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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Perturbative method for the derivation of quantum kinetic theory based on closed-time-path formalism

Jun Koide

Published Jan 3, 2002• 1 citation
Publisher

Abstract

Within the closed-time-path formalism, a perturbative method is presented, which reduces the microscopic field theory to the quantum kinetic theory. In order to make this reduction, the expectation value of a physical quantity must be calculated under the condition that the Wigner distribution function is fixed, because it is the independent dynamical variable in the quantum kinetic theory. It is shown that when a nonequilibrium Green function in the form of the generalized Kadanoff-Baym ansatz is utilized, this condition appears as a cancellation of a certain part of contributions in the diagrammatic expression of the expectation value. Together with the quantum kinetic equation, which can be derived in the closed-time-path formalism, this method provides a basis for the kinetic-theoretical description.

DOI
10.1103/physreve.65.026101

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Within the closed-time-path formalism, a perturbative method is presented, which reduces the microscopic field theory to the quantum kinetic theory. In order to make this reduction, the expectation value of a physical quantity must be calculated under the condition that the Wigner distribution function is fixed, because it is the independent dynamical variable in the quantum kinetic theory. It is shown that when a nonequilibrium Green function in the form of the generalized Kadanoff-Baym ansatz is utilized, this condition appears as a cancellation of a certain part of contributions in the diagrammatic expression of the expectation value. Together with the quantum kinetic equation, which can be derived in the closed-time-path formalism, this method provides a basis for the kinetic-theoretical description.
Utilizing a non-equilibrium Green function like the generalized Kadanoff-Baym ansatz, a systematic perturbative method is presented to calculate the expectation value of an arbitrary physical quantity under the restriction that the Wigner distribution function is fixed. It is shown that, in the diagrammatic expression of the quantity, a certain part of contributions can be eliminated due to the restriction. Together with the quantum kinetic equation, this method provides a basis for the kinetic-theoretical description.
The inversion method to derive the quantum kinetic equation is extended to the case of an inhomogeneous system. Within the closed-time-path formalism, the source is introduced to probe the (Fourier transform of the) Wigner distribution function and the equation of motion for the Wigner function is derived perturbatively by the inversion formulae. The equation of motion has the form of the generalized Boltzmann equation, which is non-local in space and time. For the obtained kinetic equation, the local conservation laws of number, energy and momentum are confirmed.
A systematic derivation of the quantum kinetic equation is presented in the framework of a closed-time-path formalism. Introducing a probe, the expectation value of the number operator is calculated as a functional of the probing source. Then, solving for the source by inverting the relation, the removal of the source leads to the quantum kinetic equation as the equation of motion for the number, which gives a generalization of the Boltzmann equation including memory. The inversion formula is used in the course of the derivation. The calculation is presented up to third order in interaction, and the effect of initial correlations is also considered.
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.
This paper describes perturbative framework, on the basis of the closed-time-path formalism, in terms of quasiparticle picture for studying quasiuniform relativistic quantum field systems near equilibrium and nonequilibrium quasistationary systems. Two calculational schemes are introduced, the one is formulated on the basis of the initial-particle distribution function and the one is formulated on the basis of the ``physical''-particle distribution function. It is shown that both schemes are equivalent and lead to a generalized kinetic or Boltzmann equation. Concrete procedure of computing a generic amplitude is presented.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.