j}^{(q)}$ is acted upon exactly once – yet another way of seeing that we
obtain collections of trees with density vertices as roots. As a final remark,
the lower bound for the time integration is due to our initial conditions
$\boldsymbol{x}$ being set at $t=t_{i}$.666We arguably have been somewhat
cavalier with integration boundaries above which we excuse with the aim of
keeping equations compact. In equation (9) a lower bound $t\geq t_{i}$ applies
to the time integration which carries through. In fact, there is technically
also an upper bound which we simply denoted by $t$ in equation (9) and which
needs to be at least as large as the time coordinate at which we calculate
observables. Here we have simply send it to infinity.

For concreteness, we list the diagrams and expressions for the density, i.e.,
the case $r=1$, up to first order below. In zeroth order we have zero
interaction vertices and therefore simply obtain the free density

$\displaystyle\tilde{\rho}_{0}(k,t;\boldsymbol{x})$ $\displaystyle=\
\leavevmode\hbox to22.04pt{\vbox to18.8pt{\pgfpicture\makeatletter\hbox{\hskip
11.02153pt\lower-9.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@rect{-10.62154pt}{-9.0pt}{21.24307pt}{18.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.62154pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$k$, $t$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
=\sum_{m_{1}}\exp\left(-i\,k\cdot{\bar{\varphi}}_{m_{1}}^{(q)}(t;\boldsymbol{x})\right)\,.$
(63) As this corresponds to the interactions being completely absent, it is no
surprise that we evaluate the (Fourier transform) of the density function
along the free trajectory. The first order correction is given by the diagram
$\displaystyle\tilde{\rho}_{1}(k,t;\boldsymbol{x})$ $\displaystyle=\
\leavevmode\hbox to77.33pt{\vbox to18.8pt{\pgfpicture\makeatletter\hbox{\hskip
66.30551pt\lower-9.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{-47.90552pt}{0.0pt}\pgfsys@curveto{-47.90552pt}{4.97063pt}{-51.93489pt}{9.0pt}{-56.90552pt}{9.0pt}\pgfsys@curveto{-61.87614pt}{9.0pt}{-65.90552pt}{4.97063pt}{-65.90552pt}{0.0pt}\pgfsys@curveto{-65.90552pt}{-4.97063pt}{-61.87614pt}{-9.0pt}{-56.90552pt}{-9.0pt}\pgfsys@curveto{-51.93489pt}{-9.0pt}{-47.90552pt}{-4.97063pt}{-47.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-56.90552pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.40552pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@rect{-10.62154pt}{-9.0pt}{21.24307pt}{18.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.62154pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$k$, $t$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-47.50552pt}{0.0pt}\pgfsys@lineto{-11.02153pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (64)
$\displaystyle=-\sum_{m_{1}}\exp\left(-i\,k\cdot{\bar{\varphi}}_{m_{1}}^{(q)}(t;\boldsymbol{x})\right)\sum\limits_{l_{1}\neq
m_{1}}\int_{t_{i}}^{\infty}\mathrm{d}t_{1}^{\prime}\,g_{qp}(t,t_{1}^{\prime})$
$\displaystyle\hskip
43.36464pt\int\frac{\mathrm{d}^{3}k_{1}^{\prime}}{(2\pi)^{3}}\,m\,v(k_{1}^{\prime},t_{1}^{\prime})\,k_{1}^{\prime}\cdot
k\,\exp\left(i\,k_{1}^{\prime}\cdot\left({\bar{\varphi}}_{m_{1}}^{(q)}(t_{1}^{\prime};\boldsymbol{x})-{\bar{\varphi}}_{l_{1}}^{(q)}(t_{1}^{\prime};\boldsymbol{x})\right)\right)\,.$
(65)

In addition, let us list the diagrams which appear in second and third order.
The corresponding expressions follow from the Feynman rules and become rather
lengthy quite quickly.

$\displaystyle\tilde{\rho}_{2}(k,t;\boldsymbol{x})$ $\displaystyle=\
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to41.56pt{\pgfpicture\makeatletter\hbox{\hskip
66.30551pt\lower-20.78091pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-47.90552pt}{11.38092pt}\pgfsys@curveto{-47.90552pt}{16.35155pt}{-51.93489pt}{20.38092pt}{-56.90552pt}{20.38092pt}\pgfsys@curveto{-61.87614pt}{20.38092pt}{-65.90552pt}{16.35155pt}{-65.90552pt}{11.38092pt}\pgfsys@curveto{-65.90552pt}{6.4103pt}{-61.87614pt}{2.38092pt}{-56.90552pt}{2.38092pt}\pgfsys@curveto{-51.93489pt}{2.38092pt}{-47.90552pt}{6.4103pt}{-47.90552pt}{11.38092pt}\pgfsys@closepath\pgfsys@moveto{-56.90552pt}{11.38092pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.40552pt}{8.1587pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{-33.67914pt}{-11.38092pt}\pgfsys@curveto{-33.67914pt}{-6.4103pt}{-37.70851pt}{-2.38092pt}{-42.67914pt}{-2.38092pt}\pgfsys@curveto{-47.64977pt}{-2.38092pt}{-51.67914pt}{-6.4103pt}{-51.67914pt}{-11.38092pt}\pgfsys@curveto{-51.67914pt}{-16.35155pt}{-47.64977pt}{-20.38092pt}{-42.67914pt}{-20.38092pt}\pgfsys@curveto{-37.70851pt}{-20.38092pt}{-33.67914pt}{-16.35155pt}{-33.67914pt}{-11.38092pt}\pgfsys@closepath\pgfsys@moveto{-42.67914pt}{-11.38092pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.17914pt}{-14.60313pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
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}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@rect{-10.62154pt}{-9.0pt}{21.24307pt}{18.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.62154pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$k$, $t$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
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}{}\pgfsys@moveto{-47.68813pt}{9.53725pt}\pgfsys@lineto{-11.02153pt}{2.20293pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-33.59657pt}{-8.95863pt}\pgfsys@lineto{-11.02153pt}{-2.93785pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ +\ \leavevmode\hbox
to105.78pt{\vbox to18.8pt{\pgfpicture\makeatletter\hbox{\hskip
94.75827pt\lower-9.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{-76.35828pt}{0.0pt}\pgfsys@curveto{-76.35828pt}{4.97063pt}{-80.38765pt}{9.0pt}{-85.35828pt}{9.0pt}\pgfsys@curveto{-90.3289pt}{9.0pt}{-94.35828pt}{4.97063pt}{-94.35828pt}{0.0pt}\pgfsys@curveto{-94.35828pt}{-4.97063pt}{-90.3289pt}{-9.0pt}{-85.35828pt}{-9.0pt}\pgfsys@curveto{-80.38765pt}{-9.0pt}{-76.35828pt}{-4.97063pt}{-76.35828pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-85.35828pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-87.85828pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-75.95828pt}{0.0pt}\pgfsys@lineto{-52.07913pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-33.27914pt}{0.0pt}\pgfsys@lineto{-11.02153pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \,,$ (66)
$\displaystyle\tilde{\rho}_{3}(k,t;\boldsymbol{x})$ $\displaystyle=\
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{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-118.63742pt}{0.0pt}\pgfsys@lineto{-94.75827pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-75.95828pt}{0.0pt}\pgfsys@lineto{-52.07913pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-33.27914pt}{0.0pt}\pgfsys@lineto{-11.02153pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (67)

We explain in detail why these are the correct sets of diagrams momentarily.
Note that there is a certain amount of freedom in defining equivalence of
diagrams which in turn induces some symmetry factors. The convention we choose
here avoids symmetry factors altogether and allows for easy numerical
evaluation of the resulting expressions. There is a minimal change in the
Feynman rules to achieve this. If one were to use the rules as written down
above, symmetry factors $\frac{1}{2}$ and $1$ for the diagrams of
$\tilde{\rho}_{2}$ and factors $\frac{1}{6}$, $\frac{1}{3}$, $\frac{1}{3}$,
$\frac{1}{3}$, $\frac{1}{2}$ and $1$ for $\tilde{\rho}_{3}$ would have to be
put into the expressions.

###  4.3 Symmetry Factors and Shot-Noise

As pointed out above, any discussion of symmetry factors needs to start out
with defining what we mean by equivalence of diagrams. We shall use here the
simple notion that two diagrams are equivalent if and only if they only differ
by the labels of the vertices. In particular, we distinguish between
topologically equal graphs with different horizontal ordering of the vertices
(an example are diagrams two to four in (4.2)). Equivalent diagrams only need
to be listed once in the sum over all diagrams. Naively, the cardinality of
each such equivalence class appears to $n!$, where $n$ is the number of
vertices. This seems to conveniently cancel the factor $\frac{1}{n!}$ coming
from the Taylor expansion. However, one has to be slightly more careful.

As an example, consider the first diagram in equation (66). This graph
actually only appears once in the Taylor expansion, because there is only one
way of connecting all interaction vertices to the density vertex. As a
consequence, the cardinality of the corresponding equivalence class is less
than $n!$ in such cases. However, this can precisely be compensated for by a
minor change in the Feynman rules. Namely, we demand that all vertices are
time-ordered, i.e., the time coordinates should satisfy $t_{i}\leq t_{1}\leq
t_{2}\leq\cdots\leq t_{n}\leq t$ for diagrams of order $n$. This can easily be