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

Quantum probability distributions in the early Universe. I. Equilibrium properties of the Wigner equation

Frank R. Graziani

Published Aug 15, 1988• 24 citations
Publisher

Abstract

This is the first in a series of papers dealing with quantum or Wigner probability distributions and the dynamics of early-Universe phase transitions. In this paper we argue that the inflaton field (a real gauge-singlet scalar field) when spatially averaged over a causal horizon behaves as if it were a dissipative quantum-mechanical system in one dimension. Realizing that the mathematical phase-space or stochastic description of Wigner is the only consistent way to describe these systems and still preserve the canonical commutation relations we devote this paper to investigating various properties of the Wigner formalism. This framework rests on the quasiprobability distribution and its evolution equation: a generalized Fokker-Planck equation. We first show that this generalized Fokker-Planck equation is equivalent to Schr\"odinger's equation for nondissipative pure states. We then show how one computes ground-state energies within this formalism (for nondissipative systems) by calculating ground-state energy levels for a variety of anharmonic potentials. Finally we compute the quantum-mechanical effective potential in one dimension to order g\ensuremath{\Elzxh}.

DOI
10.1103/physrevd.38.1122

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 15, 1988•539 citations
This is the first in a series of papers dealing with quantum or Wigner probability distributions and the dynamics of early-Universe phase transitions. In this paper we argue that the inflaton field (a real gauge-singlet scalar field) when spatially averaged over a causal horizon behaves as if it were a dissipative quantum-mechanical system in one dimension. Realizing that the mathematical phase-space or stochastic description of Wigner is the only consistent way to describe these systems and still preserve the canonical commutation relations we devote this paper to investigating various properties of the Wigner formalism. This framework rests on the quasiprobability distribution and its evolution equation: a generalized Fokker-Planck equation. We first show that this generalized Fokker-Planck equation is equivalent to Schr"odinger's equation for nondissipative pure states. We then show how one computes ground-state energies within this formalism (for nondissipative systems) by calculating ground-state energy levels for a variety of anharmonic potentials. Finally we compute the quantum-mechanical effective potential in one dimension to order g\ensuremath{\Elzxh}.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
We formulate stochastic inflation in an open quantum system framework. The field coarse-grained in a patch of fixed physical size, and the total momentum of that patch, form a canonical pair and act on a one-mode Fock space which we identify as the "bulk". At each time step, new comoving modes join the coarse-grained patch and the bulk has to be redefined. This redefinition produces an entangled mode that is traced over, yielding a non-unitary evolution equation for the bulk's density matrix. For a free test field in de Sitter, one obtains GKLS dynamics, generated by an effective Hamiltonian and a single non-Hermitian Lindblad operator, hence diffusion and Hubble friction originate from the same quantum channel. The Wigner-Weyl transform of the GKLS equation leads to a Fokker-Planck equation for the Wigner function, which matches the one that applies to the classical phase-space distribution of stochastic inflation. We also provide several schemes under which one can unravel the GKLS dynamics into stochastic Schrodinger equations when continuous measurements of the decoupled mode are performed, making contact with Langevin formulations of stochastic inflation. In the light-field regime, an additional overdamped reduction can be performed by integrating out the momentum variable in the Wigner distribution, leading to Starobinsky's slow-roll Fokker-Planck equation. In that regime, the purity of the patch is strongly suppressed. In contrast, for heavy fields, field diffusion is suppressed and the coarse-grained patch remains close to a pure underdamped oscillator, which prevents a classical stochastic treatment.
We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
A nonlinear Schrodinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner–Fokker–Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schrodinger equation through a suitable gauge transformation that allows one to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behavior of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE.
We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrödinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrödinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.
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.