Intrinsic Entropy of Squeezed Quantum Fields and Nonequilibrium Quantum Dynamics of Cosmological Perturbations.

Jen-Tsung Hsiang,Bei-Lok Hu

doi:10.3390/e23111544

Jen-Tsung Hsiang, Bei-Lok Hu

Open Access

https://doi.org/10.3390/e23111544

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Abstract

Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus, the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, different from previous treatments, we tackle this issue with a fully developed nonequilibrium quantum field theory formalism for such systems. We compute the covariance matrix elements of the parametric quantum field and solve for the evolution of the density matrix elements and the Wigner functions, and, from them, derive the von Neumann entropy. We then show explicitly why the entropy for the squeezed yet closed system is zero, but is proportional to the particle number produced upon coarse-graining out the correlation between the particle pairs. We also construct the bridge between our quantum field-theoretic results and those using the probability distribution of classical stochastic fields by earlier authors, preserving some important quantum properties, such as entanglement and coherence, of the quantum field.

Highlights

Entropy of quantum cosmological perturbations is an important topic which has clocked almost three decades of investigations [1,2,3]
We focus on the specific main points on the entropy of cosmological perturbations, analyzed here from a squeezed open quantum system perspective using techniques from nonequilibrium quantum field theory and relate them to earlier work
Our narrative takes on four steps, showing: (1) that the cosmological expansion results in the squeezing of the quantum field; (2) the nonequilibrium dynamics of a squeezed quantum field as a closed system, seen through the evolution of the density matrix operator in terms of the covariance matrix elements; (3) that the von Neumann entropy is zero in this closed system, as expected, and proportional to the particle numbers when the quantum correlation between the particle pair is coarse-grained away

Summary

Introduction

Entropy of quantum cosmological perturbations is an important topic which has clocked almost three decades of investigations [1,2,3]. There are two aspects in this theme: (1) the quantum field theory component depicting particle creation from the vacuum and quantum cosmological perturbations; (2) the nonequilibrium statistical mechanics aspect describing the evolutionary dynamics of a quantum many-body system—the fluctuations in, and the dissipation of, a quantum field These two components when combined make up the quantum field theory of nonequilibrium systems [23,24,25], the two major paradigms being the Boltzmann correlation hierarchy and the Langevin open systems [26]. Saving the discussions of the two theoretical factors for the section, we focus here on the cosmological aspect; (II) The cosmological factor has two components They are: (1) gravitational perturbation theory which describes how weak classical perturbations of scalar (density contrast) vector (vorticity) and tensor (gravitational waves) types evolve in a dynamical spacetime; and (2) quantum matter field processes such as particle creation from vacuum fluctuations amplified by the expansion of the universe and their consequences

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