# Quantum Statistical Field Theory in Gravitation and Cosmology

###### Abstract

We describe how the concepts of quantum open systems and the methods of closed-time-path (CTP) effective action and influence functional (IF) can be usefully applied to the analysis of statistical mechanical problems involving quantum fields in gravitation and cosmology. In the first lecture we discuss in general terms the relevance of open system concepts in the description of a variety of physical processes, and outline the basics of the CTP and IF formalisms. In the second lecture we illustrate the IF method with a model of two interacting quantum fields, deriving the influence action via a perturbative expansion involving the closed-time-path Green functions. We show how noise of quantum fields can be defined and derive a general fluctuation- dissipation relation for quantum fields. In the third lecture we discuss the problem of backreaction in semiclassical gravity with the example of a scalar field in a Bianchi Type-I universe. We show that the CTP effective action not only yields a real and causal equation of motion with a dissipative term depicting the effect of particle creation, as was found earlier, it also contains a noise term measuring the fluctuations in particle number and governing the metric fluctuations. The particle creation-backreaction problem can be understood as a manifestation of a fluctuation-dissipation relation for quantum fields in dynamic spacetimes, generalizing Sciama’s observation for black hole Hawking radiation. A more complete description of semiclassical gravity is given by way of an Einstein-Langevin equation, the conventional theory based on the expectation value of the energy momentum tensor being its mean field approximation. We also mention related problems of interest, including the dissipative nature of effective field theory and the theory of fluctuations, instability and phase transition as applied to the problem of transition from general relativity to quantum gravity.

Lectures given at the Canadian Summer School for Theoretical Physics and

the Third International Workshop on Thermal Field Theories and Applications

Banff, Canada, Alberta, August 15-28, 1993

(umdpp 94-45)

## Outline and Summary

The purpose of these lectures is to develop a quantum field theory suitable for the description of non-equilibrium quantum systems for problems in gravitation and cosmology. This is not a review, but only an exposition and summary of our recent work and some thoughts for future development.

### 0.1 Aim of Program

Schwinger-Keldysh On the theoretical side,
I will summarize the relation of the closed-time-path (CTP, or
Schwinger-Keldysh)
[1, 2, 3, 4, 5, 6]
functional formalism with the influence-functional (IF, or Feyman-Vernon)
[7, 8, 9, 10, 11, 12] method.
The CTP method is now quite well received and applied to particle physics
problems
involving thermal fields, as is witnessed in this meeting. After its inception
in the 60’s it was used in selected condensed matter problems.
In the first paper Esteban Calzetta and I
wrote on the CTP formalism [4] we adapted it to quantum fields in
curved spacetime [13] and applied it to the backreaction problem of
particle production in the early universe [14, 15] as the highlight
of
semiclassical gravity theory [16].
As we expected, to get a real and causal effective equation of motion
for these systems,
this method was not only neat, but absolutely indispensible. Only after going
through our own initiation process
^{1}^{1}1I became aware of this subject rather late, after hearing a talk at
the 1980 Guangzhou Particle Physics Meeting by the authors of the well-known
review [2].
I knew even then that this is the method which can render the complex
geometry obtained from the Schwinger-DeWitt (‘in-out’) effective action
formalism [17] real. It is also most suitable for treating
quantum statistical processes in cosmology, which is prevalently
non-equibrium in nature. But I wanted to see the whole picture for myself so
I started from ground level by first learning (and ended up constructing a
good part of) finite temperature quantum field theory in curved, and
especially,
dynamic spacetimes [18]. Such was my initiation to thermal field theory.
had we learned from Jordan’s paper that Bryce DeWitt had
been advocating this method for some time [3]. (We should have
guessed!) Our second paper was the development of a quantum kinetic theory
(in flat spacetime) with the CTP formalism by combining the multiple-source
(Cornwall-Jackiw-Tomboulis [19]) formalism for treating correlation
functions and the Wigner function [20] techniques to derive
the kinetic equations [5]. That was our first venture into
nonequilibrium statistical
field theory. An extension to curved spacetime was done with Habib [21].
Paz later applied the CTP formalism to cosmological backreaction problem with
finite temperature fields, including a nice calculation of the
viscosity function for reheating in the inflationary cosmology [22].

Feynman-Vernon
At about the same time Schwinger and Keldysh [1] wrote their
seminal papers which established the CTP formalism, Feynman and Vernon
[7] wrote a very important though lesser known paper which
introduced the influence functional formalism. Both were about harmonic
oscillators and Brownian motion.
The influence functional formalism was popularized by the work of
Calderia and Leggett in 1983 [8].
My initiation, or rather, reintroduction to this method
^{2}^{2}2 Even though the IF formalism was
discussed in detail in Feynman and Hibbs’ classic 1965 book,
I did not pay attention to it in my first reading of that book, because
I read it only for understanding the path integral method and never bothered
with
the statistical mechanics part. I think I am not alone in this kind of
regreted oversight and missed opportunity.
came from my insistence to understand the true statistical
properties of quantum fields in general [23] and to uncover
the missing noise term in the cosmological backreaction problem in particular.
Dissipation in the dynamics of a system arising from particle creation of
a quantum field has been shown by many researchers before [14, 15],
the last definitive result was obtained
from the calculation via the CTP formalism [4]. But the statistical
meaning of dissipation in these contexts was not clear and
the expected accompanying effects of fluctuation and noise were apparently
nowhere to be seen in the CTP formalism. I summarized these inquiries
in the First Thermal Field Workshop and gave my tentative replies to them
in the form of three conjectures [23]:

1) That colored noise associated with quantum field fluctuations is generally
expected in gravitation and cosmology;

2) That the backreaction of particle creation in a dynamical spacetime
can be viewed as the manifestation of a generalized fluctuation-dissipation
relation; and

3) That all effective field theories, including semiclassical gravity or
even quantum gravity (to the extent that it could be viewed as an effective
field theory), are intrinsically dissipative in nature.

These conjectures are logical consequences of the open system concept
[24]
applied to quantum fields in curved spacetime. Asking such questions
also led me to a closer examination of the relationship between the influence
functional and the CTP formalisms.
I will discuss these issues and methodology in what follows.
The search and discovery which led to a formulation of
quantum field theory in curved spacetime [13] in terms of
non-equilibrium statistical mechanics are also described in
[25, 26].

On the physical side, the range of problems in gravitation and cosmology which statistical field theory is particularly useful for is described in two recent reviews: Ref. [26] discusses quantum statistical processes in the early universe, with special emphasis on the entropy of quantum fields and spacetimes; Ref. [27] describes the origin and nature of noise in quantum fields, and how it is related to particle creation in black holes and the early unverse.

### 0.2 Topics of my Three Lectures at the Summer School

The titles and contents of my three lectures presented in this summer school are:

1. Quantum Decoherence and Uncertainty: The influence functional formalism and application to foundational problems in quantum mechanics and statistical mechanics.

2. Quantum Origin of Noise and Fluctuations: Application to galaxy formation problems in inflationary cosmology.

3. Dissipation, Fluctuation and Backreaction: Application to problems in semiclassical gravity and quantum cosmology.

Let me summarize the main points of these lectures.

In Lecture 1

a) Decoherence
We discussed the issue of decoherence in the problem of quantum to
classical transition using the quantum master equation derived via the
influence functional formalism. It is based on the work of Hu, Paz and Zhang
on quantum Brownian motion (QBM) in a general environment [10]. For the
bigger picture, see the recent reviews [28] on the consistent or
decoherent history formulation [29] and the environment-induced
decoherence program [30] of quantum mechanics.

b) Uncertainty
We derived the uncertainty principle at finite temperature,
pointing out that under general conditions the decoherence time
is the same as the time when thermal fluctuations overtake quantum
fluctuations.
This is based on the work of Hu and Zhang [31].
A review of the issue of decoherence and
uncertainty in terms of quantum and thermal noise is given in my talk at
Third Drexel Workshop on Quantum Dynamics of Chaotic Systems
[32]. See also the recent paper on this topic via
information theory by Anderson and Halliwell [33].

In Lecture 2:

a) Quantum Noise We showed how noise can be identified from the
imaginary part of the influence action, and how its characteristics depends on
the coupling of the system with the environment, the spectral density of
the environment and other factors. We studied a model of the Brownian particle
coupled
nonlinearly with a bath of harmonic oscillators. We deduced the noise
autocorrelation functions and a generalized fluctuation-dissipation relation
for colored and multiplicative noise. The details are contained in the second
QBM paper of Hu, Paz and Zhang [11].

b) Quantum Fluctuations We suggested a proper way to deduce noise from
quantum fluctuations of the inflaton field in the quantum theory of
galaxy formation, and showed why the conventional way of simply treating the
short wavelength sector of a free field as the source of a classical
Langevin equation is flawed [34]. We also examined
the preconditions for a classical stochastic
equation to emerge from the wave equations of quantum fields, e.g., that the
long wave-length sector would have to decohere for this to be possible. We
illustrated how to tackle these two basic issues with two coupled
fields. In the course of this exposition we also
pointed out how the closed-time-path Green functions enter in the
perturbative calculation of the influence functional.
The details of this problem worked out by Hu, Paz and Zhang are recorded
in my talk at the Chateau du Pont O’ye Meeting on the Origin of Structure in
the Universe [12].

In Lecture 3

a) Backreaction and Dissipation
For semiclassical gravity we discussed the backreaction of particle
production on the dynamics of a spacetime, using the anisotropy dissipation
in a Bianchi Type-I universe as example. This was done primarily for the
purpose
of illustrating the use of the closed-time-path method and to point out
how the statistical mechanical meaning of dissipation can indeed be sought with
the quantum open system concept. It was in asking these questions which
led us to the discovery of the connection between the CTP and IF formalisms.
First derived by Calzetta and Hu [4, 6], the result was
reviewed in my talk at the First Thermal Field Workshop [23].

b) Minisuperspace as Open System
For quantum cosmology we discussed the question of the validity of
the minisuperspace approximation [35],
wherein only the homogeneous cosmologies
are quantized and the inhomogeneous cosmologies ignored [36].
We used an interacting quantum field model and calculated the effect of
the inhomogeneous modes on the homogeneous mode via the CTP effective action.
This effect manifests in the effective equation of motion for the system as a
dissipative term. For quantum cosmology, this backreaction
turns the Wheeler-De Witt (WDW) [37]
equation for the full superspace into an effective
WDW equation for the minisuperspace with dissipation. This is detailed in
the work by Sinha and Hu [38, 39, 25]. I also discussed
the possibility of defining a gravitational entropy based on these concepts
at the Waseda Conference on Quantum Physics and the Universe [26].

### 0.3 Recent Results

In the time between the conference and the submission of this report some new results were obtained by Calzetta, Matacz, Sinha and I. They fall under the subject matters of Lectures 2 and 3, which I add here as topics 2c and 3c:

2c) Thermal Radiance As a model for an open system in the environment of a quantum scalar field, we studied a Brownian particle interacting with a bath of parametric oscillators. The single oscillator can be a particle detector, a mode for the field (e.g., the homogeneous inflaton mode), or the scale factor of the universe. We derived an expression for the influence functional in terms of the Bogolubov coefficients relating the amplitudes of the bath modes under parametric amplification, which in the second quantized sense represents particle creation in a dynamical spacetime. With this result one can discuss the statistical mechanical meaning and origin of thermal radiance observed in an accelerated detector, a black hole and the de Sitter universe, known as the Unruh and the Hawking effects [40, 41, 42], in terms of the excitation of vacuum fluctuations or quantum noise in the quantum field. This study which sets the stage for the influence functional formalism approach to quantum field theory in curved spacetime is detailed in Hu and Matacz [43]. The nature and effects of quantum noise in gravitation and cosmology is discussed in my recent talk at the Fluctuation and Order Workshop at Los Alamos [27].

3c) Einstein-Langevin Equation and Fluctuation-Dissipation Relation Calzetta and I continued our earlier investigation of the backreaction problem in semiclassical gravity to show how noise and fluctuations can also be obtained with the CTP formalism in addition to dissipation and decoherence [44]. We derived an expression for the CTP effective action in terms of the Bogolubov coefficients and showed how noise is related to the fluctuations in particle number. Matacz and I used a cumulant expansion on the influence functional [45] to extract the noise associated with the matter field and to derive an Einstein-Langevin equation as the equation of motion for the the dynamics of spacetime as an open system. Through these work, we have extended the old framework of semiclassical gravity based on the mean field theory of the Einstein equation with a source driven by the expectation value of the energy momentum tensor, to that based on a Langevin equation which describes also the fluctuations of matter fields and spacetime. The backreaction effect of matter fields on the dynamics of spacetime exemplified by the anisotropy damping due to particle creation can be viewed as a manifestation of the fluctuation-dissipation relation [46, 47, 48]. This conjecture of mine [23] inspired by Sciama’s observation on Hawking radiation in black holes was proven by Sinha and I recently [49]. We show this result in Part III below.

### 0.4 Content of this Report

Instead of repeating or rewording the already published work,
I prefer to select a few topics in quantum field theory in flat
and curved spacetimes from our current research work which
carry some representative meaning and discuss how this new approach via
quantum statistical field theory can provide new insights and directions.
The following topics are covered in this report:

Part I: Concepts and Methods in Statistical Field Theories

1) Quantum Open Systems: Coarse-Graining and Backreaction

2) The Influence Functional Approach to Statistical Field Theory

3) The Closed-Time-Path Functional Formalism in Quantum FIeld Theory

4) The Consistent Histories Formulation of Quantum Mechanics

Part II: Noise and Fluctuations in Quantum Fields

1) Perturbation Theory

2) Noise and Fluctuations

3) Langevin Equation and Fluctuation-Dissipation Relation

4) Related Problems

Part III: Einstein-Langevin Equation in Semiclassical Gravity

1) Backreaction and Dissipation

2) Noise

3) Einstein-Langevin Equation and Fluctuation-Dissipation Relation

4) Related Problems

These parts are based on ongoing (unpublished) work with Esteban Calzetta [44] (Part I), Juan Pablo Paz and Yuhong Zhang [50, 51] (Part II) and Sukanya Sinha [49] (Part III). The reader is referred to these work for further details.

Part I

## 1 Concepts and Methods in Statistical Field Theories

We first give a physical reason for treating familiar quantum processes with the conceptual framework of open systems. We then give a brief summary of the relation of the closed-time-path, the influence functional and the decoherence functional formalisms used in the path-integral approach to quantum and statistical mechanics and field theory.

### 1.1 Quantum Open Systems: Coarse-Graining and Backreaction

In physical systems containing many degrees of freedom one often attempts to select out a small set of variables to render the problem technically tractable while preserving its essence. Familiar examples are abound, e.g., thermodynamics from statistical mechanics, hydrodynamic limit of kinetic theory, collective dynamics in nuclear physics [52]. When one starts from the microscopic picture, one distinguishes the variables which depict the system of interest from those which can affect the system but whose detail is otherwise of lesser interest or no importance. To make a sensible distinction involves recognizing and devising a set of criteria to separate the relevant from the irrelavant variables. This procedure is simplified when the two sets of variables possess very different characteristic time or length or energy scales or interaction strengths. An example is the slow-fast variables separation in the Born-Oppenheimer approximation in molecular physics where the nuclear variables are assumed to enter adiabatically as parameters in the electronic wave function. Similar separation is possible in quantum cosmology between the ‘heavy’ gravitational sector and the ‘light’ matter sector characterized by the Planck mass (see, e.g., [53] and references therein). In statistical physics this separation can be made formally with projection operator techniques [54]. This usually results in a nonlinear integro-differential equation for the relevant variables, which contains the causal and correlational information from their interaction with the irrelavant variables.

Apart from finding some way of separating the overall closed system (the “universe”) into a ‘relevant’ part of primary interest (the open system) and an ‘irrelevant’ part of secondary interest (the environment) in order to render calculations possible, one also needs to devise some averaging scheme to reduce and reconstitute the detailed information of the environment such that its effect on the system can be traced to some simple macroscopic functions. This involves introducing certain coarse-graining measures. It is by the imposition of such measures that an environment is turned into a bath, and certain macroscopic characteristics such as temperature, chemical potential can be devised for its simple description. A coarse-grained description of the effect of the environment on the system (in terms of, say, the transport coefficients in hydrodynamics) is qualitatively very different from the detailed description (in terms of the underlying microscopic dynamics). A familiar example in many body theory used for simplifying the effect of the environment is the independent particle model, where, say, a nucleon is assumed to be affected by all other nucleons only via an averaged two-body potential. Mean field approximation in quantum field theory shares the same spirit, where the effect of quantum fluctuations of fields is described in terms of a renormalization of the basic physical parameters of the system.

How the environment affects the open system is called the backreaction effect. By referring to an effect as backreaction, it is implicitly assumed that a system of interest is preferentially identified, that one cares much less about the details of the other sector (the ‘irrelevant’ variables, the ‘environment’, etc). The backreaction can be significant, but should not be too overpowering, so as to invalidate the separation scheme. To what extent one views the interaction of the two sectors as interaction or as backreaction is reflexive of the degree one decides to keep or discard the information in one versus the other. It also depends on their interaction strength. The behavior of the complete (closed) system requires a self-consistent solution of equations governing both sectors. Examples are the Hartree-Fock approximation in atomic physics or nuclear physics, where the system could be described by the wave function of the electrons obeying the Schrödinger equations with a potential determined by the charge density of the electrons themselves via the Poisson equation [55]. In a cosmological backreaction problem, the system is a classical spacetime, whose dynamics is determined by Einstein’s equations with sources given by particles produced by the vacuum excited by the dynamics of spacetime and depicted by wave equations in curved spacetime [13]. Much of the physics of open systems is concerned with the practicality and validity of these procedures. They are, 1) the identification and separation of the physically interesting variables which make up the open system – oftentimes one needs to come up with the appropriate collective variables, 2) the ‘averaging’ away of the environment or irrelevant variables – how different coarse-graining measures affect the final result is important, and 3) the evaluation of the averaged effect of the environment on the system of interest. We will refer to these procedures as separation, coarse-graining and backreaction for short.

These considerations surrounding an open system are common and essential
not only to well-posed and well-studied examples of many-body systems
like molecular, nuclear and condensed-matter physics, they also
bear on some basic
issues at the foundations of quantum mechanics and statistical mechanics,
such as decoherence and the existence of classical limit [56],
the emergence of time and spacetime from quantum gravity [57].
Our recent work is concerned with the general problem of how these
considerations can be applied to both classical and quantum
gravitational systems. This includes problems
in Einstein’s classical theory of general relativity,
quantum field theory in curved
spacetime (semi-classical gravity) and quantum gravity, as well as their
related problems in relativistic cosmology and quantum cosmology.
Gravitational systems are of interest not only because of the special
characteristics of gravity, but also because
they dwell on extreme conditions such as that which exist in the black holes
and the early universe, often pushing the physical laws to their limits.
Inquiries on the quantum and statistical properties of spacetime and fields
raise many new issues in fundamental physics,
whose resolution can provide interesting new insights into the nature of
physical laws and the structure of the universe [58, 59, 60].

* * *

In terms of methodology, two formalisms have been used effectively for the description of quantum open systems, especially pertaining to the coarse-graining and backreaction problems: the closed time path effective action (CTP, or Schwinger-Keldysh) formalism [1, 2, 3, 4, 5, 6] for obtaining a real and causal equation of motion with dissipation; and the influence functional (IF, or Feynman-Vernon [7, 8, 9, 10, 11, 12]) formalism for identifying the noise and fluctuations in the environment. We give here a brief description of these formalisms and their interconnection. We also sketch the decoherent history formulation of quantum mechanics [29] as we will use this conceptual framework to apply the IF and CTP formalisms to the analysis of semiclassical gravity theory. The following summary is adapted from [44].

### 1.2 The Influence Functional Approach to Statistical Field Theory

The IF approach [7] is designed to deal with a situation in which the system described, say, by the fields is interacting with an environment , described by the fields. The full quantum system is described by a density matrix . If we are only interested in the state of the system as influenced by the overall effect, but not the precise state, of the environment, then the reduced density matrix would provide the relevant information. (The subscript stands for reduced.) It is propagated in time from by the propagator :

Assuming that the action of the coupled system decomposes as , and that the initial density matrix factorizes (i.e., takes the tensor product form), , the propagator for the reduced density matrix is given by

where

(1.1) |

is the full effective action and is the influence action. (They are called and in [10] ) The influence functional is defined as

(1.2) |

is typically
complex; its real part , containing the dissipation kernel
, contributes to the renormalization of , and
yields the dissipative terms in the effective equations of motion.
The imaginary part ,
containing the noise kernel , provides the information about
the fluctuations induced on the system through its coupling to the environment.
Since
the connection between these kernels and their effect on the physical processes
of dissipation and fluctuation has been discussed at lenght elsewhere
(cfr. Ref. [10]), we shall limit
ourselves here only to a schematic summary. ^{3}^{3}3This
simplified schematic discussion is really just for the illustration of main
ideas, not for precision or completeness. The reader is referred to
[10, 11, 12, 27, 43] for details on the discussion of the
process of decoherence in quantum to classical transition,
the origin and nature of quantum noise, the fluctuation-dissipation relation
and the explicit derivations of the master, Fokker-Planck and Langevin
equations
for the models of a Brownian particle in a general environment and interacting
quantum fields in cosmological spacetimes.

The main features of the influence action follow from the elementary properties and , which can be deduced from its definition, and derived in the final analysis from the unitarity of the underlying quantum theory of the closed system. If we decompose in its real and imaginary parts, , then , , and . Keeping only quadratic terms, we may write

(1.3) |

where and stand for the real dissipation and noise kernels respectively (, , in the notations of [10]). It is convenient to express as , and rewrite

(1.4) |

The physical meaning of the kernel may be elucidated by deriving the mean field equation of motion for the mean value of the system variable . It is

(1.5) |

The term containing represents the backreaction of the environment on the system. It causes the dissipation of energy from the system by an amount (integrated over the whole history of the system)

(1.6) |

Thus we see that the even part of the kernel is associated with dissipation, while the odd part can be assimilated to a nondissipative environment-induced change in the system dynamics. In quantum field-theoretic applications, the odd part of will contain formally infinite terms which can be absorbed in the classical action for the system via standard renormalization procedures [13]. For simplicity, we shall assume that only the even part of is left after renormalization has been carried out.

In general, the and kernels are nonlocal; however, their main features are manifest already under the local approximation , . The influence action then takes the form

(1.7) |

Assuming an action functional of the simple form , it is straightforward to derive the master equation for the reduced density matrix [7, 8]

(1.8) |

(1.9) |

where . The master equation (1.8) implies (to lowest order in a Kramers-Moyal expansion) the Fokker-Planck equation [62, 63]

(1.10) |

(where ). From this equation one can see clearly the stochasticity in the semiclassical dynamics. The Fokker-Planck equation admits the equilibrium solution

(1.11) |

which depicts a finite-temperature stationary state for the system. In particular, a fluctuation-dissipation theorem can be easily derived. If the environment acts as a heat bath, then , and this reduces to the Einstein-Kubo formula for the dispersion coefficient.

### 1.3 The Closed-Time-Path Functional Formalism in Quantum Field Theory

In the CTP approach, our goal is not to follow the dynamics of the full density matrix, or even the system part, but only the expectation values of the fields as they unfold in time. This evolution is governed by a real and causal equation of motion, which is obtained from the CTP effective action by a variational principle.

Let be the quantum fields in the theory, and their expectation values for any given initial state. Consider pairs of histories defined on all spacetime, with the property that for a given very large time (in practice, ). Assume for simplicity (more general choices are also possible [5]) that the fields were originally in their vacuum state and that the bath fields are linearly coupled to external sources . A closed time path is the path which runs from the initial field configuration at time to the final field configuration at time through the positive time branch linked to , then returns to the field configuration at time through the negative time branch linked to . The CTP generating functional for the field over the vacuum state is given by

(1.12) | |||||

(1.13) |

where denotes temporal order, denotes anti-temporal order and is the field configuration corresponding to the vacuum state of the field. Observe that the generating functional is totally defined once the in state is chosen and that whenever . Now introduce the path integral representation

(1.14) | |||||

(1.15) |

The expectation values can be obtained as

(1.16) |

The physically relevant situation under consideration corresponds to setting .

The full CTP effective action is just the Legendre transform of

(1.17) |

where now the sources are thought of as functionals of the background fields . In particular, the equations of motion are the inverses of Eqs. (1.16)

(1.18) |

The physical situations correspond to solutions of the homogeneous equations at . These equations are real and causal. Moreover, , and . As the generating functional itself, the CTP effective action is totally defined once the initial quantum state is given.

To apply this formalism to the situation above, we should substitute the field by the pair . When the physical situation requires treating the and fields asymmetrically, as is the case when, say, only the system field is relevant, we do not couple the field to an external source. (In a perturbative evaluation of the CTP generating functional, this means discarding all graphs with fields on some external leg.) Comparing the path integral expression for the generating functional with the IF approach described earlier (1.1), we find

(1.19) |

Conversely, we may describe the full IF effective action as the full CTP effective action

(1.20) |

for the quantum fields interacting with external c-number fields specialized to the expectation values of its arguments.

In the semiclassical approximation, one can neglect Feynman graphs containing closed field loops corresponding to quantum effects of the fields. Then the path integral and the Legendre transformation may be computed explicitly, yielding

(1.21) |

This equation shows the equivalence between the (full) CTP effective action and the (full) IF effective action, or and . From this one may derive the semiclassical equations of motion for the expectation values of the field. We see that the noise kernel does not contribute to these equations, because, it being even under the exchange of and , its variation vanishes at the coincidence point. However, as we shall see below, and is also clear from the master equation point of view [10], the noise kernel determines the dynamics governing the deviations from the expectation value.

### 1.4 The Consistent Histories Formulation of Quantum Mechanics

Let us now relate these concepts and techniques in statistical field theory to the quantum to classical transition problem via the consistent histories formulation of quantum mechanics [29, 56, 64].

In the consistent or decoherent histories approach, the complete description of a coupled system is given in terms of fine-grained histories . These histories are quantum in nature, i. e. it is possible in principle to observe interference effects between different generic histories. A classical description is acceptable only at the level of coarse-grained histories, and to the extent that interference effects between these histories become unobservable. Let us adopt the simple coarse-graining procedure of leaving the field unspecified. Then each coarse-grained history is labelled by a possible evolution of the field, and the interference effects between histories are measured by the decoherence functional (DF)

(1.22) |

which is the fundamental object of the theory. (For a more formal definition see [29].) The coarse-grained histories can be described classically if and only if the decoherence functional is approximately diagonal, that is, whenever . The conditions leading to this in quantum mechanics is the focus of many current studies, to which we refer the readers for the details. For quantum cosmology the issue is complicated by the problem of time, and there even the definition of the decoherence functional can be ambiguous [57]. In the problem of transition from quantum cosmology to semiclassical gravity, a WKB time is usually assumed. Using the minisuperspace quantum cosmology of Bianchi Type-I universe [35] as example, Paz and Sinha [64] showed that an influence functional appears naturally from a reduced density matrix by tracing out the matter fields. They discussed the decoherence between WKB branches of the wave function and tried to relate it to the notion of decoherence between spacetime histories. In our discussion of semiclassical gravity in Part III, we will assume that this essential task can be accomplished in some satisfactory way, from which one can write down the decoherent functional [64, 44].

(1.23) |

Already at this formal level notice that decoherence can occur only when the noise kernel is nonzero, which signals the presence of spontaneous fluctuations in the system.

For an observer confined (by necessity or by choice) to the level of coarse grained descriptions, dynamical evolution must be described in terms of mutually exclusive histories, all interference effects having been suppressed below the accuracy of his observation devices. For example, if he chooses to describe the evolution of the system in terms of its Wigner Function , he will now interpret it as an actual ensemble average, describing the joint evolution of the bundle of coarse-grained histories. Correspondingly, he will regard Eq. (1.10) as a classical Fokker-Planck equation. Now the classical random process described by Eq. (1.10) is not deterministic; rather, it describes the evolution of an ensemble of particles whose individual orbits obey the Langevin-type equations

(1.24) |

where represents a noise term with autocorrelation . (The ordinarily assummed gaussian and white nature of the noise follows only from a quadratic and local noise kernel, which describes rather special cases in cosmological situations, see [12, 27]). Thus, the observer confined to a coarse-grained history will conclude that semiclassical evolution is stochastic. Note that the statistical properties of this random evolution are totally determined by the decoherence functional; no ad hoc assumptions on the behavior of quantum fluctuations are necessary.

To see how noise arises from the IF formalism, one can rewrite the part in the influence action containing the noise kernel as

(1.25) |

Therefore the action of the environment on the system may be described by adding the external source term to the system action , and averaging over external sources with the proper weight [50, 11, 43]. Variation of this effective action directly yields the Langevin equations (1.24). This is how noise can be understood as a stochastic force from the environment acting on the system.

In Part II we will apply these formalisms to discuss noise and fluctuations from interacting quantum fields in flat space. Then in Part III we will discuss the problem of dissipation and fluctuation in a curved spacetime setting.

Part II

## 2 Noise and Fluctuations in Quantum Fields

Consider two independent self-interacting scalar fields in Minkowsky spacetime: depicting the system, and depicting the bath. The classical action for these two fields are given respectively by:

(2.1) |

(2.2) |

where are the self-interaction potentials. For a interaction,

(2.3) |

and similarly for . Here, and are the bare masses and and are the bare self-coupling constants for the and fields respectively. In (2.2) we have written in terms of a free part and an interacting part which contains . Assume these two scalar fields interact via a polynomial coupling of the form

(2.4) |

where is the vertex function with coupling constant , which we assume to be small and of the same order as , .

The total classical action of the combined system is

(2.5) |

The total density matrix of the combined system plus bath field is defined by

(2.6) |

where and are the eigenstates of the field operators and , namely,

(2.7) |

Since we are primarily interested in the behavior of the system, and of the environment only to the extent in how it influences the system, the object of interest is the reduced density matrix defined by

(2.8) |

For technical convenience, let us assume that the total density matrix at an initial time is factorized, i.e., that the system and bath are statistically independent,

(2.9) |

where and are the initial density matrix operator of the and field respectively, the former being equal to the reduced density matrix at by this assumption. The reduced density matrix of the system field evolves in time following

(2.10) |

where is the evolutionary operator of the reduced density matrix:

(2.11) |

where

(2.12) |

is the full IF effective action and is the influence action. The influence functional is defined as

(2.13) |

which summarizes the averaged effect of the bath on the system. We have seen from Sec. 1.3 that for a zero-temperature bath (i.e., the environment field is in a vacuum state, ), the influence functional is formally equivalent to the CTP vacuum generating functional, and the influence action in (2.12) is the usual CTP vacuum effective action.

### 2.1 Perturbation Theory

The above is the formal framework we shall work with. Let us now develop a perturbation theory for evaluating the influence action. If and are assumed to be small parameters, the influence functional can be calculated perturbatively by making a power expansion of . Up to second order in , and first order in (one-loop), the influence action is given by

(2.14) |

where the quantum average of a physical variable over the unperturbed action is defined by

(2.15) |

Here, is the influence functional of the free bath field, assuming a linear coupling with external sources and :

(2.16) | |||||

(2.17) |

Let us define the following free propagators of the field

(2.18) |

(2.19) |

(2.20) |

We see that these are just the familiar Feynman, Dyson and positive-frequency Wightman propagators of a free scalar field given respectively by

(2.21) |

(2.22) |

(2.23) |

The perturbation calculation by means of Feynman diagrams for theory in the CTP formalism has been carried out before for quantum fluctuations [4] and for coarsed-grained fields [66, 38]. For bi-quadratic coupling,

For a general polynomial-type coupling with given by (2.4), the renormalized full effective action to second order in is given by [50, 12]

(2.26) |

where the subscript on denotes the order of in the perturbative expansion on the action. Here is the renormalized action of the field, now with physical mass and physical coupling constant , namely,

(2.27) |

For the bi-quadratic interaction case the potential renormalization is

(2.28) |

which is symmetric; and and are real nonlocal kernels

(2.29) |

(2.30) |

(2.31) |

Renormalization of the potential which arises from the contribution of the bath appears only for even couplings. For the case the above result is exact. This is a generalization of the result obtained in [11, 12] where it was shown that the non-local kernel is associated with dissipation or the generalized viscosity function that appears in the corresponding Langevin equation and is associated with the time correlation function of the stochastic noise term. In general is nonlocal, which gives rise to colored noises. Only at high temperatures would the noise kernel become a delta function, which corresponds to a white noise source. Let us elaborate on the meaning of the noise kernel.

### 2.2 Noise and Fluctuations

The real part of the influence functional comes from the imaginary part of the influence action which contains the noise kernel. It is given by

(2.32) |

where is redefined by absorbing the . This term can be rewritten using a functional Gaussian identity [7] to be:

(2.33) |

where

(2.34) |

is the functional distribution of and is a normalization factor given by

(2.35) |

The terms in the effective action describing the coupling between the noise and the system field is then

(2.36) |

In this way the reduced density matrix can be rewritten as

(2.37) |

Therefore we can view as a nonlinear external stochastic force and the reduced density matrix is calculated by taking a stochastic average over the distribution of this source.

Since the expansion of the action is to the quadratic order, the associated noise is Gaussian. It is completely characterized by

(2.38) |

We see that the non-local noise kernel is just the two point auto-correlation function of the external stochastic source (multiplied by ).

In this framework, the expectation value of any functional operator of the field is then given by

(2.39) | |||||

This provides the physical interpretation of as a noise or fluctuation kernel of the quantum field.

### 2.3 Langevin Equation and Fluctuation-Dissipation Relation

We will now derive the semiclassical equation of motion generated by the influence action . Define a “center-of-mass” function and a “relative” function as follows

(2.40) |

The equation of motion for is derived by demanding (cf. [4])

(2.41) |

which gives

(2.42) |

We see that this is in the form of a Langevin equation with a nonlinear stochastic force

(2.43) |

This corresponds to a multiplicative noise arising from a nonlinear field coupling (additive if ). is the renormalized effective Lagrangian of the system action . The nonlocal kernel defined by

(2.44) |

is responsible for nonlocal dissipation. Interaction with the environment field imparts a dissipative force in the effective dynamics of the system field given by

(2.45) |

Only in special cases like a high temperature ohmic environment when this kernel becomes a delta function will the dissipation become local.

In the biquadratic coupling example the corresponding stochastic force is

(2.46) |

The dissipation kernel is

(2.47) |

and the dissipative force is

(2.48) |

As discussed in [11], we can show that a general fluctuation-dissipation relation exists between the (th order) dissipation and the (th order coupling) noise kernels in the form

(2.49) |

where the kernel