# Effective dynamics of quantum fluctuations in field theory: with
applications to cosmology

Ding Ding111Corresponding author. E-mail<EMAIL_ADDRESS>State Key
Laboratory of Surface Physics, Department of Physics, Center for Field Theory
and Particle Physics, and Institute for Nanoelectronic devices and Quantum
computing, Fudan University, Shanghai 200433, China  Yu Zhao222E-mail:
<EMAIL_ADDRESS>State Key Laboratory of Surface Physics, Department of
Physics, Center for Field Theory and Particle Physics, and Institute for
Nanoelectronic devices and Quantum computing, Fudan University, Shanghai
200433, China  Yidun Wan333Corresponding author. E-mail<EMAIL_ADDRESS>State Key Laboratory of Surface Physics, Department of Physics, Center for
Field Theory and Particle Physics, and Institute for Nanoelectronic devices
and Quantum computing, Fudan University, Shanghai 200433, China

###### Abstract

We develop a novel framework for describing quantum fluctuations in field
theory, with a focus on cosmological applications. Our method uniquely
circumvents the use of operator/Hilbert-space formalism, instead relying on a
systematic treatment of classical variables, quantum fluctuations, and an
effective Hamiltonian. Our framework not only aligns with standard formalisms
in flat and de Sitter spacetimes, which assumes no backreaction, demonstrated
through the $\varphi^{3}$-model, but also adeptly handles time-dependent
backreaction in more general cases. The uncertainty principle and spatial
symmetry emerge as critical tools for selecting initial conditions and
understanding effective potentials. We discover that modes inside the Hubble
horizon _do not_ necessarily feel an initial Minkowski vacuum, as is commonly
assumed. Our findings offer fresh insights into the early universe’s quantum
fluctuations and potential explanations to large-scale CMB anomalies.

###### Contents

  1. 1 Introduction
    1. 1.1 Key results
    2. 1.2 Motivations and background
  2. 2 Review of effective dynamics in quantum mechanics
    1. 2.1 Basic elements of fluctuation dynamics
    2. 2.2 Quantum backreaction and effective potentials
  3. 3 Fields on a flat spacetime
    1. 3.1 Effective Poisson brackets and evolution
    2. 3.2 Translational and rotational invariance
    3. 3.3 Uncertainty principle and energy-minimization
    4. 3.4 Solutions
      1. 3.4.1 Quantum fluctuations for free theory
      2. 3.4.2 Quantum fluctuations at tree-level
      3. 3.4.3 Quantum fluctuations at quantum-level
  4. 4 Fields on cosmological spacetimes
    1. 4.1 Free field
      1. 4.1.1 Different choices of variables and initial conditions
      2. 4.1.2 The equations of motion and its solution
    2. 4.2 Interacting field: non-Gaussianities
      1. 4.2.1 Finding the solution
  5. 5 Conclusions and discussions
  6. A Poisson bracket for fluctuations of arbitrary order
    1. A.1 The derivation
    2. A.2 Examples of using the generic BCH-PB formula
    3. A.3 Poisson brackets for quadratic and cubic fluctuations
  7. B Tree-level dynamics of higher-order fluctuations

##  1 Introduction

The framework of inflation explains elegantly how quantum fluctuations of the
early universe can seed a Gaussian primordial power spectrum. This primordial
power spectrum would then grow to form the inhomogeneous structures of our
universe — the temperature correlations of the cosmic microwave background
(CMB) and the Large Scale Structure (LSS) of galaxies.

###  1.1 Key results

In this paper, we develop a framework describing quantum fluctuations and its
backreaction on classical variables. It is an extension of [1]—which mainly
considers quantum mechanical systems— to the case of generic interacting
fields.

Our framework is novel in that it consistently describes evolution and initial
conditions without resorting to any operator/Hilbert-space language.

Our framework leads to the following results:

  * •

Prescriptions on how to describe the evolution of quantum fluctuations to
arbitrary order in interaction strength, how to find suitable initial
conditions, and how to utilize spatial symmetries to solve the equations of
motion by explicitly decoupling the irrelevant quantum fluctuations.

  * •

A new way of understanding effective potentials and selecting initial
conditions by means of the uncertainty principle and symmetries in position-
space.

Albeit devoid of operators and Hilbert spaces, our framework aligns with the
standard operator/Hilbert-space formalism. We demonstrate this in cases free
of backreaction, as is required for a fixed background metric, using the
$\varphi^{3}$-model on both Minkowski and de Sitter spacetimes as an example.
Nevertheless, our framework can coherently encompass time-dependent
backreaction, as to be discussed in Section 2.2. In the context of cosmology,
non-vanishing backreaction will affect the preservation of background
homogeneity, which can be analyzed adeptly in our framework.

###  1.2 Motivations and background

The improved precision of cosmological measurements [2] in the past two
decades has allowed physicists to peek into the early conditions of our
universe and understand their subsequent evolution. This in turn helped
theorists establish cosmological perturbation theory as the preferred
description for the observed Large Scale Structure and cosmic microwave
background. In essence, the theory tells us how small inhomogeneous
perturbations evolve on a homogeneous background, assuming some initial
conditions for the inhomogeneities. Based on the temperature correlations in
the CMB, the initial condition is thought to obey (classical) Gaussian
statistics, from which there are no strong hints of deviation.

But this is not the end of the story, as our universe is unlikely to have
started with purely classical initial conditions. Evolving our universe
backward in time, energy will increase to the point where quantum effects
deserve serious consideration. We must then ask: What are the quantum
conditions that seeded the classical initial inhomogeneities? The theory of
inflation [3], which was originally devised to solve cosmological puzzles
largely in the homogeneous sector, answers this question elegantly [4]. In the
inflationary framework, the universe exists in a quasi-de Sitter background
and (in most models) undergoes roughly 70 $e$-folds of rapid expansion before
transferring all its energy into the preheating phase [5], giving rise to the
“hot big bang”. During this quasi-de Sitter phase, the quantum fluctuations of
inhomogeneities exit the Hubble horizon and classicalize (through some yet-to-
be known mechanism). They re-enter the Hubble horizon post-inflation as
classical inhomogeneities to seed the Gaussian statistics that is required to
explain our CMB observations. In other words, inflation builds a coherent
picture of the origin of structure based on early quantum fluctuations.

The inflationary framework thus offers an opportunity to bridge quantum theory
and classical physics, incorporating the interplay between matter fields and
gravity. Unfortunately, since we do not have a full theory of quantum gravity,
how one should translate theory to observation via first principle is easier
said than done. Even the compromise of using the framework of quantum field
theory in curved spacetime entails major obstacles. (For de Sitter spacetime,
see for example [6, 7, 8].) Such operator/Hilbert-space formalisms therefore
suffer from both conceptual and practical difficulties. Fortunately, we can
turn to effective methods that aim to approximate quantum effects with a
(modified) classical system.

In fact, since the infancy of the inflationary paradigm, the use of effective
potentials has already proved to be successful in fixing the inflation
potential in model-building. The idea is that instead of analyzing the full
quantum system, one integrates out quantum fluctuations in the path-integral
to obtain a modified classical potential [9, 10, 11]. In recent years further
development in effective methods—using classical variables and effective
classical dynamics for expectation values—has experienced a modest resurgence.
(As an incomplete list, see for example [12, 13, 14, 15, 16, 17]).

In this paper, we introduce a systematic way to effectively describe an
interacting quantum field through three elements: “classical” variables,
quantum fluctuations, and an effective Hamiltonian to generate their dynamics.
It is found that the uncertainty principle and position-space symmetries allow
us to solve (in some cases analytically) the equations of motion for quantum
fluctuations. The quantum mechanical version of the method was conceptualized
in [1]. In field theory context, attempts to derive effective potentials were
made in [18]. The Coleman-Weinberg potential was obtained via the
_assumptions_ of an adiabatic approximation and the saturation of the
uncertainty principle. Nevertheless, the key role of position-space symmetries
was yet to be uncovered. In our present work, we will argue that position-
space symmetries are consequences of minimal backreaction, as is required for
cosmology if the background metric is to stay fixed. In turn, position-space
symmetries will decouple the evolution of quantum fluctuation from each other
to an extent that allows us to find solutions either analytically (in flat
space) or numerically. Additionally, saturation of the uncertainty principle
for systems dictated by low-energy effective potentials will arise as a
_consequence_ rather than an assumption.

Our framework is shown [17] to share connections with the variational
interpretation of the effective action approach developed by Jackiw [11]. At
its Gaussian level, it coincides with the method of classical-quantum
correspondence [15]. But in [16] it was argued that our method has the
advantage of being more systematic and works for more generic situations. The
dynamics of our method at leading order also share similarities with the well-
known transport method [12]. As we will show later, however, the two differ
both conceptually and practically. Furthermore, our method also provides a
natural way of treating quantum backreaction since the mixed dynamics between
(both homogeneous and inhomogeneous) classical variables and quantum
fluctuations are naturally embedded in our framework, as was shown in [16].
Finally, attempts to extend [1] to inflation also include [19, 20, 21], which
uses quantum constraints (Wheeler-DeWitt equations) as a starting point for
discussing quantum effects. In these works, the inhomogeneous part of the
Hamiltonian is kept to second order. In particular, the effects of higher-
power interaction terms are not discussed. While their results shed light on
leading-order quantum backreaction effects for constrained systems, our
present work provides a systematical prescription for dealing with quantum
fluctuations of generic field systems.

The paper is organized as follows: Section 2 reviews our effective framework
in a quantum mechanical setting and makes contact with the variational
approach to effective potentials. In particular, we will show how one could
arrive at the celebrated Coleman-Weinberg potential as a leading-order result.
Section 3 demonstrates the field version of our method in flat spacetime. We
discover that the uncertainty principle and symmetries are pivotal in
selecting minimal-energy Hamiltonians. Section 4 applies our method to
cosmological spacetimes and computes the power spectrum for quadratic systems.
We find, with the aid of the uncertainty principle, that the large-wavelength
modes no longer see a Minkowski vacuum when initial conditions are set at
finite times in the past. In particular, initial conditions differ physically
for different choices of variables related to canonical transformations. (A
similar conclusion was argued for with operator methods in [22].) Section 4.2
then generalizes our method to interacting fields and discusses how one could
compute non-Gaussianities. For a $\varphi^{3}$-model, we obtain the same
superhorizon limit for the bispectrum as that derived by standard operator
methods. Section 5 summarizes the paper. The paper ends with discussing the
challenges and potential directions for future work.

Unless explicitly stated otherwise, natural units where $\hbar=c=G_{N}=1$ is
assumed throughout the paper.

##  2 Review of effective dynamics in quantum mechanics

To understand the philosophy behind our effective framework, consider first
the classical case. Given a classical physical system governed by a
Hamiltonian $H[x,p]$ with the canonical pair $(x,p)$, one can describe its
evolution completely with two functions of time $x(t)$ and $p(t)$, or
equivalently, two degrees of freedom.

The quantum counterpart of such a classical system however requires infinitely
many complex degrees of freedom. In textbook quantum mechanics (Schrodinger
picture), these are the time-dependent expansion coefficients of a generic
state when expanded in a basis of the typically infinite-dimensional Hilbert
space.

One could also choose a different parameterization for quantum states by
opting for fluctuations (also known as moments of a wavefunction) as opposed
to these expansion coefficients [1]. If a state is “semi-classical”, or if one
is only interested in leading order quantum effects, the quantum system can
often be consistently approximated with finitely many fluctuation parameters.
For Gaussian-type states, early attempts to apply this philosophy include
going beyond the 1PI effective potential [11] and describing quantum tunneling
in quantum chemistry [23, 24]. These works typically deal with second-order
fluctuations and, when extended to cosmology, are directly relevant to the
production of the primordial power spectrum. Despite differences in
appearance, this class of methods was shown [16] to be equivalent to the more
recently proposed “quantum-classical correspondence”, which can also be used
to describe inhomogeneities generated by backreaction in cosmology and black
holes [15, 16, 25, 26].

Going beyond Gaussian systems, a systematic prescription of using quantum
fluctuations to approximate a state was developed by Bojowald and
collaborators [1, 27, 28]. A key input that was not previously discovered was
how one could construct an effective Hamiltonian in such a way that it drove
the dynamics of arbitrary-order fluctuations consistently. Once such a
prescription was found, the parameterization of quantum dynamics using
fluctuations could be done to arbitrary orders for generic Hamiltonians, at
least in principle.

In practice, one cannot afford to keep track of arbitrary orders of
fluctuations, nor is it needed for describing observationally relevant quantum
fluctuations in cosmology. A truncation or hierarchy 444In this paper, we will
use the term “hierarchy” to refer to any relation among fluctuations of
different types. Finding a hierarchy is intended to help reduce the number of
fluctuations relevant to some physical process, so that calculations are
tractable. For example, for a set of fluctuations obeying a Gaussian
hierarchy, all even-order fluctuations are related to (products of) quadratic-
order ones, whereas all odd-order fluctuations vanish. In this case, we only
need to keep track of the quadratic-order fluctuations. among different orders
of fluctuations can be assumed. For example, one could assume on physical
grounds that higher-order fluctuations are suppressed by $O(\hbar)$ and thus
negligible. One could, on the other hand, also assume a Gaussian hierarchy,
where Wick’s theorem reduces higher-order fluctuations to products of
quadratic fluctuations. In both cases, only a finite number of fluctuations is
needed to describe the system consistently.

In the remainder of this paper, we shall first illustrate how to implement our
effective framework for generic quantum mechanical Hamiltonians. Then we
restrict to a system with standard kinetic terms to acquire some physical
intuition. We will show that the well-known Coleman-Weinberg effective
potential can be thought of as a leading-order result in our effective
framework. This part of the paper is largely a review of [1, 18, 27].

The contribution of our paper to the field version of the effective framework
will be presented in Sections 3 and 4. These two sections will show that one
could describe the quantum fluctuations systematically for an interacting
quantum field on both flat and cosmological spacetimes. We also highlight the
role of spatial symmetries, which is absent in the quantum mechanical case.

###  2.1 Basic elements of fluctuation dynamics

To showcase, let us first restrict to a system of one canonical pair $(x,p)$.
Given its quantum Hamiltonian $H[\hat{x},\hat{p}]$, one can construct three
objects: (1) fluctuation parameters (or “fluctuations” for short), denoted as
$\Delta(\dots)$, (2) an effective Hamiltonian $H_{\Delta}$ associated with
these fluctuations, and (3) an effective Poisson bracket $\\{\dots,\dots\\}$
associated with the phase space of fluctuations and classical variables as
$\langle\hat{x}\rangle$ and $\langle\hat{p}\rangle$.

Mathematically, for some arbitrary state $|\psi\rangle$, the three objects are
defined as

$\displaystyle\Delta(x^{n-m}p^{m})\equiv$
$\displaystyle\langle(\hat{x}-\langle\hat{x}\rangle)^{n-m}(\hat{p}-\langle\hat{p}\rangle)^{m}\rangle_{W}\,,$
$\displaystyle H_{\Delta}\equiv$ $\displaystyle\langle
H[\hat{x},\hat{p}]\rangle_{W}-H[\langle\hat{x}\rangle,\langle\hat{p}\rangle]\,,$
$\displaystyle\\{\langle\hat{A}\rangle,\langle\hat{B}\rangle\\}\equiv$
$\displaystyle\frac{1}{i\hbar}\langle[\hat{A},\hat{B}]\rangle\,.$ (2.1)

In the first line, the subscript $W$ refers to Weyl-ordering: an operation
that symmetrizes the operators and then divides the result by the number of
possible permutations 555For example, $\langle x^{2}p\rangle_{W}=\langle
xxp\rangle_{W}=\langle xxp+xpx+pxx+xxp+xpx+pxx\rangle/3!=\langle
x^{2}p+xpx+px^{2}\rangle/3$, where we have viewed $x^{2}$ as the product of
two distinct operators $xx$. Here, we have used the shorthand
$x=\hat{x},p=\hat{p}$.. In particular, the ordering of operators in the
$\Delta(\dots)$ notation does not matter. In the last line of (2.1), $\hat{A}$
and $\hat{B}$ can be arbitrary powers of product-operators (which do not
include any $c$-numbers such as $\hat{A}=\langle\hat{x}\rangle$). We demand
that the Poisson brackets follow the Leibniz rule when one of its slots has
products of expectation values. We will also refer to the power $n$ of
product-operators in $\Delta(\dots)$ as the order of the fluctuation, denoted
as $\Delta^{(n)}$ 666For example, both $\Delta(x^{2})$ and $\Delta(xp)$ are
quadratic-order fluctuations, which can be collectively denoted as
$\Delta^{(2)}$.. Hereafter, we shall drop the hats on operators when there is
no confusion.

Let us obtain some intuition for the dynamics of a system by restricting to
the Hamiltonian

$H[x,p]=\frac{1}{2}p^{2}+V(x)\,.$ (2.2)

To obtain $H_{\Delta}$, we take an expectation value and expand the operators
around their averages

$\langle H[x,p]\rangle_{W}=\frac{1}{2}\langle p\rangle^{2}+V(\langle
x\rangle)+\underbrace{\frac{1}{2}\Delta(p^{2})+\sum_{n=2}\frac{1}{n!}\partial_{x}^{n}V|_{x=\langle
x\rangle}\Delta(x^{n})}_{H_{\Delta}}\,.$ (2.3)

The quantum effects of the system are captured by $H_{\Delta}$ in the sense
that it augments the classical equations of motion through fluctuations
$\Delta$

$\displaystyle\dot{\langle x\rangle}=$ $\displaystyle\\{\langle
x\rangle,\langle H[x,p]\rangle_{W}\\}=\langle p\rangle\,,$
$\displaystyle\dot{\langle p\rangle}=$ $\displaystyle\\{\langle
p\rangle,\langle H[x,p]\rangle_{W}\\}=-\partial_{x}V(x)|_{x=\langle
x\rangle}+{\\{\langle p\rangle,H_{\Delta}\\}}\,.$ (2.4)

The second term on the RHS of the second line represents additional quantum
evolution sourced by the quantum fluctuations $\Delta$. These terms, which
couple classical variables to quantum fluctuations, can be interpreted as
backreaction on the classical system dictated by $\langle x\rangle$ and
$\langle p\rangle$.

The backreaction in our framework comes from terms in $H_{\Delta}$ that are
products between classical variables and quantum fluctuations. Note that
however, backreaction is not induced by the effective Poisson bracket between
classical variables and fluctuations. Indeed, one can show [1] that at any
order $n$ of fluctuations $\Delta^{(n)}$, one has $\\{\langle
x\rangle,\Delta^{(n)}\\}=0$ and similarly for $\langle p\rangle$. This implies
that fluctuations $\Delta$ make up a subspace that is symplectically
orthogonal to the classical phase space spanned by $\langle x\rangle$ and
$\langle p\rangle$. Therefore by including the fluctuation parameters
$\Delta$, we obtain an extended phase space on top of the classical one, via
definitions (2.1). This extension is what allows us to include quantum degrees
of freedom in our effective framework.

###  2.2 Quantum backreaction and effective potentials

Given a classical inflation (scalar) field, described by $\varphi$, a full
quantum analysis should quantize the full field: both the inhomogeneous and
the homogeneous parts of the scalar. But, since the homogeneous part of the
inflation couples to the Hubble parameter via the Einstein equations, the
quantization of the homogeneous sector involves the gravitational background,
making it extremely difficult. In practice, one could compromise by including
quantum backreaction on the classical background field, described by
$\langle\varphi\rangle$, with the use of effective potentials. This allows us
to integrate out fluctuations like $\Delta(\varphi\varphi)$ to effectively
couple them to $\langle\varphi\rangle$ with a modification to the original
classical potential of $\varphi$.

The price to pay for such a compromise is at least twofold. First, the time
dependence of the backreaction is either partially lost or, at the very least,
made implicit. One uses the same function(al) — the effective potential — to
describe backreaction at all times. Second, the dynamics of quantum
fluctuations are lost, too, as they are integrated into the path-integral. But
the evolution of these quantum fluctuations is precisely what generates the
primordial power spectrum in the inflationary framework. One would ultimately
still need to analyze them. In the process, one runs into the danger of “re-
quantizing” the inflaton, as the quantum fluctuations now evolve on an
effective background that was obtained via their “initial” quantization.

Our effective framework based on (2.1) can avoid making these compromises. The
“classical” variables and quantum fluctuations are described consistently in a
single framework. From (2.4), we see how the extended part of the phase space
provides quantum backreaction to the classical variables $\langle x\rangle$
and $\langle p\rangle$ through $H_{\Delta}$. The quantum backreaction is also
dynamic in the sense that the fluctuations evolve according to

$\dot{\Delta}(\dots)=\\{\Delta(\dots),\langle
H[x,p]\rangle\\}=\\{\Delta(\dots),H_{\Delta}\\}\,.$

Notice that since $H_{\Delta}$ may contain classical variables $\langle
x\rangle$ and $\langle p\rangle$ through $\partial_{x}^{n}V(x)|_{x=\langle
x\rangle}$, the evolution of $\Delta$ is also affected by the classical
background. Our framework captures the two-way communication between classical
variables and quantum effects in a time-dependent manner.

When applying cosmological perturbation theory to inflation, one often sets
apart the classical portion of the full system from the sub-system we wish to
quantize. The legitimacy of this operation relies on the assumption of minimal
quantum backreaction [4]. So when is quantum backreaction — loosely speaking —
minimal? It is minimal in the case of quadratic systems, which is incidentally
how one describes cosmological perturbations in inflation at the leading
order. This answer suggests that one needs to be very careful when treating
non-Gaussian systems during inflation, for the sake of keeping backreaction
small. We will show in Section 3 how spatial symmetries help us minimize
backreaction and control the growth of inhomogeneity.

That quadratic systems lead to minimal backreaction is most easily seen in a
quantum mechanical toy model, with a potential $V(x)=m^{2}x^{2}/2$ for some
parameter $m$. one finds that quantum fluctuations and classical variables are
decoupled:

$\displaystyle\langle H[x,p]\rangle=$ $\displaystyle H[\langle
x\rangle,\langle
p\rangle]+\frac{1}{2}\Delta(p^{2})+\frac{1}{2}m^{2}\Delta(x^{2})$
$\displaystyle\dot{\langle x\rangle}=$ $\displaystyle\langle p\rangle$
$\displaystyle\dot{\langle p\rangle}=$
$\displaystyle-\partial_{x}V(x)|_{x=\langle x\rangle}\,.$

The evolution of the quadratic fluctuation $\Delta(x^{2})\,,\;\Delta(xp)$, and
$\Delta(p^{2})$ also decouples from the classical variables due to the form of
$\langle H[x,p]\rangle$.

For anharmonic systems, it is possible to capture leading-order quantum
effects by truncating $\langle H[x,p]\rangle$ beyond the quadratic order
$\Delta^{(2)}$ in fluctuations.

$\langle H[x,p]\rangle=H[\langle x\rangle,\langle
p\rangle]+\frac{1}{2}\Delta(p^{2})+\frac{1}{2}V^{\prime\prime}(\langle
x\rangle)\Delta(x^{2})+\cancel{\sum_{n=3}\partial_{x}^{n}V(x)|_{x=\langle
x\rangle}\Delta(x^{n})}\,.$ (2.5)

With such a truncated Hamiltonian, denoted as

$\langle H[x,p]\rangle_{2}=H[\langle x\rangle,\langle
p\rangle]+\frac{1}{2}\Delta(p^{2})+\frac{1}{2}V^{\prime\prime}(\langle
x\rangle)\Delta(x^{2})\,,$ (2.6)

one can connect our effective framework to the conventional description of
quantum backreaction that is based on the idea of effective actions and time-
dependent variational principle [11]. At the leading order (where non-local
terms are absent), the conventional result is commonly referred to as the
Coleman-Weinberg effective potential [10].

Specifically, our effective system can be reduced to one with a Coleman-
Weinberg effective potential by varying $\langle H[x,p]\rangle$ with respect
to the fluctuations. This is a manifestation of the variational interpretation
of effective potentials. In this procedure, an obstacle to overcome is
implementing the consistency with the Heisenberg uncertainty principle, which
dictates the quantum fluctuations. To this end, it is useful to introduce a
pair of canonical variables — constructed from the fluctuation parameters —
that automatically satisfies the uncertainty principle: [28]

$\displaystyle\Delta(x^{2})=Q^{2}\,,\;\Delta(xp)=QP\,,\;\Delta(p^{2})=P^{2}+\frac{U}{Q^{2}}\,,$
$\displaystyle U=\Delta(x^{2})\Delta(p^{2})-\Delta^{2}(xp)\,,$ (2.7)

where $Q$ and $P$ are canonical pairs satisfying $\\{Q,P\\}=1$. The parameter
$U$ is time-independent when one restricts to systems that contain up to only
second-order fluctuations — the parameter $U$ is the Casimir of the Poisson
algebra of fluctuations $(\Delta(xp),\Delta(x^{2}),\Delta(p^{2}))$. For
minimal-uncertainty states, it is simply $U=\frac{\hbar^{2}}{4}$. One can
check that the map (2.7) is consistent with the Poisson bracket algebra
defined by (2.1).

With such a change of variables, the effective Hamiltonian we wish to
extremize becomes

$\langle H[x,p]\rangle=H[\langle x\rangle,\langle
p\rangle]+\frac{1}{2}P^{2}+\frac{1}{2}V^{\prime\prime}(\langle
x\rangle)Q^{2}+\frac{U}{2Q^{2}}\,.$ (2.8)

Note that due to the uncertainty principle, $\Delta(p^{2})$ contributes a
$U/Q^{2}$ term that acts as a “repulsive” potential stopping the variance of
position $\Delta(x^{2})=Q^{2}$ from reaching zero.

By extremizing the effective Hamiltonian for $Q,P$, which represents the
fluctuation parameters, we obtain the minimized Hamiltonian

$\displaystyle\mathrm{Extr}[\langle H[x,p]\rangle]=H[\langle x\rangle,\langle
p\rangle]+\sqrt{UV^{\prime\prime}(\langle x\rangle)}$ $\displaystyle\text{when
}P^{2}=0\,,\;Q^{2}=\sqrt{\frac{U}{V^{\prime\prime}(\langle
x\rangle)}}\Leftrightarrow\Delta(p^{2})=V^{\prime\prime}(\langle
x\rangle)\Delta(x^{2})\,,\;\Delta(xp)=0\,.$ (2.9)

Setting $U=\hbar^{2}/4$ leads to the Coleman-Weinberg result in quantum
mechanics [29].

Let us recap what we did. We arrived at the Coleman-Weinberg effective
potential by applying two principles: (1) Extremization of our effective
Hamiltonian $\langle H\rangle$ with respect to the quantum fluctuations and
(2) quantum fluctuations complying with the uncertainty principle. The two
principles entail a variational problem with inequality constraints. For a
quantum mechanical system with one canonical pair, it is easy enough to find a
canonical map (2.7) that embodies the uncertainty relation, making the
constraints trivial. Direct extremization of $\langle H\rangle$ with respect
to the canonical variables $Q$ and $P$ gives the Coleman-Weinberg potential.

In field theory, there are infinitely many canonical pairs (seen most easily
in Fourier space). Finding a map from fluctuations to canonical variables
$\\{\Delta\\}\rightarrow\\{Q_{1},Q_{2},\dots,P_{1},P_{2},\dots\\}$ is all but
impossible. Nevertheless, one could still apply principles (1) and (2) to
obtain the Coleman-Weinberg potential, as we will show in Section 3.3.

To summarize, we have shown that our effective framework centered around
quantum fluctuations can recover the results in an operator formalism. The
leading order the effective action, viz the Coleman-Weinberg effective
potential, can be recovered by keeping up to second order in fluctuation
parameters $\Delta^{(2)}$ and demanding extremization of the effective
Hamiltonian. This allows us to interpret the Coleman-Weinberg result
physically as a low-energy effective system that is stable to quantum
fluctuations. In situations beyond such an approximation, our method provides
a systematic way to include higher-order quantum fluctuations in a time-
dependent way. In particular, our method applies to the case of interacting
fields, which we discuss in Sections 3 and 4.

##  3 Fields on a flat spacetime

Our effective framework also applies to systems with infinite degrees of
freedom, namely fields. Nevertheless, compared with quantum mechanics, three
key aspects become more consequential: (1) calculating Poisson brackets
between various fluctuations for fields, (2) global symmetries of the system,
especially the translational and rotational invariance of expectation values
and fluctuations, and (3) determining initial values based on the uncertainty
principle and an assumption of initial Gaussianity 777One could forgo the
Gaussianity assumption and replace it with some other hierarchy between
higher-order fluctuations and lower-order ones. The underlying idea is that we
want a way to fix the initial values for all the fluctuations by specifying
only a finite number of them (for example the quadratic-order ones). To this
end, Gaussianity — or the hierarchy behind Wick’s theorem — is the most common
realization in traditional quantum field theory..

We will demonstrate these aspects through a massive real $\varphi^{3}$-model
in the Minkowski spacetime. Nevertheless, almost all of our conclusions in
this section also hold for more general cosmological spacetimes. We will
highlight the exceptions as they appear. Our present example will also help
clarify the notations used in our effective framework for field applications.

The action of a real scalar field $\varphi(t,\mathbf{x})$ with mass $m$ is

$S_{\text{Mink.}}[\varphi]\equiv\int
dtd^{3}x\left[\frac{1}{2}(\dot{\varphi}^{2}-\delta^{ij}\partial_{i}\varphi\partial_{j}\varphi)-\frac{1}{2}m^{2}\varphi^{2}-\frac{\mu}{3}\varphi^{3}\right]\,.$
(3.1)

Here, $\mu\neq 0$ is the coupling constant of the $\varphi^{3}$-interaction.
The term “Mink.” refers to the Minkowski spacetime. The associated Hamiltonian
is

$\displaystyle H_{\text{Mink.}}$ $\displaystyle=\int
d^{3}x\left[\frac{1}{2}\pi^{2}+\frac{1}{2}\delta^{ij}\partial_{i}\varphi\partial_{j}\varphi+\frac{1}{2}m^{2}\varphi^{2}+\frac{\mu}{3}\varphi^{3}\right]$
(3.2)
$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{2}\left[\pi_{\mathbf{k}}\pi_{\mathbf{-k}}+\omega_{k}^{2}\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}}\right]$
$\displaystyle\quad+\frac{\mu}{3}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d^{3}p}{(2\pi)^{3}}\frac{d^{3}q}{(2\pi)^{3}}(2\pi)^{3}\delta^{3}(\mathbf{k+p+q})\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\,,$
(3.3)

where we have denoted the conjugate momentum and effective frequency of the
scalar field by

$\pi(t,\mathbf{x})\equiv\dot{\varphi}(t,\mathbf{x}),\qquad\omega_{k}^{2}\equiv
k^{2}+m^{2},\qquad k=\sqrt{\mathbf{k}\cdot\mathbf{k}}.$

The commutator between the field operators $\varphi(t,\mathbf{x})$ and
$\pi(t,\mathbf{x})$ is

$\left[\varphi(t,\mathbf{x}),\pi(t,\mathbf{y})\right]=i\hbar\delta^{3}(\mathbf{x-y})\,.$

In Eq. (3.3), we rewrite the Hamiltonian in terms of the momentum space
representations of the field operators:

$\varphi_{\mathbf{k}}(t)\equiv\int
d^{3}x\varphi(t,\mathbf{x})e^{-i\mathbf{k}\cdot\mathbf{x}},\qquad\pi_{\mathbf{k}}(t)\equiv\int
d^{3}x\pi(t,\mathbf{x})e^{-i\mathbf{k}\cdot\mathbf{x}},$

satisfying

$\left[\varphi_{\mathbf{k}},\pi_{\mathbf{p}}\right]=i\hbar(2\pi)^{3}\delta^{3}(\mathbf{k+p}).$

To simplify the notations, we omit the time argument when there is no risk of
confusion:
$\varphi_{\mathbf{k}}\equiv\varphi_{\mathbf{k}}(t),\pi_{\mathbf{k}}\equiv\pi_{\mathbf{k}}(t)$.

In the effective dynamics of quantum fields, calculations often involve
numerous integrals and delta functions of momenta-sums. For brevity, we
introduce the following conventions:

$\int_{k,p,q,\dots}\equiv\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d^{3}p}{(2\pi)^{3}}\frac{d^{3}q}{(2\pi)^{3}}\dots\qquad\tilde{\delta}_{\mathbf{k,p,q,}\dots}\equiv(2\pi)^{3}\delta^{3}(\mathbf{k+p+q}+\dots)\,.$

Using these notations, the Hamiltonian simplifies to

$H_{\text{Mink.}}[\varphi,\pi]=\int_{k}\frac{1}{2}\left[\pi_{\mathbf{k}}\pi_{\mathbf{-k}}+\omega_{k}^{2}\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}}\right]+\frac{\mu}{3}\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}.$
(3.4)

For any state $\ket{\psi}$ in Fock space, the effective Hamiltonian is

$H_{\text{Mink.},Q}\equiv\langle
H_{\mathrm{Mink.}}[\varphi,\pi]\rangle=H_{\text{Mink.}}[\langle\varphi\rangle,\langle\pi\rangle]+H_{\text{Mink.},\Delta}.$
(3.5)

The second term on the RHS above is the quantum-fluctuation part of our
effective Hamiltonian and is is given by:

$\displaystyle H_{\text{Mink.},\Delta}=$
$\displaystyle\int_{k}\frac{1}{2}\left[\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})+\omega_{k}^{2}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})\right]$
$\displaystyle\
+\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\left[\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})\langle\varphi_{\mathbf{q}}\rangle+\frac{1}{3}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\right].$
(3.6)

###  3.1 Effective Poisson brackets and evolution

One needs to calculate the Poisson brackets between different fluctuations
before one can obtain their evolution. Given that
$\varphi_{\mathbf{k}},\pi_{\mathbf{p}}$ with different $\mathbf{k},\mathbf{p}$
are distinct operators, the direct calculation of higher-order Poisson
brackets for Weyl-ordered expectation values is a complex task. Nevertheless,
inspired by [1], we have developed a less tedious method for calculating the
Poisson bracket between any two fluctuations via the Baker-Campbell-Hausdorff
formula. (See Appendix A.)

Specifically, the equations of motion for fluctuations in the
$\varphi^{3}$-model only involve Poisson brackets containing fluctuations in
$H_{\text{Mink.},\Delta}$, namely, the fluctuations
$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}})$,
and $\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})$.
The relevant nonvanishing Poisson brackets are computed by the following
matching rules:

$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(X\pi_{\mathbf{a}})\\}$
$\displaystyle=\ \cdots\
+\tilde{\delta}_{\mathbf{k,a}}\Delta(X\varphi_{\mathbf{p}})+\tilde{\delta}_{\mathbf{p,a}}\Delta(X\varphi_{\mathbf{k}}),$
(3.7)
$\displaystyle\\{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}}),\Delta(X\varphi_{\mathbf{a}})\\}$
$\displaystyle=\ \cdots\
-\tilde{\delta}_{\mathbf{k,a}}\Delta(X\pi_{\mathbf{p}})-\tilde{\delta}_{\mathbf{p,a}}\Delta(X\pi_{\mathbf{k}}),$
(3.8)
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(X\pi_{\mathbf{a}})\\}$
$\displaystyle=\ \cdots\
+\tilde{\delta}_{\mathbf{k,a}}\Big{[}\Delta(X\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})-\Delta(X)\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Big{]}$
$\displaystyle\qquad\quad\
+\tilde{\delta}_{\mathbf{p,a}}\Big{[}\Delta(X\varphi_{\mathbf{q}}\varphi_{\mathbf{k}})-\Delta(X)\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}})\Big{]}$
$\displaystyle\qquad\quad\
+\tilde{\delta}_{\mathbf{q,a}}\Big{[}\Delta(X\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})-\Delta(X)\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})\Big{]},$
(3.9)
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(X\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$
$\displaystyle=\ \cdots\
-\frac{\hbar^{2}}{4}\Delta(X)\Big{(}\tilde{\delta}_{\mathbf{k,a}}\tilde{\delta}_{\mathbf{p,b}}\tilde{\delta}_{\mathbf{q,c}}+\tilde{\delta}_{\mathbf{k,a}}\tilde{\delta}_{\mathbf{p,c}}\tilde{\delta}_{\mathbf{q,b}}+\tilde{\delta}_{\mathbf{k,b}}\tilde{\delta}_{\mathbf{p,a}}\tilde{\delta}_{\mathbf{q,c}}$
$\displaystyle\qquad\qquad+\tilde{\delta}_{\mathbf{k,b}}\tilde{\delta}_{\mathbf{p,c}}\tilde{\delta}_{\mathbf{q,a}}+\tilde{\delta}_{\mathbf{k,c}}\tilde{\delta}_{\mathbf{p,a}}\tilde{\delta}_{\mathbf{q,b}}+\tilde{\delta}_{\mathbf{k,c}}\tilde{\delta}_{\mathbf{p,b}}\tilde{\delta}_{\mathbf{q,a}}\Big{)},$
(3.10)

In the above expressions, $X$ represents the product of multiple field
operators $\varphi_{\mathbf{a_{i}}}$ and $\pi_{\mathbf{b_{j}}}$ (We define
$\Delta(1)=1$). The ellipsis “…” indicates the summation of all other possible
matches in the above forms (3.7), (3.8), (3.9), and (3.10). For instance,

$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\\}=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{b}}\pi_{\mathbf{p}})+\tilde{\delta}_{\mathbf{k,b}}\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{p}})+\tilde{\delta}_{\mathbf{p,a}}\Delta(\varphi_{\mathbf{b}}\pi_{\mathbf{k}})+\tilde{\delta}_{\mathbf{p,b}}\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{k}}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}=\tilde{\delta}_{\mathbf{k,a}}\Big{[}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Delta(\pi_{\mathbf{b}}\pi_{\mathbf{c}})\Big{]}$
$\displaystyle+\tilde{\delta}_{\mathbf{k,b}}\Big{[}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\pi_{\mathbf{c}}\pi_{\mathbf{a}})-\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Delta(\pi_{\mathbf{c}}\pi_{\mathbf{a}})\Big{]}+\tilde{\delta}_{\mathbf{k,c}}\Big{[}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\pi_{\mathbf{a}}\pi_{\mathbf{b}})-\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\Big{]}$
$\displaystyle+\tilde{\delta}_{\mathbf{p,a}}\Big{[}\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}})\Delta(\pi_{\mathbf{b}}\pi_{\mathbf{c}})\Big{]}+\tilde{\delta}_{\mathbf{p,b}}\Big{[}\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}}\pi_{\mathbf{c}}\pi_{\mathbf{a}})-\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}})\Delta(\pi_{\mathbf{c}}\pi_{\mathbf{a}})\Big{]}$
$\displaystyle+\tilde{\delta}_{\mathbf{p,c}}\Big{[}\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}}\pi_{\mathbf{a}}\pi_{\mathbf{b}})-\Delta(\varphi_{\mathbf{q}}\varphi_{\mathbf{k}})\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\Big{]}+\tilde{\delta}_{\mathbf{q,a}}\Big{[}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})\Delta(\pi_{\mathbf{b}}\pi_{\mathbf{c}})\Big{]}$
$\displaystyle+\tilde{\delta}_{\mathbf{q,b}}\Big{[}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\pi_{\mathbf{c}}\pi_{\mathbf{a}})-\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})\Delta(\pi_{\mathbf{c}}\pi_{\mathbf{a}})\Big{]}+\tilde{\delta}_{\mathbf{q,c}}\Big{[}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\pi_{\mathbf{a}}\pi_{\mathbf{b}})-\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\Big{]}$
$\displaystyle-\frac{\hbar^{2}}{4}\Big{(}\tilde{\delta}_{\mathbf{k,a}}\tilde{\delta}_{\mathbf{p,b}}\tilde{\delta}_{\mathbf{q,c}}+\tilde{\delta}_{\mathbf{k,a}}\tilde{\delta}_{\mathbf{p,c}}\tilde{\delta}_{\mathbf{q,b}}+\tilde{\delta}_{\mathbf{k,b}}\tilde{\delta}_{\mathbf{p,a}}\tilde{\delta}_{\mathbf{q,c}}+\tilde{\delta}_{\mathbf{k,b}}\tilde{\delta}_{\mathbf{p,c}}\tilde{\delta}_{\mathbf{q,a}}+\tilde{\delta}_{\mathbf{k,c}}\tilde{\delta}_{\mathbf{p,a}}\tilde{\delta}_{\mathbf{q,b}}+\tilde{\delta}_{\mathbf{k,c}}\tilde{\delta}_{\mathbf{p,b}}\tilde{\delta}_{\mathbf{q,a}}\Big{)},$
(3.11)

where the first five lines of
$\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$
are derived from matching the expression in (3.9). In contrast, the last line
that is proportional to $\hbar^{2}$ results from applying the matching rule in
(3.10). The complete set of Poisson brackets relevant to the equations of
motion is in Appendix A.3. These brackets are instrumental in deriving the
dynamics of quadratic- and cubic-order fluctuations, $\Delta^{(2)}$ and
$\Delta^{(3)}$, within the $\varphi^{3}$-model.

For further clarity, a few comments are in order:

  1. 1.

The Poisson structure of fluctuations is model-independent. The Poisson
brackets we have listed above can be directly applied to a $\varphi^{3}$-model
in a cosmological spacetime, or to any other real scalar field systems.

  2. 2.

The Poisson brackets involving quadratic fluctuations $\Delta^{(2)}$ can be
directly “read out” using the Leibniz rule and intuitions from classical
Poisson brackets:

$\\{\Delta(A_{0}A_{1}),\Delta(X_{1}X_{2}\cdots
X_{n})\\}=\Delta\Bigg{(}\frac{1}{i\hbar}\Big{[}A_{0}A_{1},\ X_{1}X_{2}\cdots
X_{n}\Big{]}\Bigg{)},$

where $A_{i}$ and $X_{j}$ can be either $\varphi_{\mathbf{k}}$ or
$\pi_{\mathbf{p}}$ operators. Nevertheless, when dealing with brackets
including cubic fluctuations and higher-order ones, such as the cubic-cubic
bracket
$\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$,
unexpected terms may emerge, as seen in the last line of (3.11). These terms
are pivotal in distinguishing between classical and quantum theories because
they are the remnants of non-commuting variables. When higher-order
fluctuations are present, it is usually better to use the Baker-Campbell-
Hausdorff formula in Appendix A to calculate the brackets.

  3. 3.

The Hamiltonian’s Equation $\dot{\Delta}=\\{\Delta,H_{\text{Mink.},\Delta}\\}$
gives the equations of motion:

$\displaystyle\dot{\Delta}(X\varphi_{\mathbf{a}})$ $\displaystyle=\ \cdots\ +\
\Delta(X\pi_{\mathbf{a}}),$ $\displaystyle\dot{\Delta}(X\pi_{\mathbf{a}})$
$\displaystyle=\
\cdots-\omega_{a}^{2}\Delta(X\varphi_{\mathbf{a}})-\mu\int_{p}\left[\Delta(X\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})-\Delta(X)\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})\right],$
$\displaystyle\dot{\Delta}(X\pi_{a}\pi_{b}\pi_{c})$ $\displaystyle=\ \cdots\
+\frac{\mu\hbar^{2}}{2}\tilde{\delta}_{\mathbf{a,b,c}}\Delta(X),$ (3.12)

Here, $X$ represents a combination of several $\varphi_{\mathbf{a_{i}}}$ and
$\pi_{\mathbf{b_{j}}}$ operators, and in “$\cdots$” we sum over all possible
matches. (See Appendix A for more details.) We have also assumed that
$\langle\varphi_{\mathbf{k}}\rangle=\langle\pi_{\mathbf{k}}\rangle=0$ — their
proof will be discussed later in Section 3.4.3.

  4. 4.

When perturbatively expanding $\Delta$’s in terms of the powers of $\mu$:

$\Delta(\cdots)=\Delta_{0}(\cdots)+\Delta_{1}(\cdots)+\cdots+\Delta_{n}(\cdots)+\cdots,$

where $\Delta_{n}(\cdots)$ is at the $\mu^{n}$ order, the equations of motion
(3) suggest that the dynamics of $\Delta^{(k)}_{n}$ are influenced by
fluctuations that are lower-order in either the power of product-operators or
in the powers of $\mu$. Therefore, if all $\Delta_{0}(\cdots)$’s at any time
are known, it is possible in principle to compute all $\Delta^{(k)}_{n}$
sequentially, in the order of powers of $\mu\hbar^{2}$. The zeroth order
$\Delta_{0}(\cdots)$’s evolve as free fields:

$\Delta_{0}(X\varphi_{\mathbf{a}})=\cdots+\Delta_{0}(X\pi_{\mathbf{a}}),\qquad\Delta_{0}(X\pi_{\mathbf{a}})=\cdots-\omega_{a}^{2}\Delta_{0}(X\varphi_{\mathbf{a}}),$

whose dynamics are exactly solvable using Wick’s theorem. We will illustrate
two applications of this in later sections.

  5. 5.

In the second line of Eqs. (3), the interaction perturbation is represented as

$-\mu\int_{p}\Big{[}\Delta(X\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})-\Delta(X)\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})\Big{]}.$

When assuming that the state $\ket{\psi}$ is a harmonic vacuum state $\ket{0}$
where Wick’s theorem applies, the “bubble term” in
$\int_{p}\Delta(X\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})$ is canceled by
$-\int_{p}\Delta(X)\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})$,
leaving only “connected contributions” that include fluctuations with fields
involving the external momenta in $X$ as well as the internal momenta
$\mathbf{p}$ and $\mathbf{a-p}$. For instance, with
$X\equiv\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}$, Wick’s theorem gives

$\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})-\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})=\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{p}})\Delta(\varphi_{\mathbf{c}}\varphi_{\mathbf{a-p}})+\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{a-p}})\Delta(\varphi_{\mathbf{c}}\varphi_{\mathbf{p}}).$

In this paper, for demonstration, we primarily focus on the dynamics of
quadratic and cubic fluctuations. Our goal is to deduce the tree-level
solutions at the $\mu^{0}$ and $\mu^{1}$ orders, and then compare our findings
in Minkowski spacetime with those in de Sitter cosmological spacetime in
Section 4. Incorporating Eqs. (3), we present the equations of motion for
quadratic fluctuations $\Delta^{(2)}$ as follows:

$\displaystyle\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})$
$\displaystyle=\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})+\Delta(\varphi_{\mathbf{b}}\pi_{\mathbf{a}}),$
$\displaystyle\dot{\Delta}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})$
$\displaystyle=\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})-\omega_{b}^{2}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})-\mu\int_{p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{k,b}}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),$
$\displaystyle\dot{\Delta}(\pi_{\mathbf{a}}\pi_{\mathbf{b}})$
$\displaystyle=-\omega_{a}^{2}\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})-\omega_{b}^{2}\Delta(\varphi_{\mathbf{b}}\pi_{\mathbf{a}})-\mu\int_{p}\Bigg{[}\Delta(\pi_{\mathbf{a}}\varphi_{\mathbf{p}}\varphi_{\mathbf{b-p}})+(\mathbf{a\leftrightarrow
b})\Bigg{]}.$ (3.13)

For cubic fluctuations $\Delta^{(3)}$, the equations of motion are:

$\displaystyle\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})=$
$\displaystyle\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{a}})+\Delta(\varphi_{\mathbf{c}}\varphi_{\mathbf{a}}\pi_{\mathbf{b}})+\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}),$
$\displaystyle\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})=$
$\displaystyle\Delta(\varphi_{\mathbf{b}}\pi_{\mathbf{a}}\pi_{\mathbf{c}})+\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{c}^{2}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})$
$\displaystyle-$
$\displaystyle\mu\int_{p}\Big{[}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{p}}\varphi_{\mathbf{c-p}})-\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{c-p}})\Big{]}$
$\displaystyle\dot{\Delta}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})=$
$\displaystyle\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{b}^{2}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{c}^{2}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{c}}\pi_{\mathbf{b}})$
$\displaystyle-$
$\displaystyle\mu\int_{p}\Bigg{\\{}\Big{[}\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{p}}\varphi_{\mathbf{c-p}})-\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{c-p}})\Big{]}+(\mathbf{b\leftrightarrow
c})\Bigg{\\}}$
$\displaystyle\dot{\Delta}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})=$
$\displaystyle-\omega_{a}^{2}\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{b}^{2}\Delta(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{c}^{2}\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})$
$\displaystyle-$
$\displaystyle\mu\int_{p}\Bigg{\\{}\Big{[}\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{p}}\varphi_{\mathbf{c-p}})-\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{c-p}})\Big{]}+(\mathbf{a\to
b\to c})\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{\mu}{2}\tilde{\delta}_{\mathbf{a,b,c}}\,.$ (3.14)

Here, we set $\hbar=1$. The notation $(\mathbf{a\leftrightarrow b})$ denotes a
term identical to the one preceding it but with the momenta $\mathbf{a}$ and
$\mathbf{b}$ interchanged. Similarly, $(\mathbf{k\rightarrow p\rightarrow q})$
indicates two additional terms, obtained from the preceding term by cyclically
permuting the momenta $\mathbf{k},\mathbf{p}$, and $\mathbf{q}$ .

Finally, we note down the equations of motion of the classical variables
$\langle\varphi_{\mathbf{a}}\rangle$ and $\langle\pi_{\mathbf{a}}\rangle$:

$\displaystyle\dot{\langle\varphi_{\mathbf{a}}\rangle}$
$\displaystyle=\langle\pi_{\mathbf{a}}\rangle,$
$\displaystyle\dot{\langle\pi_{\mathbf{a}}\rangle}$
$\displaystyle=-\omega_{a}^{2}\langle\pi_{\mathbf{a}}\rangle-\mu\int_{k}\Bigg{[}\langle\varphi_{\mathbf{k}}\rangle\langle\varphi_{\mathbf{a-k}}\rangle+\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{a-k}})\Bigg{]}.$
(3.15)

These are obtained by the Poisson brackets

$\\{\langle\varphi_{\mathbf{a}}\rangle,\langle\pi_{\mathbf{b}}\rangle\\}=\tilde{\delta}_{\mathbf{a,b}},\qquad\\{\langle\varphi_{\mathbf{a}}\rangle,\Delta(\cdots)\\}=\\{\langle\pi_{\mathbf{a}}\rangle,\Delta(\cdots)\\}=0,$

in view of the effective Hamiltonian $H_{Q}$ (3.5). Notably, as opposed to
Eqs. (3) that relies on the assumption that
$\langle\varphi_{\mathbf{a}}\rangle=\langle\pi_{\mathbf{a}}\rangle=0$, Eq.
(3.15) does not make this assumption. Nevertheless, when translational and
rotational invariances are imposed, it will follow that
$\langle\varphi_{\mathbf{a}}\rangle=\langle\pi_{\mathbf{a}}\rangle=0$ for any
momentum $\mathbf{a}$, such that the classical variables in Eq. (3.15) will
vanish.

###  3.2 Translational and rotational invariance

Although specific realizations of field configurations might not always be
homogeneous or isotropic (for instance, our universe has inhomogeneous
structures), we generally expect the quantum average of our system to exhibit
translational and rotational invariance, provided that the initial conditions
adhere to these symmetries. This assumption leads to significant
simplifications in the effective equations of motion in three ways:

  1. 1.

At the leading order, a given fluctuation couples to only a finite number of
other fluctuations. In particular, $\delta$-functions arise in such a way that
at $\mu^{1}$-order — with no “loop” integrals — terms in the equations of
motion will be proven to be local in the Fourier space despite objects like
$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})$
originally triply integrated.

  2. 2.

The Weyl-ordered expectation values within fluctuations remain real, although
$\varphi_{\mathbf{k}}$ and $\pi_{\mathbf{k}}$ are not Hermitian operators.

  3. 3.

For quantum averages associated with the inhomogeneous part of the field,
terms proportional to $\langle\varphi_{\mathbf{k}}\rangle$ can be disregarded
in the equations of motion.

The last simplification above leads to minimal backreaction, a crucial
requirement for (quantum) perturbation theory to be consistent in cosmology.
Similar to the discussion in Section 2.2, this simplification enables us to
decouple quantum fluctuations from the classical background, while minimizing
the growth of inhomogeneities of the background, thus, preserving a
homogeneous background metric. The second simplification will be important for
establishing a connection between our effective Hamiltonian and traditional
effective potentials.

We showcase the above three simplifications using our $\varphi^{3}$-model. We
first show that simplifications (1) to (3) are the consequences of
translationally and rotationally invariant initial conditions. Then, we will
argue that these simplifications are maintained throughout the evolution
generated by $H_{\text{Mink.},\Delta}$.

For any expectation value that is translationally and rotationally invariant,
we find that for arbitrary products of Fourier-space operators $A,B,C,\ldots$:

$\langle
A(\mathbf{k_{1}})B(\mathbf{k_{2}})C(\mathbf{k_{3}})\dots\rangle=(2\pi)^{3}\delta^{3}(\mathbf{k_{1}+k_{2}+k_{3}\dots})D(k_{1},k_{2},k_{3},\dots;\\{\mathbf{k}_{i}\cdot\mathbf{k}_{j}\\})\,.$
(3.16)

For cubic-products, rotational invariance makes the function $D$ on the RHS
depend only on the magnitudes of the three relevant momenta, i.e.
$D=D(k,p,q)$. Nonetheless, for any higher power of product operators, the
function $D$ may depend also on the angles between certain momenta, as
indicated by the collective notation
$\\{\mathbf{k_{i}}\cdot\mathbf{k}_{j}\\}$. As a specific case of (3.16), modes
with non-vanishing momenta obey $\langle\varphi_{\mathbf{k}}\rangle=0$ for
$\mathbf{k}\neq\mathbf{0}$. These modes represent the inhomogeneous
contributions. To carry our discussion over to inhomogeneities in cosmology,
we will proceed with the assumption that all external momenta
$\mathbf{a,b,c,d,}\ldots$ are non-zero.

The first question one can ask is: Do our effective equations of motion
preserve translational and rotational invariance in the sense of (3.16)? Yes.
For linear operators,

$\displaystyle\dot{\langle\varphi_{\mathbf{a}}\rangle}=$
$\displaystyle\dot{\langle\pi_{\mathbf{a}}\rangle}$
$\displaystyle\dot{\langle\pi_{\mathbf{a}}\rangle}=$
$\displaystyle-\omega_{a}^{2}{\langle\varphi_{\mathbf{a}}\rangle}-\mu\int_{p,q}\tilde{\delta}_{-\mathbf{a},\mathbf{p},\mathbf{q}}\langle\varphi_{\mathbf{p}}\rangle\langle\varphi_{\mathbf{q}}\rangle-\mu\int_{p,q}\tilde{\delta}_{\mathbf{-a,p,q}}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\,.$

If the initial conditions satisfy (3.16), i.e.,
$\langle\pi_{\mathbf{a}}\rangle_{t=0}\propto\tilde{\delta}_{\mathbf{a}}=\tilde{\delta}_{\mathbf{a,0}}$
and
$\Delta_{t=0}(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\propto\tilde{\delta}_{\mathbf{p,q}}$,
$\langle\pi_{\mathbf{a}}\rangle$ will not evolve for $\mathbf{a}\neq 0$ (the
inhomogeneous part of the field) because its initial rate of change will be
proportional to
$\tilde{\delta}_{\mathbf{-a,p,q}}\tilde{\delta}_{\mathbf{p,q}}\sim\tilde{\delta}_{\mathbf{a,0}}=0$.
Therefore, we also have $\langle\varphi_{\mathbf{a}}\rangle=0$ for
$\mathbf{a}\neq 0$ at the next “time-step”. Physically, spatial symmetry
guarantees that there is no backreaction from quantum fluctuations that
promote growth in background inhomogeneity—an important consistency condition
for our later application to cosmology.

Let us now examine the quadratic and cubic fluctuations based on equation sets
(3.13) and (3.14). Consider, say,
$\dot{\Delta}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})$ and
$\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})$ as
representatives.

Consider the equations of motion for
$\dot{\Delta}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})$ and
$\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})$ in
(3.13) and (3.14). The terms without an explicit $\mu$ factor are all
initially proportional to $\tilde{\delta}_{\mathbf{a,b}}$ or
$\tilde{\delta}_{\mathbf{a,b,c}}$ if they started from initial conditions
(3.16). (The terms proportional to $\langle\varphi\rangle$ are absent because
we have argued that linear expectation values vanish initially.) Their
integral terms (with the explicit $\mu$ factor) are all proportional to delta-
functions of sums over external momenta:

$\displaystyle\text{Integral term of
}\dot{\Delta}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})=$
$\displaystyle-\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{b,k}}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\sim\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{b,k}}\tilde{\delta}_{\mathbf{a,p,q}}(\dots)$
$\displaystyle\sim\int_{k}\tilde{\delta}_{\mathbf{a,b}}(\dots),$
$\displaystyle\text{Integral term of
}\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})=$
$\displaystyle-\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{c,k}}\Big{(}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})-\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Big{)}$
$\displaystyle\sim\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{c,k}}(\tilde{\delta}_{\mathbf{a,b,p,q}}-\tilde{\delta}_{\mathbf{a,b}}\delta_{\mathbf{p,q}})(\dots)\sim\int_{k}\Big{(}\tilde{\delta}_{\mathbf{-a,-b,-c}}+\tilde{\delta}_{\mathbf{c,0}}\delta_{\mathbf{a,b}}\Big{)}(\dots)$
$\displaystyle\sim\int_{k}\tilde{\delta}_{\mathbf{-a,-b,-c}}(\dots)\,.$

The bracket $(\dots)$ consists of the functions $D$ and/or their products.

One notices that the resulting delta-functions above,
$\tilde{\delta}_{\mathbf{a,b}}$ or $\tilde{\delta}_{\mathbf{-a,-b,-c}}$,
always come with an integral over internal momenta such as $\int_{k}(\dots)$.
Integrals of this type are undesirable in an equation of motion for they make
the evolution non-local in the Fourier space. For the evolution of cubic
fluctuations, one can get rid of these integrals if there is a hierarchy
between the fourth-order and second-order fluctuations, such as
$\Delta^{(4)}\sim\Delta^{(2)}\Delta^{(2)}$, via Wick’s theorem. The integrand
is then proportional to the product of two delta-functions
$\delta(\dots)\delta(\dots)$. Such a simplification can be realized
consistently if one starts with states obeying Wick’s theorem and assumes it
is preserved at leading order in $\mu^{1}$ throughout evolution. For example,
for $\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})$,
applying Wick’s theorem we have

$\displaystyle\text{Integral term of
}\dot{\Delta}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})=$
$\displaystyle-\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{c,k}}\Big{(}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})-\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Big{)}$
$\displaystyle\approx$
$\displaystyle-\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{c,k}}\Big{(}\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})+\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{p}})\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{q}})$
$\displaystyle+$
$\displaystyle\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{q}})\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{p}})-\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Big{)}+O(\mu^{2})$
$\displaystyle\approx$
$\displaystyle-\mu\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\tilde{\delta}_{\mathbf{c,k}}\Big{(}\tilde{\delta}_{\mathbf{a,p}}\tilde{\delta}_{\mathbf{b,q}}(\dots)+\tilde{\delta}_{\mathbf{a,q}}\tilde{\delta}_{\mathbf{b,p}}(\dots)\Big{)}$
$\displaystyle\sim$
$\displaystyle-\mu\tilde{\delta}_{\mathbf{a,b,c}}(\dots)\,,$

which is local in momentum space because the integration is eliminated. This
result follows from that our $\varphi^{3}$-model admits tree-level (i.e.
$O(\mu)$ order) three-point vertices. It also follows that one cannot get rid
of the integral over internal momenta for quadratic fluctuations in (3.13).
The cubic fluctuations contribute to quadratic fluctuations at the
$O(\mu^{2})$ order via “loop” integrals because cubic fluctuations per se are
already $O(\mu)$ if they evolved from Gaussian-type initial conditions (i.e.
ones obeying Wick’s theorem). That is, the $O(\mu)$-integrals in (3.13)
represent quantum contributions to 2-pt functions, which naturally come with
integrals over internal momenta.

Let us now finalize our discussion on how (3.16) is preserved. We have shown
that if (3.16) is satisfied at fixed initial point $t=t_{0}$, then it can be
consistently satisfied at the next “time step” $t=t_{0}+\delta t$ because our
equations are all first-order in $d/dt$. Iterating this logic for all time
steps, the preservation of (3.16) is thus proven for any $t$.

In the end, let us briefly address rotational invariance. Rotational
invariance on top of translational invariance guarantees that the functions
$D(k,p,q)$ are real functions for Weyl-ordered averages of product operators
that are Hermitian in position space. Consider for example a Weyl-ordered
cubic expectation value

$\displaystyle\langle A(\mathbf{k})B(\mathbf{p})C(\mathbf{q})\rangle_{W}^{*}=$
$\displaystyle\int\int\int
d^{3}xd^{3}yd^{3}ze^{+i(\mathbf{k}\mathbf{x}+\mathbf{p}\mathbf{y}+\mathbf{q}\mathbf{z})}\langle
A(\mathbf{x})B(\mathbf{y})C(\mathbf{z})\rangle_{W}$ $\displaystyle=$
$\displaystyle\int\int\int
d^{3}x^{\prime}d^{3}y^{\prime}d^{3}z^{\prime}e^{-i(\mathbf{k}\mathbf{x}^{\prime}+\mathbf{p}\mathbf{y}^{\prime}+\mathbf{q}\mathbf{z}^{\prime})}\langle
A(-\mathbf{x}^{\prime})B(-\mathbf{y}^{\prime})C(-\mathbf{z}^{\prime})\rangle_{W}$
$\displaystyle=$ $\displaystyle\int\int\int
d^{3}x^{\prime}d^{3}y^{\prime}d^{3}z^{\prime}e^{-i(\mathbf{k}\mathbf{x}^{\prime}+\mathbf{p}\mathbf{y}^{\prime}+\mathbf{q}\mathbf{z}^{\prime})}\langle
A(\mathbf{x}^{\prime})B(\mathbf{y}^{\prime})C(\mathbf{z}^{\prime})\rangle_{W}$
$\displaystyle=$ $\displaystyle\langle
A(\mathbf{k})B(\mathbf{p})C(\mathbf{q})\rangle_{W}\,.$

In the first line, since the Weyl-ordering was symmetric to begin with, it
ensures that a complex-conjugation on the expectation value did not reshuffle
the operator ordering. We renamed our integration variables in position-space
in the second line and used rotational invariance in the third line. One can
easily generalize the proof to arbitrary powers of product-operators to show
that functions $D$ are real for any Weyl-ordered expectation values (of
position-space Hermitian operators).

We have therefore proven simplifications (1) to (3). These simplifications
remove in the equations of motion terms linear in $\langle\varphi\rangle$ and
$\langle\pi\rangle$ — representing backreaction in our model — and strip away
integrals over internal momenta at tree level. Conversely, classical variables
associated with background inhomogeneities will also decouple from quantum
fluctuations, retaining the fixed background geometry. It is also worth noting
the realness property of fluctuations will also help us find minimal-energy
initial conditions in Section 3.3.

###  3.3 Uncertainty principle and energy-minimization

We need a set of initial conditions for fluctuations. It is natural to
consider minimal-energy configurations as the prime candidates. To find them,
we look for the set of fluctuation-values that minimize $H_{\Delta}$.
Nevertheless, quantum fluctuations must obey the uncertainty principle
(including its generalization to higher-order fluctuations). We must derive
them in our context. We will then see that for a system with potential
$V(\varphi)$, the minimized effective Hamiltonian contains the Coleman-
Weinberg effective potential. In particular, the saturation of the uncertainty
principle will be a result of requiring the system to have minimal energy.
This is true even for interacting fields when quantum fluctuations are kept to
quadratic order. For systems with no interactions, the set of initial values
for fluctuations reduces to the familiar one in flat-spacetime quantum field
theory.

First, we derive the Weyl-ordered version of the well-known quadratic
uncertainty principle. Given an arbitrary state $|\psi\rangle$:

$\varphi_{\mathbf{-k}}|\psi\rangle=|u\rangle,\quad\pi_{\mathbf{-k}}|\psi\rangle=|v\rangle.$

Using the Cauchy-Schwarz inequality for $|u\rangle$ and $|v\rangle$, we have

$\displaystyle\langle\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}}\rangle\langle\pi_{\mathbf{k}}\pi_{\mathbf{-k}}\rangle\geq$
$\displaystyle|\langle\varphi_{\mathbf{k}}\pi_{\mathbf{-k}}\rangle|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\langle\varphi_{\mathbf{k}}\pi_{\mathbf{-k}}+\pi_{\mathbf{-k}}\varphi_{\mathbf{k}}+i\tilde{\delta}(\mathbf{0})\rangle\langle\varphi_{\mathbf{k}}\pi_{\mathbf{-k}}+\pi_{\mathbf{-k}}\varphi_{\mathbf{k}}+i\tilde{\delta}(\mathbf{0})\rangle^{*}$
$\displaystyle=$
$\displaystyle|\langle\varphi_{\mathbf{k}}\pi_{\mathbf{-k}}\rangle_{W}|^{2}+\mathrm{Im}[\langle\varphi_{\mathbf{k}}\pi_{\mathbf{-k}}\rangle_{W}]+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}$
$\displaystyle=$
$\displaystyle\langle\varphi_{\mathbf{k}}\pi_{\mathbf{-k}}\rangle_{W}^{2}+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}\text{
(if translationally and rotationally invariant)}.$ (3.17)

In the last line, we used the realness of
$\langle\varphi_{\mathbf{k}}\pi_{\mathbf{p}}\rangle_{W}$, which relies on
rotational and translational invariance. The appearance of
$\tilde{\delta}(\mathbf{0})\equiv(2\pi)^{3}\delta^{3}(\mathbf{0})$ originates
from the commutator
$[\varphi_{\mathbf{k}},\pi_{\mathbf{-k}}]=i(2\pi)^{3}\delta^{3}(\mathbf{k-k})$.

For inhomogeneous degrees of freedom — those with momenta $\mathbf{k}\neq 0$
and thus $\langle\varphi_{\mathbf{k}}\rangle=0$ — the fluctuations coincide
with Weyl-ordered expectation values (e.g.,
$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})=\langle\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}}\rangle_{W}$).
This generalizes to all combinations of product operators. Hence,

$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})\geq\Delta^{2}(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}.$
(3.18)

Low-energy effective potential. Let us now demonstrate how one could derive
the Coleman-Weinberg effective potential as a low-energy limit of our
effective Hamiltonian. Consider a scalar-field $\varphi$ with a canonical
kinetic term and some arbitrary potential $V(\varphi)$. At second-order in
fluctuations, the effective Hamiltonian for our field theory is

$\displaystyle\langle H_{\varphi}\rangle$
$\displaystyle=H[\langle\varphi\rangle,\langle\pi\rangle]+\int\frac{d^{3}k}{(2\pi)^{3}}\Big{[}\frac{1}{2}(\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})+k^{2}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}}))+\frac{1}{2}V^{\prime\prime}(\langle\varphi\rangle)\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})\Big{]},$
$\displaystyle[\varphi_{\mathbf{k}},\pi_{\mathbf{p}}]$
$\displaystyle=i(2\pi)^{3}\delta^{3}(\mathbf{k+p})\,.$

We will write $V^{\prime\prime}(\langle\varphi\rangle)$ as $V^{\prime\prime}$
in what follows.

Consider the fluctuation part of the effective Hamiltonian

$\displaystyle H_{\Delta}=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{2}\Big{[}\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})+(k^{2}+V^{\prime\prime})\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})\Big{]}$
$\displaystyle\geq$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\sqrt{k^{2}+V^{\prime\prime}}\sqrt{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})}$
$\displaystyle\geq$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\sqrt{k^{2}+V^{\prime\prime}}\sqrt{\Delta^{2}(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})+\frac{\hbar^{2}}{4}(2\pi)^{6}(\delta^{3}(\mathbf{k-k}))^{2}}\,.$

The first inequality uses
$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}}),\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})\geq
0$, and the second inequality is the uncertainty principle, where the realness
of $\Delta(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})$ is guaranteed by spatial
symmetries.

The last line above tells us that effective Hamiltonian is minimized in the
absence of field-momentum correlations, i.e.,
$\Delta(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})=0$, analogous to (2.9) in
quantum mechanics. The minimized effective Hamiltonian $\langle H\rangle$ then
becomes

$\mathrm{Min}[\langle
H\rangle]=H[\langle\varphi\rangle,\langle\pi\rangle]+(2\pi)^{3}\delta^{3}(\mathbf{0})\frac{\hbar}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\sqrt{k^{2}+V^{\prime\prime}}\,.$
(3.19)

The $\delta^{3}(\mathbf{0})\sim\mathrm{Vol}[\mathbb{R}^{3}]$, the volume in
position space. Divided by the volume, the integral term of (3.19) can been
shown [18] to be equivalent to the covariant form of the Coleman-Weinberg
potential (density) up to constants of $\langle\varphi\rangle$:

$V_{\mathrm{CW}}=\frac{i\hbar}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\log\Big(1+\frac{V^{\prime\prime}}{\eta_{\mu\nu}k^{\mu}k^{\nu}}\Big{missing})\,,\;\eta_{\mu\nu}=(-1,1,1,1)$
(3.20)

after integrating $\int dk^{0}$.

Let us remark on the comparison with the past derivations of effective
potentials using fluctuations [18, 27, 28]. These work either assume a
saturation of the uncertainty principle and adiabatic approximations in the
evolution or rely on canonical maps. What is new here is that we make none of
these assumptions nor rely on any canonical maps (as they have not been found
for fields). The saturation of the uncertainty principle in our work is
_derived_ by requiring that the effective Hamiltonian is stable to
fluctuations. The uncertainty principle underpins how inequalities should be
used in a constrained minimization problem. It, along with translational and
rotational symmetries, directly permits us to derive the low-energy effective
potential. The consistency between these symmetries and the evolution of
fluctuations is non-trivial and needs to be checked, as we have done in
Section (3.2). Physically, these symmetries lead to minimal backreaction in
our framework and legitimate using a fixed background metric in the
cosmological application in Section 4.

Finding an initial state. In an operator formalism, one often starts with an
initial vacuum of the “free” Hamiltonian when the scattering particles are far
away enough to not feel any interactions 888In cosmological applications, one
cannot assume initially vanishing interactions on physical grounds.
Nonetheless, in practice, this is often still assumed implicitly through an
$i\varepsilon$-deformation of the contour in time. It is the opinion of the
authors that the assumption is in most applications a reasonable compromise
when the construction and calculations involving an interacting vacuum in
curved spacetime is too difficult. This difficulty is exacerbated by the
gauge/constraint aspects of quantization when higher-order metric
perturbations are involved.. In our effective formalism, initial conditions
can be found in alignment with this philosophy.

To conform with the spirit of traditional quantum field theory, we look for a
set of values for fluctuations that minimize the energy of the “free” part of
our Hamiltonian, defined as the part with quadratic field products. For our
$\varphi^{3}$ example, it corresponds to the first line of (3.6). We will
denote the effective Hamiltonian of the “free” part as
$H_{\mathrm{Mink.},\Delta^{(2)}}$. To minimize
$H_{\mathrm{Mink.},\Delta^{(2)}}$, we seek initial fluctuations that satisfy
all inequalities while minimizing
$\sqrt{\Delta^{2}(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})+\tilde{\delta}(\mathbf{0})^{2}/4}$.
The set of fluctuations meeting this requirement is

$\displaystyle\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})=\omega_{k}^{2}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-k}})=\omega_{k}\sqrt{\Delta^{2}(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}},$
$\displaystyle\Delta(\varphi_{\mathbf{k}}\pi_{\mathbf{-k}})=0.$

In more familiar terms:

$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{-p}})=\frac{1}{2\omega_{k}}\tilde{\delta}_{\mathbf{k,p}},\qquad\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-p}})=\frac{\omega_{k}}{2}\tilde{\delta}_{\mathbf{k,p}},\qquad\Delta(\varphi_{\mathbf{k}}\pi_{\mathbf{-p}})=0\times\tilde{\delta}_{\mathbf{k,p}}.$
(3.21)

This corresponds to what one would have obtained using the conventional
operator formalism for a free-scalar vacuum in Minkowski spacetime, with

$\varphi_{\mathbf{k}}=\frac{1}{\sqrt{2\omega_{k}}}e^{-i\omega_{k}t}a_{\mathbf{k}}+\frac{1}{\sqrt{2\omega_{k}}}e^{+i\omega_{k}t}a_{\mathbf{-k}}^{\dagger},\;\pi_{\mathbf{k}}=\dot{\varphi}_{\mathbf{k}}.$
(3.22)

Here, the time-independent annihilation operator $a_{\mathbf{k}}$ annihilates
the Minkowski vacuum $|0\rangle$ associated with the $t$-slicing.

Appending initial conditions (3.21) with an initial Gaussian hierarchy, where
Wick’s theorem applies to the fluctuations, one obtains a complete set of
initial conditions for evolving arbitrary orders of fluctuations
$\Delta^{(n)}$.

###  3.4 Solutions

Having established the equations of motion (3.13) and (3.14), along with the
homogeneous and isotropic initial conditions (3.21), we are now equipped to
derive effective solutions for the $\varphi^{3}$-system perturbatively. We
will focus on the $\mu^{1}$-order solutions for quadratic and cubic
fluctuations, $\Delta^{(2)}$ and $\Delta^{(3)}$. We’ll also touch upon the
quantum contributions in this section, and more extensive cases will be
tentatively explored in Appendix B. A thorough analysis of effective solutions
at various orders will be reported elsewhere.

To determine the equations of motion for $\Delta^{(n)}$, which involves $n$
operators of $\varphi$ and/or $\pi$ fields, we assume a perturbative expansion
of $\Delta^{(n)}$ based on powers of $\mu$:

$\Delta^{(n)}=\Delta^{(n)}_{0}+\Delta^{(n)}_{1}+\Delta^{(n)}_{2}+\cdots,$

where $\Delta^{(n)}_{k}$ denotes the component of $\Delta^{(n)}$ that is of
the order $\mu^{k}$. Following our previous discussion, once all
$\Delta_{0}^{(n)}$ values — the quantum fluctuations of a free scalar field at
$O(\mu^{0})$ — have been determined, it becomes feasible to systematically
compute $\Delta^{(n)}_{k}$ at any given order in $\mu$.

####  3.4.1 Quantum fluctuations for free theory

For $\mu^{0}$-order, $\Delta^{(n)}_{0}$’s adhere to the following free
equations of motion:

$\dot{\Delta}_{0}(X\varphi_{\mathbf{a}})=\cdots+\Delta_{0}(X\pi_{\mathbf{a}}),\qquad\dot{\Delta}_{0}(X\pi_{\mathbf{a}})=\cdots-\omega_{a}^{2}\Delta_{0}(X\varphi_{\mathbf{a}}).$
(3.23)

The initial conditions (3.21) lead to static solutions for both quadratic and
cubic fluctuations such that at any time $t$:

$\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}};t)=\frac{1}{2\omega_{a}}\tilde{\delta}_{\mathbf{a,b}},\qquad\Delta_{0}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}};t)=0,\qquad\Delta_{0}(\pi_{\mathbf{a}}\pi_{\mathbf{b}};t)=\frac{\omega_{a}}{2}\tilde{\delta}_{\mathbf{a,b}}.$
(3.24)

These static solutions are a natural consequence of the initial conditions
(3.21), corresponding to the vacuum state $\ket{0}$ of the free scalar field,
which is an eigenstate of the free Hamiltonian and hence remains static.

Given the vacuum state $\ket{0}$, the higher-order fluctuations at
$\mu^{0}$-order can be derived from $\Delta_{0}^{(2)}$ (3.24) using Wick’s
theorem. For instance, the cubic fluctuations $\Delta_{0}^{(3)}$ are all $0$:

$\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}};t)=\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}};t)=\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}};t)=\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}};t)=0,$
(3.25)

and the quartic ones are:

$\displaystyle\Delta_{0}\left(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}};t\right)$
$\displaystyle=\Delta_{0}\left(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\right)\Delta_{0}\left(\varphi_{\mathbf{c}}\varphi_{\mathbf{d}}\right)+\Delta_{0}\left(\varphi_{\mathbf{a}}\varphi_{\mathbf{c}}\right)\Delta_{0}\left(\varphi_{\mathbf{b}}\varphi_{\mathbf{d}}\right)+\Delta_{0}\left(\varphi_{\mathbf{a}}\varphi_{\mathbf{d}}\right)\Delta_{0}\left(\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\right)$
$\displaystyle=\frac{1}{4\omega_{a}\omega_{c}}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}}+\frac{1}{4\omega_{a}\omega_{b}}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}}+\frac{1}{4\omega_{a}\omega_{d}}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}},$
$\displaystyle\Delta_{0}\left(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}};t\right)$
$\displaystyle=\Delta_{0}\left(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\right)\Delta_{0}\left(\pi_{\mathbf{c}}\pi_{\mathbf{d}}\right)=\frac{\omega_{c}}{4\omega_{a}}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}},$
$\displaystyle\Delta_{0}\left(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}};t\right)$
$\displaystyle=\Delta_{0}\left(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\right)\Delta_{0}\left(\pi_{\mathbf{c}}\pi_{\mathbf{d}}\right)+\Delta_{0}\left(\pi_{\mathbf{a}}\pi_{\mathbf{c}}\right)\Delta_{0}\left(\pi_{\mathbf{b}}\pi_{\mathbf{d}}\right)+\Delta_{0}\left(\pi_{\mathbf{a}}\pi_{\mathbf{d}}\right)\Delta_{0}\left(\pi_{\mathbf{b}}\pi_{\mathbf{c}}\right)$
$\displaystyle=\frac{\omega_{a}\omega_{c}}{4}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}}+\frac{\omega_{a}\omega_{b}}{4}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}}+\frac{\omega_{a}\omega_{d}}{4}\tilde{\delta}_{\mathbf{a,b}}\tilde{\delta}_{\mathbf{c,d}}$
$\displaystyle\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}};t)$
$\displaystyle=\Delta_{0}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}};t)=0.$
(3.26)

These constants are indeed static solutions to the free equations of motion
(3.23).

####  3.4.2 Quantum fluctuations at tree-level

The equations of motion (3.13) indicate that the evolutions of quadratic
fluctuations $\Delta_{1}^{(2)}$ of order $\mu$ depend on the free cubic
fluctuations $\Delta_{0}^{(3)}$. Fortunately, the free cubic
$\Delta^{(3)}_{0}=0$ at $\mu^{0}$-order. Therefore, the subleading terms of
$\Delta^{(2)}$ are at least of order $\mu^{2}$:

$\Delta^{(2)}=\Delta^{(2)}_{0}+O(\mu^{2}).$

Therefore, for tree-level solutions of $\Delta^{(2)}$ up to $\mu$-order,
focusing on the $\mu^{0}$-order — the free solution $\Delta^{(2)}_{0}$ (3.24)
— is sufficient.

Subsequently, with the $\mu^{0}$-order solutions for $\Delta_{0}^{(2)}$ s and
$\Delta_{0}^{(4)}$ s determined in (3.24) and (3.26), we can now apply them to
(3.14) to solve for $\Delta_{1}^{(3)}$. The equations of motion for
$\Delta^{(3)}_{1}$, ignoring $O(\mu^{2})$ terms, are:

$\displaystyle\dot{\Delta}_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})$
$\displaystyle=\Delta_{1}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})+\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})+\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}),$
$\displaystyle\dot{\Delta}_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle=\Delta_{1}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})+\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{c}^{2}\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})-\frac{\mu}{2\omega_{a}\omega_{b}}\tilde{\delta}_{k,p,q},$
$\displaystyle\dot{\Delta}_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle=\Delta_{1}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{b}^{2}\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{c}^{2}\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}),$
$\displaystyle\dot{\Delta}_{1}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle=-\omega_{a}^{2}\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{b}^{2}\Delta_{1}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})-\omega_{c}^{2}\Delta_{1}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})+\frac{\mu}{2}\tilde{\delta}_{\mathbf{a,b,c}}.$

Figure 1: Evolutions of
$\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})$,
$\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})$,
$\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$, and
$\Delta_{1}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$.

The above set forms a non-homogeneous linear ordinary differential equation
system involving eight $\Delta_{1}^{(3)}$ equations with the initial condition

$\Delta_{1}^{(3)}(t=0)=0.$

For this initial condition, the solution is:

$\displaystyle\omega_{a}\omega_{b}\omega_{c}\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})$
$\displaystyle=\frac{\mu\left(\cos\omega_{\rm{T}}t-1\right)}{2\omega_{\rm{T}}}\tilde{\delta}_{\mathbf{a,b,c}},$
$\displaystyle\omega_{b}\omega_{c}\Delta_{1}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})$
$\displaystyle=\omega_{c}\omega_{a}\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})=\omega_{a}\omega_{b}\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})=-\frac{\mu\sin\omega_{\rm{T}}t}{2\omega_{\rm{T}}}\tilde{\delta}_{\mathbf{a,b,c}},$
$\displaystyle\omega_{a}\Delta_{1}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle=\omega_{b}\Delta_{1}(\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{a}})=\omega_{c}\Delta_{1}(\varphi_{\mathbf{c}}\pi_{\mathbf{a}}\pi_{\mathbf{b}})=\frac{\mu(1-\cos\omega_{\rm{T}}t)}{2\omega_{\rm{T}}}\tilde{\delta}_{\mathbf{a,b,c}},$
$\displaystyle\Delta_{1}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle=\frac{\mu\sin\omega_{\rm{T}}t}{2\omega_{\rm{T}}}\tilde{\delta}_{\mathbf{a,b,c}},$
(3.27)

where $\omega_{\rm{T}}=\omega_{a}+\omega_{b}+\omega_{c}$ is the inherent
angular frequency of $\Delta^{(3)}$ s. The evolutions of these cubic
fluctuations are depicted in Fig. 1.

These solutions demonstrate that, at the tree level in Minkowski spacetime,
all $\Delta^{(3)}$ fluctuations are bounded and evolve harmonically. As
discussed in Appendix B, the bounded and harmonic nature of tree-level quantum
fluctuations is a characteristic feature of a Minkowski spacetime. In Section
4, we will observe that in a cosmological de Sitter spacetime,
$\Delta^{(3)}_{1}$ fluctuations diverge in a $\varphi^{3}$-model as the
conformal time $\eta$ approaches $0$, corresponding to the far-future of an
exponentially expanding spacetime.

####  3.4.3 Quantum fluctuations at quantum-level

To conclude this section, we outline the typical quantum effects on the
effective dynamics of the $\varphi^{3}$ model in the Minkowski spacetime. A
more exhaustive study is ongoing.

We first examine the quantum effect on the classical variables
$\langle\varphi_{\mathbf{a}}\rangle$ and $\langle\pi_{\mathbf{a}}\rangle$ at
the $\mu^{1}$-order. Given the initial condition
$\langle\varphi_{\mathbf{a}}\rangle=\langle\pi_{\mathbf{a}}\rangle=0$, their
equations of motion (3.15) become:

$\displaystyle\dot{\langle\varphi_{\mathbf{a}}\rangle}_{1}$
$\displaystyle=\langle\pi_{\mathbf{a}}\rangle_{1},$
$\displaystyle\dot{\langle\pi_{\mathbf{a}}\rangle}_{1}$
$\displaystyle=-\omega_{a}^{2}\langle\varphi_{\mathbf{a}}\rangle_{1}-\mu\int_{p}\Bigg{[}\langle\varphi_{\mathbf{p}}\rangle_{0}\langle\varphi_{\mathbf{a-p}}\rangle_{0}+\Delta_{0}(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})\Bigg{]}=-\omega_{a}^{2}\langle\varphi_{\mathbf{a}}\rangle_{1}-\frac{\mu}{2m}\tilde{\delta}_{\mathbf{a}}\int\frac{d^{3}k}{(2\pi)^{3}}.$

These equations imply that $\langle\varphi_{\mathbf{a}}\rangle\neq 0$ only for
$\mathbf{a}=0$. This scenario represents a homogeneous field strength
$\langle\varphi(\mathbf{x})\rangle$, uniform across all positions
$\mathbf{x}$. This expectation value is negative (under the assumption that
$\mu>0$) and exhibits UV divergence. This divergence is anticipated, given
that the $\mu\varphi^{3}$ term in the Hamiltonian, being unbounded from below,
naturally pushes the field $\varphi$ towards infinite descent.

In quantum field theory, such a divergent field strength
$\langle\varphi\rangle$ correlates with tadpole diagrams. Drawing from
textbook quantum field theory, we know that tadpole terms break Lorentz
invariance. It is conventional to redefine the field operator $\varphi$ by re-
expanding it around its new expectation value $\langle\varphi\rangle$:

$\varphi\to\varphi-\langle\varphi\rangle.$

With this redefined field, the equations of motion for quantum fluctuations
remain unchanged, resulting in

$\langle\varphi_{\mathbf{a}}\rangle(t)=\langle\pi_{\mathbf{a}}\rangle(t)=0.$

This is the reason for omitting the term
$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})\langle\varphi_{\mathbf{q}}\rangle$
in the fluctuation-part of the effective Hamiltonian,
$H_{\text{Mink.},\Delta}$, as indicated in equation (3.6). This omission leads
to the simplified form of the equations of motion presented in equation (3).

Then, as a simple yet nontrivial example, we consider an observable quantum
effect — the $\mu^{2}$-order correction, $\Delta^{(2)}_{2}$, to the quadratic
quantum fluctuations. Having determined all $\Delta_{1}^{(3)}$ in Eqs. (3.27),
we can now express the $\mu^{2}$-order equations of motion for quadratic
quantum fluctuations as follows:

$\displaystyle\dot{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})$
$\displaystyle=\Delta_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}})+\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}),$
$\displaystyle\dot{\Delta}_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})$
$\displaystyle=\Delta_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}})-\omega_{b}^{2}\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})-\mu\int_{p}{(2\pi)^{3}}\Delta_{1}(\varphi_{\mathbf{a}}\varphi_{\mathbf{p}}\varphi_{\mathbf{b-p}}),$
$\displaystyle=\Delta_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}})-\omega_{b}^{2}\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})+\frac{\mu^{2}}{2}\tilde{\delta}_{\mathbf{a,b}}\int_{p}\frac{1-\cos\omega_{\rm{T}}t}{\omega_{\rm{T}}\omega_{a}\omega_{p}\omega_{|\mathbf{a+p}|}}$
$\displaystyle\dot{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}})$
$\displaystyle=-\omega_{a}^{2}\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})-\omega_{b}^{2}\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})-\mu\int_{p}\left[\Delta_{1}(\pi_{\mathbf{a}}\varphi_{\mathbf{p}}\varphi_{\mathbf{b-p}})+\Delta_{1}(\pi_{\mathbf{b}}\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})\right]$
$\displaystyle=-\omega_{a}^{2}\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})-\omega_{b}^{2}\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})+\mu^{2}\tilde{\delta}_{\mathbf{a,b}}\int_{p}\frac{\sin\omega_{\rm{T}}t}{\omega_{\rm{T}}\omega_{a}\omega_{|\mathbf{a+p}|}},$

where

$\omega_{\rm{T}}=\omega_{a}+\omega_{p}+\omega_{\mathbf{|\mathbf{a+p}|}}.$

The RHS of these equations exhibit $\ln\Lambda$ UV divergence, where $\Lambda$
is the UV cutoff. Renormalization is necessary to address this divergence,
resulting in $\mu^{2}$-order corrections to quadratic $\Delta^{(2)}_{0}$ in
(3.24). The renormalization procedure remains a subject of ongoing
investigations.

##  4 Fields on cosmological spacetimes

We have seen in the previous section that our framework can provide a
consistent effective description of quantum fields in flat spacetime. The
description allowed for analytical solutions to the $\varphi^{3}$-model
without relying on any operator-method input..

For cosmological spacetimes, such as the exponentially expanding spacetime of
inflation, solving the effective equations for an interacting field is not
easy. Results from the in-in method [30] already suggest that for cubic
fluctuations, the solution can at best only be written in terms of non-
elementary complex integrals that integrate to the (conformal) time of
interest $\eta$. It is only in the superhorizon limit can one hope to express
the solutions in terms of elementary functions.

In our effective framework, we will soon see the obstruction to obtaining
analytical solutions manifest in the form of time-dependent interaction
strengths. In the $\varphi^{3}$-model on de Sitter, these will appear as
functions $\Omega_{k}^{2}(\eta)$ for the quadratic interaction and
$\lambda(\eta)$ for the cubic interaction. The analogue of (3.14) in an
expanding spacetime will typically not be solvable analytically when these
interaction strengths depend on $\eta$.

Fortunately, following the strategy outlined in Section 3.4, one could still
numerically describe the evolution of 3-pt correlators. One simply needs to
organize the solution in cubic-interaction strength and use the uncertainty
principle to find the desired minimal-energy initial conditions. At the
zeroth-order in cubic-interaction strength, the analogue of (3.13) can still
be solved analytically despite the quadratic-interaction strength being time-
dependent. As we will see, however, this process sees additional subtleties.
The expansion of spacetime permits different choices of canonical variables
that correspond to different minimal-energy initial conditions, even at the
zeroth-order in cubic-interaction strength 999This issue is separate from the
ambiguity of vacuum in curved-spacetime quantum field theory. Here, we are not
comparing vacua associated with different time-slicings but minimal energies
associated with different canonical variables of a fixed time-slicing.. The
differences vanish if one sets initial conditions at past-infinity. But for
inflation, it has been debated whether too much expansion could lead to the
“trans-Planckian” problem [31]. Settling the debate is beyond the scope of
this work. For demonstration purposes, we will instead simply pick a set of
canonical variables to show that our effective framework is self-consistent
and that one could arrive at the correct value for the cubic fluctuation
$\Delta(\delta\varphi\delta\varphi\delta\varphi)$ (in the superhorizon limit)
when compared with the in-in method outlined in [32].

###  4.1 Free field

Let us warm up by analyzing a free field on a fixed de Sitter background. The
action of such a system is

$S_{2}=\int d\eta
d^{3}x\frac{1}{2}\sqrt{|g|}\Big{[}-g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi\Big{]}\,.$
(4.1)

We can split the field into a background (homogeneous) part and a perturbation
(inhomogeneous) part

$\varphi(\eta,\mathbf{x})=\bar{\varphi}(\eta)+\delta\varphi(\eta,\mathbf{x})\,.$

Substituting this separation into action (4.1), and upon imposing the
background equations of motion (for $\bar{\varphi}(\eta)$), the
$O(\delta\varphi)$ term vanishes. We thus have an action where perturbation
decouples from the background system

$S_{2}|_{\mathrm{OS}}=\int d\eta d^{3}xa^{2}\frac{1}{2}\bar{\varphi}^{2}+\int
d\eta
d^{3}xa^{2}\frac{1}{2}\Big{[}(\delta\varphi^{\prime})^{2}-\frac{1}{2}\delta^{ij}\partial_{i}\delta\varphi\partial_{j}\delta\varphi\Big{]}\,,$

where the subscript “$\mathrm{OS}$” means on-shell for the background system.

Now consider the $O(\delta\varphi^{2})$-part of the action

$S_{\mathrm{Free}}=\int d\eta
d^{3}xa^{2}\Big{[}\frac{1}{2}(\delta\varphi^{\prime})^{2}-\frac{1}{2}\delta^{ij}\partial_{i}\delta\varphi\partial_{j}\delta\varphi\Big{]}\,,$
(4.2)

which we call the “free” part to separate it from the interacting part
consisting of $O(\delta\varphi^{3})$-order and higher. (We wish to emphasize
that the notion of the “free” part is ambiguous when there are
$\delta\varphi^{\prime}$-type interactions, but it is not an issue here.)

The prescription for obtaining an effective Hamiltonian and its equations of
motion is similar to the one outlined in Section 3. But there is a nuance in
the case of a cosmological spacetime: There are at least three “natural” sets
of canonical variables [22]. We must choose one before we can begin canonical
quantization. Different choices cannot be expected to be physically equivalent
[33], as we now address.

####  4.1.1 Different choices of variables and initial conditions

We claim that there are three “natural” sets of configuration variables
because of two observations. First, canonically quantizing (4.2) “as is” and
imposing an initial vacuum (associated with $\eta$-time slicing) lead to
divergences at early times, which may make quantization inconsistent. Second,
action (4.2) is not one that collects the familiar simple harmonic oscillators
in Fourier space due to the $a^{2}$ factor. If one ignores these two issues,
one could insist that the variables $\delta\varphi$ and
$\pi=a^{2}\delta\varphi^{\prime}$ are natural and carry out the standard
canonical quantization. Otherwise, one could find two other sets of variables,
which we now introduce.

At the present level, the two issues highlighted above are mostly aesthetic,
assuming that one imposes initial conditions at past infinity
($\eta\rightarrow-\infty$). The observables will not be affected.
(Nevertheless, the situation will worsen a lot for generic non-Gaussian
systems.) One might be tempted to make the system appear more like harmonic
oscillators and its initial conditions more convergent with a change of
variables to $f=a\delta\varphi$. The “$f$-variable” enables us to do two
things. One, we can trade initial divergences for late-time divergences. The
late time divergences are not as problematic because the system will have
classicalized and one would not have to deal with infinities in Hilbert space.
The correlators associated with $\delta\varphi$ will also remain finite. Two,
we may refurbish our system into simple harmonic oscillators with one catch:
We need to integrate by parts. The act of integration by parts affects nothing
if one imposes quantum initial conditions at past infinity. But the situation
will be different otherwise.

Let us be more specific. The first of the three natural variables we can pick
is the original $\delta\varphi$ and its canonical momentum
$\pi_{\delta\varphi}=\delta\varphi^{\prime}$. The other two options arise
after changing to the variable $f=a\delta\varphi$. After switching to
$f$-variables, one is faced with two possible actions, differentiated by our
willingness to ignore a boundary term

$\displaystyle\text{Option 1: }S_{\mathrm{free},1}=\int d\eta
d^{3}x\frac{1}{2}\Big{[}(f^{\prime})^{2}-2\mathcal{H}ff^{\prime}+\mathcal{H}^{2}f^{2}-\delta^{ij}\partial_{i}f\partial_{j}f\Big{]}$
$\displaystyle\text{Canonical momentum:
}\pi_{1}(\eta,\mathbf{x})=f^{\prime}(\eta,\mathbf{x})-\mathcal{H}f$
$\displaystyle\text{Option 2: }S_{\mathrm{free},2}=\int d\eta
d^{3}x\frac{1}{2}\Big{[}(f^{\prime})^{2}+\frac{a^{\prime\prime}}{a}f^{2}-\delta^{ij}\partial_{i}f\partial_{j}f\Big{]}-\cancel{\frac{1}{2}\int
d\eta d^{3}x(\mathcal{H}f^{2})^{\prime}}$ $\displaystyle\text{Canonical
momentum: }\pi_{2}(\eta,\mathbf{x})=f^{\prime}(\eta,\mathbf{x})\,,$ (4.3)

where $\mathcal{H}=a^{\prime}/a$ is the Hubble parameter in conformal time.
The two options will introduce two distinct canonical momenta due to
differences in terms with “velocity” $f^{\prime}$ in the Lagrangian.

The action from option 1 is completely equivalent to the original one (4.2),
whereas the action from option 2 — which ignores the boundary term — resembles
a set of simple harmonic oscillators, albeit with time-dependent
“frequencies”. So in terms of picking a “natural” starting point for canonical
quantization, neither of the options offers a significant merit over the
other.

Does this mean one can pick any one of the two options? No, unless one sets
initial conditions at past infinity. If one follows the standard canonical
quantization and assumes an initial minimal-energy or Gaussian state (which
includes the usual Bunch-Davies vacuum), what matters are the initial values
for the quadratic expectation values

$\langle ff\rangle\,,\;\langle f\pi\rangle\,,\;\langle\pi\pi\rangle\,,$ (4.4)

where the $\pi$ above can be either $\pi_{1}$ or $\pi_{2}$. Clearly, since
$\pi_{2}-\pi_{1}=\mathcal{H}f\sim-\eta^{-1}$ on de Sitter, the differences in
quadratic expectation values for the two choices of momentum vanish only when
$\eta\rightarrow-\infty$ at past infinity. (The situation will worsen when
there is a “velocity” type non-Gaussian terms in the Lagrangian — the
differences in initial expectation values may even diverge for different
choices of canonical momentum.)

Our effective framework offers a more intuitive way to understand the
discrepancy: Different canonical pairs correspond to different effective
Hamiltonians, and hence, different minimal-energy conditions. In our case, the
differences in minimal energy states vanish in the infinite-past limit.

In Fourier space, by applying our prescription for obtaining the effective
Hamiltonian, we have

$\displaystyle\text{Option 1:
}H_{1,\Delta}=\int_{k}\Big{[}\frac{1}{2}\Delta(\pi_{1,\mathbf{k}}\pi_{1,\mathbf{-k}})+\mathcal{H}\Delta(f_{\mathbf{k}}\pi_{1,\mathbf{-k}})+\frac{1}{2}k^{2}\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})\Big{]}$
$\displaystyle\text{Option 2:
}H_{2,\Delta}=\int_{k}\Big{[}\frac{1}{2}\Delta(\pi_{2,\mathbf{k}}\pi_{2,\mathbf{-k}})+\frac{1}{2}(k^{2}+\frac{a^{\prime\prime}}{a})\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})\Big{]}\,.$

In de Sitter, both $a^{\prime\prime}/a$ and $\mathcal{H}$ vanish in the
infinite-past limit ($\eta\rightarrow-\infty$), so the difference between the
two effective Hamiltonians vanishes. One can use our uncertainty-principle
prescription in Section 3.3 to find a common set of initial values for the two
options — they correspond to the Minkowski vacuum in the operator/Hilbert-
space Language. This is what modes feel in the subhorizon limit under Bunch-
Davies initial conditions because their scales are too small to feel any
effects from curvature.

But for a finite initial time $|\eta_{i}|<\infty$, there is no common set of
initial values for the two options anymore. To find their minimal-energy
conditions, one should use the uncertainty principle, per Section 3.3. The
prescription outlined there tells us that the minima of the two Hamiltonians
are

$\displaystyle\text{Option 1:
}H_{1,\Delta}\geq\int_{k}\Big{[}\mathcal{H}\Delta(f_{\mathbf{k}}\pi_{1,\mathbf{-k}})+k\sqrt{\Delta^{2}(f_{\mathbf{k}}\pi_{1,\mathbf{-k}})+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}}\Big{]}\,.$
$\displaystyle\text{Option 2:
}H_{2,\Delta}\geq\int_{k}\Big{[}(k^{2}-\frac{2}{\eta^{2}})^{1/2}\sqrt{\Delta^{2}(f_{\mathbf{k}}\pi_{2,\mathbf{-k}})+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}}\Big{]}\,.$

Note that
$1/4\tilde{\delta}(\mathbf{0})^{2}\rightarrow\hbar^{2}/4\tilde{\delta}(\mathbf{0})^{2}$
if we restore $\hbar$, and $\tilde{\delta}(\mathbf{0})$ comes from
$\tilde{\delta}(\mathbf{k-k})$. The smallest values of the two lower bounds
are reached when

Option 1: $\displaystyle
k^{2}\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})=\Delta(\pi_{1,\mathbf{k}}\pi_{1,\mathbf{-k}})=k\sqrt{\Delta^{2}(f_{\mathbf{k}}\pi_{1,\mathbf{-k}})+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}}\,,\;$
$\displaystyle\Delta^{2}(f_{\mathbf{k}}\pi_{1,\mathbf{-k}})=\frac{1}{4}\frac{1}{(k/\mathcal{H})^{2}-1}\tilde{\delta}(\mathbf{0})^{2}\,.$
Option 2:
$\displaystyle(k^{2}-\frac{2}{\eta^{2}})\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})=\Delta(\pi_{2,\mathbf{k}}\pi_{2,\mathbf{-k}})=(k^{2}-\frac{2}{\eta^{2}})^{1/2}\sqrt{\Delta^{2}(f_{\mathbf{k}}\pi_{2,\mathbf{-k}})+\frac{1}{4}\tilde{\delta}(\mathbf{0})^{2}}\,,\;$
$\displaystyle\Delta^{2}(f_{\mathbf{k}}\pi_{2,\mathbf{-k}})=0\,.$ (4.5)

In particular, when the effective Hamiltonians are minimized, the initial 2-pt
functions,

$\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})|_{\text{option
1}}=\frac{1}{2}(k^{2}-\frac{1}{\eta_{i}^{2}})^{-1/2}\tilde{\delta}(\mathbf{0})\,,\;\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})|_{\text{option
2}}=\frac{1}{2}(k^{2}-\frac{2}{\eta_{i}^{2}})^{-1/2}\tilde{\delta}(\mathbf{0})\,,$
(4.6)

differ, significantly for long-wavelength modes. Nevertheless, they coincide
when the initial time is taken to the $\eta_{i}\rightarrow-\infty$ limit.

Suppose one demands that the effective system (for either option 1 or option
2) rests in its minimal-energy configuration initially according to (4.5). Not
all modes “feel” a Minkowski vacuum, as is commonly assumed in an
operator/Hilbert-space framework [34]. The modes are actually in squeezed
states [35], where the squeezing parameter depends on both the wavenumber $k$
and initial time $\eta_{i}$ in a non-polynomial way.

The use of non-Bunch-Davies initial conditions has observational consequences.
This was discussed in [22] by employing different canonical variables and
using an operator/Hilbert-space language. Alternatively, our effective
framework provides an easier parameterization of initial conditions using
quantum fluctuations as dynamical quantities. As a potential application in
this direction, we would like to point out the possibility of explaining the
large-angle anomaly in the CMB [36]. We see, for example in (4.6), that large-
wavenumber modes deviate from the Minkowski vacuum the most. To explain the
weakness of large-angle temperature correlations, we would require modes with
large wavenumbers to contribute weaker power spectra compared with the
standard Bunch-Davies result. In [22], squeezed initial conditions akin to
(4.5) lead to a modified power spectrum that is ambiguous in whether it
amplifies or weakens the large-angle sector of the power spectrum. So a
squeezed initial state may not be enough to explain the anomaly. Using our
method, however, one could parameterize more general hierarchies between
fluctuations in an $\eta_{i}$\- or $k$-dependent way. The ratio between the
fluctuations $\Delta(ff)$ and $\Delta(\pi\pi)$, such as the ones in (4.5), are
among the simplest parameterizations of a fluctuation-hierarchy — Our method
provides a way to probe more complicated ones, potentially involving higher-
order fluctuations. The examination of CBM anomalies using our effective
framework is the subject of an ongoing investigation, which will be reported
elsewhere.

####  4.1.2 The equations of motion and its solution

To demonstrate our effective framework without having to worry about ambiguity
in variables, we will assume that initial conditions can be set at past
infinity so that we are free to pick any one of the aforementioned canonical
momenta. We will use option 2 in (4.3) for easier comparison with the flat-
spacetime case of Section 3. That is, our effective Hamiltonian in Fourier
space will be

$H_{\mathrm{free},\Delta}=\int_{k}\frac{1}{2}\Big{[}\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})+\Omega_{k}^{2}(\eta)\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})\Big{]}\,,\;\pi_{\mathbf{k}}=f^{\prime}_{\mathbf{k}}\,,\;\Omega_{k}^{2}(\eta)=k^{2}-\frac{2}{\eta^{2}}\,,$
(4.7)

where we have used a fixed de Sitter solution for $a^{\prime\prime}/a$.

To obtain initial conditions we again assume that the initial fluctuations
follow a Gaussian hierarchy (where Wick’s theorem applies) and minimizes
$H_{\mathrm{free},\Delta}$. The only initial free parameters we must find are
the quadratic expectation values, which can be found with the uncertainty-
principle prescription as before

$\displaystyle\Delta(f_{\mathbf{p}}\pi_{\mathbf{q}})=$ $\displaystyle
0\times\tilde{\delta}_{\mathbf{p,q}}$
$\displaystyle\Delta(f_{\mathbf{p}}f_{\mathbf{q}})=$
$\displaystyle\frac{1}{2p}\tilde{\delta}_{\mathbf{p,q}}$
$\displaystyle\Delta(\pi_{\mathbf{p}}\pi_{\mathbf{q}})=$
$\displaystyle\frac{p}{2}\tilde{\delta}_{\mathbf{p,q}}\,.$ (4.8)

They look exactly like the Minkowski version, as is expected from a Bunch-
Davies initial condition in the subhorizon limit.

With the Poisson bracket rules for the fluctuations, we can derive the
equations of motion for $H_{\mathrm{free},\Delta}$ to be

$\displaystyle\Delta^{\prime}(f_{\mathbf{p}}f_{\mathbf{q}})=$
$\displaystyle\Delta(f_{\mathbf{p}}\pi_{\mathbf{q}})+\Delta(f_{\mathbf{q}}\pi_{\mathbf{p}})$
$\displaystyle\Delta^{\prime}(f_{\mathbf{p}}\pi_{\mathbf{q}})=$
$\displaystyle\Delta(\pi_{\mathbf{p}}\pi_{\mathbf{q}})-\Omega_{q}^{2}(\eta)\Delta(f_{\mathbf{p}}f_{\mathbf{q}})$
$\displaystyle\Delta^{\prime}(\pi_{\mathbf{p}}\pi_{\mathbf{q}})=$
$\displaystyle-\Omega_{q}^{2}(\eta)\Delta(\pi_{\mathbf{p}}f_{\mathbf{q}})-\Omega_{p}^{2}(\eta)\Delta(\pi_{\mathbf{q}}f_{\mathbf{p}})\,.$
(4.9)

When one assumes statistical homogeneity and isotropy, the quadratic
expectation values above are delta-functions times real functions of $\eta$.
Imposing our initial conditions, these equations of motion can be solved by
hand to be

$\displaystyle\Delta(f_{\mathbf{p}}f_{\mathbf{q}})=$
$\displaystyle\Big{(}\frac{1}{2p}+\frac{1}{2p^{3}\eta^{2}}\Big{)}\tilde{\delta}_{\mathbf{p,q}}$
$\displaystyle\Delta(f_{\mathbf{p}}\pi_{\mathbf{q}})=$
$\displaystyle-\frac{1}{2(p\eta)^{3}}\tilde{\delta}_{\mathbf{p,q}}$
$\displaystyle\Delta(\pi_{\mathbf{p}}\pi_{\mathbf{q}})=$
$\displaystyle\Big{(}\frac{p}{2}-\frac{1}{2p\eta^{2}}+\frac{1}{2p^{3}\eta^{4}}\Big{)}\tilde{\delta}_{\mathbf{p,q}}\,.$
(4.10)

These solutions coincide with the expectation values obtained in an operator
formalism, assuming Bunch-Davies initial conditions.

In particular, in the spatially-flat gauge, the comoving curvature
perturbation $\mathcal{R}$ and the perturbation $\delta\varphi$ are related by

$\mathcal{R}=\frac{\mathcal{H}}{\bar{\varphi}^{\prime}}\delta\varphi\,.$

This gives the power spectrum of $\mathcal{R}$ to be

$\langle\mathcal{R}_{\mathbf{p}}\mathcal{R}_{\mathbf{q}}\rangle=(\frac{\mathcal{H}}{a\bar{\varphi}^{\prime}})^{2}\langle
f_{\mathbf{p}}f_{\mathbf{q}}\rangle\longrightarrow\frac{H^{2}}{\dot{\bar{\varphi}}^{2}}\frac{H^{2}}{2p^{3}}(2\pi)^{3}\delta^{3}(\mathbf{p+q})\,,$
(4.11)

where the arrow indicates the superhorizon limit.

###  4.2 Interacting field: non-Gaussianities

In this section, we apply our formalism to the case of an interacting field on
fixed de Sitter background. This system serves as a good approximation if one
considers up to leading order in slow roll and assumes sufficient decoupling
between metric and matter perturbations. Our goal is to compute the 3-pt
function, or non-Gaussianity, of the theory. In passing, we demonstrate that
our method is self-contained in the sense that nothing (from equations of
motion to initial values to solution schemes) necessarily requires any
operator-formalism input. The techniques developed in Section 3 suffice.

Let us first consider a generic matter action with potential $V(\varphi)$

$S[\varphi]=\int d\eta
d^{3}x\sqrt{|g|}\Big{[}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V(\varphi)\Big{]}\,.$
(4.12)

We will again assume the following splitting and change of variables

$\varphi(\eta,\mathbf{x})=\bar{\varphi}(\eta)+\delta\varphi(\eta,\mathbf{x})\,,\;f=a\delta\varphi(\eta,\mathbf{x})\,.$
(4.13)

The action can then be split into a background part, containing only
$\bar{\varphi}$, and perturbation part containing terms $O(f^{2},f^{3})$ or
higher.

$S[\varphi]|_{\mathrm{OS}}=S_{0}[\bar{\varphi}]+\int d\eta
d^{3}x[\frac{1}{2}(f^{\prime})^{2}+\frac{1}{2}(\frac{2}{\eta^{2}}f^{2}-\delta^{ij}\partial_{i}f\partial_{j}f)-a^{4}\sum_{n=2}\frac{1}{n!}(\partial_{\phi}^{(n)}V)|_{\phi_{0}}(\frac{f}{a})^{n}]$
(4.14)

To arrive at the above action, we have imposed the equation of motion for the
background field $\bar{\varphi}$, as implied by the subscript “$\mathrm{OS}$”
for on-shell. This makes terms linear in $\delta\varphi$ vanish and explains
why the sum in the integration starts at $n=2$. Additionally, integration by
parts is performed.

Now, consider the case for $V(\varphi)=\lambda_{0}\varphi^{3}/3$ where
$\lambda_{0}$ is a time-dependent small parameter controlling the magnitude of
non-Gaussianity. (The smallness is also necessary from the point of view of
background spacetime stability.) Up to leading order in slow-roll, the
$V_{\varphi\varphi}$ term in (4.14) is negligible compared to other terms,
giving us the action

$S[\varphi]|_{\mathrm{OS}}\approx S_{0}[\bar{\varphi}]+\int d\eta
d^{3}x\Bigg{[}\frac{1}{2}(f^{\prime})^{2}+\frac{1}{2}\Big{(}\frac{2}{\eta^{2}}f^{2}-\delta^{ij}\partial_{i}f\partial_{j}f\Big{)}+\frac{\lambda_{0}}{3H}\frac{1}{\eta}f^{3}\Bigg{]}\,.$
(4.15)

Viewed in Fourier space, the action resembles that of a $\varphi^{3}$-model on
flat spacetime but with an effective time-dependent mass and cubic-interaction
(for a mode with wavenumber $k=|\mathbf{k}|$)

$\Omega_{k}^{2}(\eta)=k^{2}-\frac{2}{\eta^{2}}\,,\;\lambda(\eta)=-\frac{\lambda_{0}}{H\eta}.$
(4.16)

The sign change for the coupling strength is due to the translation from the
scale factor to conformal time, $a(\eta)=-1/(\eta H)$, on fixed de Sitter.

To apply our formalism to the system (4.15), we first notice that its
$O(f^{2},f^{3})$-part does not “backreact” on $S_{0}[\bar{\varphi}]$ in the
sense that $\bar{\varphi}$ and $f$ do not mix in the Lagrangian and the
background spacetime is fixed regardless of how $f$ evolves. So in a
Hamiltonian formalism, we only need to examine the $O(f^{2},f^{3})$-part of
the action if we are only interested in the evolution of $f$.

Thus the Lagrangian and Hamiltonian governing variable $f$ are (in the Fourier
space)

$\displaystyle S_{f^{3}}\equiv$ $\displaystyle
S[\varphi]|_{\mathrm{OS}}-S_{0}[\bar{\varphi}]$ $\displaystyle=$
$\displaystyle\int
d\eta\int_{k}\Big{[}\frac{1}{2}f^{\prime}_{\mathbf{k}}f^{\prime}_{\mathbf{-k}}+\frac{1}{2}\Omega_{k}^{2}(\eta)f_{\mathbf{k}}f_{\mathbf{-k}}\Big{]}$
$\displaystyle+$ $\displaystyle\int
d\eta\int_{k,p,q}\frac{\lambda(\eta)}{3}f_{\mathbf{k}}f_{\mathbf{p}}f_{\mathbf{q}}\tilde{\delta}_{\mathbf{k,p,q}}\,,$
$\displaystyle H_{f^{3}}=$ $\displaystyle\int
d\eta\int_{k}\Big{[}\frac{1}{2}\pi_{\mathbf{k}}\pi_{\mathbf{-k}}+\frac{1}{2}\Omega_{k}^{2}(\eta)f_{\mathbf{k}}f_{\mathbf{-k}}\Big{]}$
$\displaystyle+$ $\displaystyle\int
d\eta\int_{k,p,q}\frac{\lambda(\eta)}{3}f_{\mathbf{k}}f_{\mathbf{p}}f_{\mathbf{q}}\tilde{\delta}_{\mathbf{k,p,q}}\,.$

Since there are no velocity-type interactions, the canonical conjugate to
$f_{\mathbf{k}}$ is simply $\pi_{\mathbf{k}}=f^{\prime}_{\mathbf{k}}$, with
Fourier-space normalization
$\\{f_{\mathbf{k}},\pi_{\mathbf{p}}\\}=\tilde{\delta}_{\mathbf{k,p}}$.

Upon employing our effective formalism, the fluctuation part of the effective
Hamiltonian is

$\displaystyle H_{\Delta}[H_{f^{3}}]\equiv$ $\displaystyle
H_{Q}[H_{f^{3}}]-H_{0}[H_{f^{3}}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{k}\Big{[}\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{-k}})+\Omega_{k}^{2}(\eta)\Delta(f_{\mathbf{k}}f_{\mathbf{-k}})\Big{]}$
$\displaystyle+$
$\displaystyle\int_{k,p,q}\frac{\lambda(\eta)}{3}\Big{[}3\Delta(f_{\mathbf{k}}f_{\mathbf{p}})\langle
f_{\mathbf{k}}\rangle+\Delta(f_{\mathbf{k}}f_{\mathbf{p}}f_{\mathbf{q}})\Big{]}\tilde{\delta}_{\mathbf{k,p,q}}\,,$
(4.17)

where we have re-shuffled the integration variables to get the factor of the
$O(\langle f\rangle)$ term in the third line.

For solutions respecting translational and rotational invariance on the level
of quantum averages, linear-operator expectation values like $\langle
f_{\mathbf{k}}\rangle,\langle\pi_{\mathbf{k}}\rangle$ will vanish. The
cubic-$\Delta$ equations of motion are therefore

$\displaystyle\Delta^{\prime}(f_{\mathbf{b}}f_{\mathbf{c}}f_{\mathbf{d}})=$
$\displaystyle\Delta(f_{\mathbf{b}}f_{\mathbf{c}}\pi_{\mathbf{d}})+\Delta(f_{\mathbf{c}}f_{\mathbf{d}}\pi_{\mathbf{b}})+\Delta(f_{\mathbf{d}}f_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle\Delta^{\prime}(f_{\mathbf{b}}f_{\mathbf{c}}\pi_{\mathbf{d}})=$
$\displaystyle\Delta(f_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})+\Delta(f_{\mathbf{c}}\pi_{\mathbf{b}}\pi_{\mathbf{d}})-\Omega_{d}^{2}(\eta)\Delta(f_{\mathbf{b}}f_{\mathbf{c}}f_{\mathbf{d}})$
$\displaystyle+$
$\displaystyle\lambda(\eta)\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}(-\tilde{\delta}_{\mathbf{d},\mathbf{k}})[\Delta(f_{\mathbf{b}}f_{\mathbf{c}}f_{\mathbf{p}}f_{\mathbf{q}})-\Delta(f_{\mathbf{p}}f_{\mathbf{q}})\Delta(f_{\mathbf{b}}f_{\mathbf{c}})]$
$\displaystyle\Delta^{\prime}(f_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})=$
$\displaystyle\Delta(\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-[\Omega_{c}^{2}(\eta)\Delta(f_{\mathbf{b}}f_{\mathbf{c}}\pi_{\mathbf{d}})+(\mathbf{c}\leftrightarrow\mathbf{d})]$
$\displaystyle+$
$\displaystyle\lambda(\eta)\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}[-\tilde{\delta}_{\mathbf{c,k}}(\Delta(f_{\mathbf{b}}f_{\mathbf{p}}f_{\mathbf{q}}\pi_{\mathbf{d}})-\Delta(f_{\mathbf{b}}\pi_{\mathbf{d}})\Delta(f_{\mathbf{p}}f_{\mathbf{q}}))+(\mathbf{c}\leftrightarrow\mathbf{d})]$
$\displaystyle\Delta^{\prime}(\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})=$
$\displaystyle-\Omega_{b}^{2}(\eta)\Delta(f_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\Omega_{c}^{2}(\eta)\Delta(f_{\mathbf{c}}\pi_{\mathbf{d}}\pi_{\mathbf{b}})-\Omega_{d}^{2}(\eta)\Delta(f_{\mathbf{d}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle+$
$\displaystyle\lambda(\eta)\int_{k,p,q}\tilde{\delta}_{\mathbf{k,p,q}}\Big{[}-\tilde{\delta}_{\mathbf{b,k}}(\Delta(\pi_{\mathbf{c}}\pi_{\mathbf{d}}f_{\mathbf{p}}f_{\mathbf{q}})-\Delta(\pi_{\mathbf{c}}\pi_{\mathbf{d}})\Delta(f_{\mathbf{p}}f_{\mathbf{q}}))$
$\displaystyle+$ $\displaystyle(\text{Two more terms cycling
}\mathbf{b\rightarrow c\rightarrow d})\Big{]}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\lambda(\eta)\tilde{\delta}_{\mathbf{b,c,d}}\,.$
(4.18)

These equations look formally like the equations of motion (3.14) for the
$\varphi^{3}$-model on Minkowski but with a time-dependent mass and cubic-
interaction parameter.

####  4.2.1 Finding the solution

We will try to solve the cubic fluctuations perturbatively in $O(\lambda)$.
Given a Gaussian initial condition, the cubic fluctuations vanish in
$O(\lambda^{0})$. So to find the leading order non-trivial solution, we only
need to go to $O(\lambda)$. What this means for the equation set (4.18) is
that the even-order fluctuations inside integrals only need to be kept up to
$O(\lambda^{0})$, i.e., its “free” solution.

Previously, the free solution is

$\displaystyle\Delta(f_{\mathbf{k}}f_{\mathbf{p}})=$
$\displaystyle\Big{(}\frac{1}{2k}+\frac{1}{2k^{3}\eta^{2}}\Big{)}\tilde{\delta}_{\mathbf{k,p}}$
$\displaystyle\Delta(f_{\mathbf{k}}\pi_{\mathbf{p}})=$
$\displaystyle-\frac{1}{2(k\eta)^{3}}\tilde{\delta}_{\mathbf{k,p}}$
$\displaystyle\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}})=$
$\displaystyle\Big{(}\frac{k}{2}-\frac{1}{2k\eta^{2}}+\frac{1}{2k^{3}\eta^{4}}\Big{)}\tilde{\delta}_{\mathbf{k,p}}\,,$

with all other fluctuations obtained via a Gaussian hierarchy, which in
particular means that all odd-order fluctuations vanish.

With the initial values determined and free solutions specified, one can solve
the cubic order fluctuations up to $O(\lambda)$. Because of the time
dependence in $\Omega_{b}^{2}(\eta)$ and $\lambda(\eta)$, the solutions cannot
be obtained analytically. We therefore use numerical evolution. The result for
the 3-pt correlator
$\langle\delta\varphi_{\mathbf{b}}\delta\varphi_{\mathbf{c}}\delta\varphi_{\mathbf{c}}\rangle$
is shown in Figure 2, where we also make the comparison with the superhorizon-
limit expression obtained from the “in-in” method.

Figure 2: The comparison between the solution to (4.18) (blue) and the
superhorizon limit of the ”in-in” result (yellow). (Namely, the accuracy of
the yellow line in representing the 3-pt correlator becomes worse as one
extends back into the subhorizon limit, or equivalently, larger $|\eta|$.) The
gray vertical line represents the time of horizon crossing for the mode
$k_{\mathrm{T}}\equiv|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|$, which roughly
acts as an effective wavenumber for the 3-pt correlator. The free solution
used in (4.18) are that of (4.10), which reduces in the far-past to the Bunch-
Davies initial condition commonly used for the “in-in” method. The external
momenta used were $(\mathbf{b,c,d})=(H,2H,1.5H)$, where $H$ is the Hubble
parameter. We have checked that other sets of external momenta also exhibits
excellent agreement with the “in-in” result.

Our numerical evolution shows excellent agreement with the “in-in” method in
the superhorizon limit, which exhibits a $\log[-(a+b+c)\eta]$ divergence
originating from the fixed de Sitter background. For more realistic
cosmological applications, we would treat gauge symmetry and the weak time
dependence of the Hubble parameter more seriously. The 3-pt correlator of
gauge-invariant variables would then be finite and slow-roll suppressed for
most single-field inflation models.

##  5 Conclusions and discussions

We have introduced a systematic framework for describing a quantum-field
system in an effective way, without invoking an operator-Hilbert space
language. The effective description, originally developed in [1] for quantum
mechanical systems, treats fluctuations $\Delta$ of canonical pairs as
messengers of quantum effects. Via effective Poisson brackets (2.1), one
obtains an effective Hamiltonian that governs the evolution of fluctuations.
This provides us with a time-dependent description of how quantum fluctuations
backreact on the classical part of the system. At second order in
fluctuations, minimizing the effective Hamiltonian while respecting the
uncertainty principle leads to the Coleman-Weinberg effective potential, which
can be understood as a low-energy limit of the $H_{\Delta}$ that we
constructed. We have shown that this way of obtaining the low-energy effective
potential works for both quantum mechanics and quantum fields. In particular,
the saturation of minimal uncertainty becomes a result rather than an
assumption when one wishes to find the low-energy effective potential.

The novel part of our work lies in showing how one could go beyond the low-
energy effective potential for interacting field systems by allowing
fluctuations to evolve freely instead of taking fixed values that minimize
energy. At first glance, a field system would give rise to infinitely many
equations of motion for the fluctuations. By arguing from the perspective of
minimal backreaction, we have motivated the use of translational and
rotational symmetries and shown how to make finite the number of equations.
Physically, we have thus shown that minimal backreaction helps maintain
spatial symmetries, which decouple the evolution of fluctuations from both
themselves and the inhomogeneous part of the classical background.

In flat space, one could even find analytical solutions by hand following the
strategy outlined in Section 3. The key elements used were: (1) perturbation
theory (in interaction strength); (2) a set of minimal-energy initial
conditions found using the uncertainty principle; (3) an initial Gaussian
hierarchy where Wick’s theorem applies. These three elements are also manifest
in the core techniques of textbook quantum field theory — perturbation around
a Gaussian system and the assumption of initial “free” vacuum, which together
imply Wick’s theorem. In our framework, the three key elements are enough to
completely fix the system of equations, allowing us to solve for the
fluctuations by hand. We have demonstrated this with a $\varphi^{3}$-model.
But the strategy also applies to systems with more generic interactions, even
on curved spacetimes.

In particular, we have shown that for a $\varphi^{3}$-model on a de Sitter
background our result for the 3-pt correlator (or bispectrum) agrees with the
“in-in” result in the superhorizon limit. This gives us confidence to apply it
to more realistic inflationary systems, with more complicated interactions and
additional subtleties arising from gauge invariance. The systematic
application of our method to the calculation of non-Gaussianities will be left
for future work. We end by mentioning some opportunities and challenges in
this direction.

Backreaction and stability of the background. Our framework naturally
incorporates how time-dependent quantum fluctuations backreact on classical
field variables. A preliminary investigation in this direction, using toy
models and an approximate canonical map for second-order fluctuations, was
carried out in [16]. In our work, we have extended the investigation to
higher-order fluctuations for interacting fields on static and expanding
backgrounds. We have shown that backreaction is minimized when spatially
symmetric initial conditions are used. Nevertheless, the stability of spatial
symmetries to quantum backreaction remains to be explored.

For inflationary cosmology, one could envision a scenario where a small
deviation from initial homogeneity and isotropy causes the inhomogeneous
background modes to couple to quantum fluctuations. The growth of these modes
could further enhance the coupling and consequently also enhance the
backreaction, causing more inhomogeneity growth. A positive feedback loop
would then ensue to produce too much structure when inflation ends. This
scenario is of course incompatible with the observation of a Gaussian
primordial power spectrum. Therefore it did not occur in our universe. One
would like to know why. Does its absence impose constraints on the allowed
inflaton models or on initial quantum states? Is there some other unknown
mechanism that suppresses the coupling between quantum fluctuations and the
classical inhomogeneous background, perhaps related to the quantum-to-
classical transition? The systematic framework developed in our paper can be
used in search of answers to these questions.

Non-Gaussianities and the choice of canonical variables. As we have seen in
Section 4, if one imposes initial conditions at large but finite times, the
minimal-energy state would be different for canonical variables that seem
equally “natural”. In particular, the initial values for a given fluctuation,
such as $\Delta(ff)=a^{2}\Delta(\delta\varphi\delta\varphi)$, would be
different. In this case, different choices of canonical variables, as pointed
out in [22], may give rise to different predictions. This is true even when
the canonical variables are related by only linear canonical transformations.
Therefore, we expect the issue to be worse when one considers non-
Gaussianities. The cubic interaction terms will open up the possibility of
choices of canonical variables that are non-linearly related, as can be the
case when there are “velocity-type” interactions, such as
$\delta\varphi^{\prime}(\partial_{\mu}\delta\varphi\partial^{\mu}\delta\varphi)$
interaction terms that can appear in the $P(X,\varphi)$ theories [37]. When
deriving the form of cubic interactions, the terms ignored by integration-by-
parts may also complicate our choice of variables.

Our effective framework offers a way of understanding the difference arising
from different variable choices. In the case of quadratic systems, different
choices of variables give rise to different effective Hamiltonians. The
uncertainty principle then selects out different minimal-energy configurations
for the fluctuations, which eventually grow and classicalize to seed
primordial perturbations. On the other hand, when non-Gaussian interactions
are present, different choices of canonical variables can define different
notions of a “free” Hamiltonian, around which one applies perturbation theory.
What is considered to be “free” in one Hamiltonian, viz those composed of
quadratic terms, might contain cubic interactions in a different Hamiltonian.
This can happen when the two Hamiltonians are related by non-linear canonical
transformation. The ambiguity existing in the notion of a “free” Hamiltonian
may be mitigated by assigning initial values based on a minimization of the
full effective Hamiltonian instead of just the free one.

Minimal-energy state and the uncertainty principle. If one wishes to minimize
the full effective Hamiltonian, which contains cubic fluctuations or higher,
one requires the uncertainty principle of higher-order fluctuations. These
will be inequalities similar to the celebrated Heisenberg uncertainty
principle for position and momentum, but instead contain $\Delta^{(n)}$ with
$n\geq 3$. An interesting thing is: If one again uses the Cauchy-Schwarz
inequality, the cubic moments would be on the RHS of the analogue of (3.17).
That is, cubic fluctuations would be bounded from above by products of quartic
and quadratic fluctuations, as opposed to having a lower bound. This conforms
with intuition from classical analysis, where systems with only
$\varphi^{3}$-interaction has no global minima, unless one includes a quartic
interaction. If no other inequalities are constraining cubic fluctuations, it
would seem to suggest that one should consider up to at least a fourth order
in fluctuations if one wishes to find a minimal-energy state via the
uncertainty principle. Fortunately, the inequalities constraining cubic and
quartic fluctuations only need to be derived once because they are model-
independent.

Quantum corrections to the power spectrum. Lastly, we wish to comment on
quantum corrections to second-order fluctuations, which in our context are
defined as correction terms that are, for example, $O(\mu^{2})$ or higher in
(3.13). Taking (3.13) for instance, the leading order correction to the 2-pt
correlators comes from the cubic fluctuations. To obtain the $O(\mu^{2})$
correction, one should solve the “tree-level” solution of cubic fluctuations,
namely to $O(\mu)$. For higher order corrections, say to $O(\mu^{4})$, one
ought to obtain the $O(\mu)$ solution for the quintic fluctuations, giving us
$O(\mu^{2})$ solutions of the quartic fluctuations, further giving us the
$O(\mu^{3})$ solutions of the cubic fluctuations and hence finally giving us
the $O(\mu^{4})$ solutions to the quadratic fluctuations. For arbitrarily
higher order, one simply iterates this process more, starting with the tree-
level solution of the highest required order, which is also the easiest one to
solve. See Section 3.4.

While the strategy is clear, as we substitute higher-order fluctuations into
lower order ones, we encounter our version of “loop” integrals, defined as
integrals over internal momenta that cannot be eliminated with delta-
functions. One would need to resort to certain regularization scheme and
renormalize the theory. This is difficult even at $O(\mu^{2})$ order for a
$\varphi^{3}$-model on Minkowski, as shown in Section 3.4. For generic orders
in $\mu$, we expect two main difficulties. First, different orders of
fluctuations would be mixed in the equations of motion. Unless one can find a
way to decouple equations of motion for different order of fluctuations, we
might not be able to obtain analytical expressions for the integrand inside a
loop-integral. Second, even if one obtains an analytical expression for the
integrand, the loop-integral is still not guaranteed to be expressible using
simple functions. Renormalization would remain difficult and deserve future
study.

## Acknowledgements

DD is supported by China Postdoctoral Science Foundation (Certificate No.
2023M730704). YW is supported by the General Program of Science and Technology
of Shanghai No. 21ZR1406700, and Shanghai Municipal Science and Technology