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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum fields out of thermal equilibrium

O. J. P. Éboli, R. Jackiw, So-Young Pi

Published Jun 15, 1988• 145 citations
Publisher

Abstract

The isoentropic, but energy-nonconserving, time evolution of mixed quantum states is studied in quantum mechanics and quantum field theory. A variational principle, which gives the Liouville--von Neumann equation, is implemented approximately by making a Gaussian Ansatz for the density matrix. The dynamical equations governing the parameters that define the Ansatz satisfy equations variously analogous to the Schr\"odinger equation and to mechanical problems. Interesting nonequilibrium evolution is found in special cases, as, for example, when the analog Schr\"odinger equation gives rise to reflectionless transmission. For field theory in an external, time-dependent metric we obtain equations that were previously derived in the many-field (spherical-model) limit.

DOI
10.1103/physrevd.37.3557

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The isoentropic, but energy-nonconserving, time evolution of mixed quantum states is studied in quantum mechanics and quantum field theory. A variational principle, which gives the Liouville--von Neumann equation, is implemented approximately by making a Gaussian Ansatz for the density matrix. The dynamical equations governing the parameters that define the Ansatz satisfy equations variously analogous to the Schr"odinger equation and to mechanical problems. Interesting nonequilibrium evolution is found in special cases, as, for example, when the analog Schr"odinger equation gives rise to reflectionless transmission. For field theory in an external, time-dependent metric we obtain equations that were previously derived in the many-field (spherical-model) limit.
Recently, a number of authors have begun to study the evolution of quantum fields in the early Universe characterized by a time-dependent density matrix ${\ensuremath{\rho}}_{S}$(t). All of this work is predicated on the assumption that one's ``subsystem'' of interest is in some sense ``decoupled'' from the rest of the Universe, so that ${\ensuremath{\rho}}_{S}$ satisfies a Liouville--von Neumann equation which implies, e.g., an isentropic evolution. Starting from ``first principles,'' i.e., the Schr\"odinger equation for the totality of ``subsystem'' plus surroundings (``bath''), it is shown here how such a picture can be derived as the limiting case of a more complete statistical description. Quite generally, one finds that ${\ensuremath{\rho}}_{S}$ and the ``bath'' density matrix ${\ensuremath{\rho}}_{B}$ satisfy coupled nonlinear generalizations of the Liouville--von Neumann equation and evidence a nonisentropic evolution. However, in a Vlasov-type approximation, ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{B}$ satisfy instead much simpler bilinear equations which imply an isentropic evolution. And finally, in the limit that the ``back reaction'' of ${\ensuremath{\rho}}_{S}$ on ${\ensuremath{\rho}}_{B}$ can be neglected in computing the evolution of ${\ensuremath{\rho}}_{B}$, one recovers a true Liouville--von Neumann equation for the evolution of ${\ensuremath{\rho}}_{S}$ in an external field.
One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.
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.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.
We use the so-called Liouville--von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr\"odinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal, and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Because of the spinodal instability of long wavelength modes, the two-point correlation functions lead to the t1/4{t}^{1/4}t1/4-scaling relation during the quench and the Cahn-Allen scaling relation t1/2{t}^{1/2}t1/2 after completion of the quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of the quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.