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arXiv:hep-ph/9903291 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Stochastic dynamics of correlations in quantum field theory: From the Schwinger-Dyson to Boltzmann-Langevin equation

Esteban Calzetta, B. L. Hu

Published Dec 27, 1999• 98 citations
arXivHTMLPDF

Abstract

The aim of this paper is twofold: to probe the statistical mechanical properties of interacting quantum fields, and to provide a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master $(n=\ensuremath{\infty}\mathrm{PI})$ effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the $n\mathrm{PI}$ effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called slaving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical version of which being the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum field theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.

DOI
10.1103/physrevd.61.025012

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Frontiers of Physics•May 1, 2016•50 citations
The aim of this paper is twofold: to probe the statistical mechanical properties of interacting quantum fields, and to provide a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master (n=\ensuremath∞PI)(n=\ensuremath{\infty}\mathrm{PI})(n=\ensuremath∞PI) effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the nPIn\mathrm{PI}nPI effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called slaving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical version of which being the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum field theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.
The statistical mechanical properties of interacting quantum fields in terms of the dynamics of the correlation functions are investigated. We show how the Dyson - Schwinger equations may be derived from a formal action functional, the n-particle irreducible (nPI,n→∞nPI, n \to \inftynPI,n→∞) or the `master' effective action. It is related to the decoherence functional between histories defined in terms of correlations. Upon truncation of the Dyson - Schwinger hierarchy at a certain order, the master effective action becomes complex, its imaginary part arising from the higher order correlation functions, the fluctuations of which we define as the correlation noises of that order. Decoherence of correlation histories via these noises gives rise to classical stochastic histories %driven by the flucutations of these higher correlation functions. Ordinary quantum field theory corresponds to taking the lowest order functions, usually the mean field and the 2-point functions. As such, our reasoning shows that it is an effective theory which can be intrinsically dissipative. The relation of loop expansion and correlation order as well as the introduction of an arrow of time from the choice of boundary conditions are expounded with regard to the origin of dissipation in quantum fields. Relation with critical phenomena, quantum transport, molecular hydrodynamics and potential applications to quantum gravity, early universe processes and black hole physics are mentioned.
Building upon the work of Hu, Paz, and Zhang [1,2] on open quantum systems we consider the quantum Brownian motion (QBM) model with one oscillator (position variable xxx) as the system, {\it nonlinearly} coupled to an environment of NNN harmonic oscillators (with mass mnm_nmn​, natural frequency ωnω_nωn​, position qnq_nqn​ and momentum pnp_npn​ variables) in the form ∑n(vn1(x)qnk+vn2(x)pnl)\sum_{n}\left(v_{n1}(x)q_{n}^{k}+v_{n2}(x)p_{n}^{l}\right)∑n​(vn1​(x)qnk​+vn2​(x)pnl​) where k,lk, lk,l are integers (the present work only considers the k=l=2k=l=2k=l=2 cases). The vertex functions $v_{n1}, v_{n2} $ are of the form vn1=λCn1f(x),vn2(x)=−λ Cn2mn−2ωn−2f(x)v_{n1}=λC_{n1} f(x), v_{n2}(x)=-λ\,C_{n2}m_{n}^{-2}ω_{n}^{-2}f(x)vn1​=λCn1​f(x),vn2​(x)=−λCn2​mn−2​ωn−2​f(x) where Cn1,2C_{n1,2}Cn1,2​ are the coupling constants with the nnnth oscillator, f(x)f(x)f(x) is any arbitrary function of xxx, and λλλ is a dimensionless constant. Employing the closed-time-path formalism the influence action SIFS_{IF}SIF​ is calculated using a perturbative expansion in λλλ. It is possible to identify the terms in SIFS_{IF}SIF​ quadratic or higher in Δ(s)≡f(x+(s))−f(x−(s))Δ(s)\equiv f(x_{+}(s))-f(x_{-}(s))Δ(s)≡f(x+​(s))−f(x−​(s)) to constitute the noise kernel, while terms linear in ΔΔΔ to that of the dissipation kernel. The non-Gaussian noise kernel gives rise to non-zero three-point correlation function of the corresponding stochastic force. The pathway presented here should be useful for the exploration of \textit{non-Gaussian properties of systems nonlinearly coupled with their environments}; examples in early universe cosmology and in quantum optomechanics (QOM) are mentioned. A modified fluctuation-dissipation relation (FDR) is also established, which ensures the consistency of the model and the accuracy of results even at higher perturbative orders. Another result of significance is the derivation of a nonlinear Langevin equation which is expected to be useful for many open quantum system applications.
We present a perturbative derivation of the generalized Langevin equation starting from a relativistic quantum field theory at finite temperature. Considering a complex scalar field coupled to a thermal mediator, we use a collective field formulation (Hubbard-Stratonovich transformation) to decouple the non-linear density constraint associated with a localized composite degree of freedom. By performing a saddle-point expansion and explicitly integrating out the collective fluctuations, we obtain the dynamic response kernel for the macroscopic center-of-mass coordinate. We show that localizing the density spontaneously breaks translational invariance, naturally generating a harmonic restoring force that results in an Ornstein-Uhlenbeck process. Furthermore, we resolve the vanishing friction paradox of the free-field limit by employing a resummed Dyson-Schwinger approach. We explicitly demonstrate that the thermal width of the dressed propagator generates finite Ohmic dissipation via Landau damping of the Källén-Lehmann spectral functions. This work provides a transparent, effective field theory bridge between microscopic thermal quantum fields and macroscopic open quantum systems.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0\hbar\mapsto0ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.