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

Decoherence and back reaction: The origin of the semiclassical Einstein equations

Juan Pablo Paz, Sukanya Sinha

Published Aug 15, 1991• 95 citations
Publisher

Abstract

Two basic properties defining classical behavior are "decoherence" and "correlations between co-ordinates and momenta." We study how the correlations that define the semiclassical decohering histories of the relevant cosmological variables are affected by the interaction with an environment formed by unobserved ("irrelevant") degrees of freedom. For some quantum cosmological models we analyze under what conditions the semiclassical coarse-grained histories obey the so-called semiclassical Einstein's equations (i.e., ${G}_{\ensuremath{\mu}\ensuremath{\nu}}=\ensuremath{\kappa}〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$). These equations are shown to be valid only as a description of adiabatic regions of histories for which the interference effects have been suppressed. We also discuss the problem related to the existence of divergences in the decoherence factor of various quantum cosmological models.

DOI
10.1103/physrevd.44.1038

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Two basic properties defining classical behavior are "decoherence" and "correlations between co-ordinates and momenta." We study how the correlations that define the semiclassical decohering histories of the relevant cosmological variables are affected by the interaction with an environment formed by unobserved ("irrelevant") degrees of freedom. For some quantum cosmological models we analyze under what conditions the semiclassical coarse-grained histories obey the so-called semiclassical Einstein's equations (i.e., G\ensuremathμ\ensuremathν=\ensuremathκ〈T\ensuremathμ\ensuremathν〉{G}_{\ensuremath{\mu}\ensuremath{\nu}}=\ensuremath{\kappa}〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉G\ensuremathμ\ensuremathν​=\ensuremathκ〈T\ensuremathμ\ensuremathν​〉). These equations are shown to be valid only as a description of adiabatic regions of histories for which the interference effects have been suppressed. We also discuss the problem related to the existence of divergences in the decoherence factor of various quantum cosmological models.
We study some aspects of the semiclassical limit of quantum cosmology using multidimensional minisuperspace models with scalar perturbations. Regarding the minisuperspace coordinates as the "relevant" degrees of freedom and the scalar field modes as the "irrelevant" ones, we study the reduced density matrix to analyze the existence of decoherence and correlations. For a Bianchi type-I model we show that the correlations predicted from the peak of the Wigner function associated with a decohered WKB branch are the same as those determined by the semiclassical Einstein equations. We discuss the renormalization of the divergent quantities that appear in the calculation and show that usual dimensional-regularization techniques only allow us to regulate the phase of the reduced density matrix. This illustrates the fundamental difference that exists between the divergences found in the back-reaction equations and the ones that affect the quantum coherence. We analyze a class of models with adiabatic behavior in which there are no problematic divergences affecting the decoherence factor. Finally, we discuss some aspects of the relation between our approach, based on the use of the reduced density matrix, and that formulated in terms of spacetime histories.
We sketch a quantum mechanical framework for the universe as a whole. Within that framework we propose a program for describing the ultimate origin in quantum cosmology of the quasiclassical domain of familiar experience and for characterizing the process of measurement. Predictions in quantum mechanics are made from probabilities for sets of alternative histories. Probabilities can be assigned only to sets of histories that approximately decohere. Decoherence is defined and the mechanism of decoherence is reviewed. Decoherence requires a sufficiently coarse-grained description of alternative histories of the universe. A quasiclassical domain consists of a branching set of alternative decohering histories, described by a coarse graining that is maximally refined consistent with decoherence, with individual branches that exhibit a high level of classical correlation in time. A quasiclassical domain is emergent in the universe as a consequence of the initial condition and the action function of the elementary particles. It is an important question whether all the quasiclassical domains are roughly equivalent or whether there are various essentially inequivalent ones. A measurement is a correlation with variables in a quasiclassical domain. An observer (or information gathering and utilizing system) is a complex adaptive system that has evolved to exploit the relative predictability of a quasiclassical domain. We suggest that resolution of many of the problems of interpretation presented by quantum mechanics is to be accomplished, not by further scrutiny of the subject as it applies to reproducible laboratory situations, but rather by an examination of alternative histories of the universe, stemming from its initial condition, and a study of the problem of quasiclassical domains.
In this topical review we discuss the connections between chaos, decoherence and quantum cosmology.We understand chaos as classical chaos in systems with a finite number of degrees of freedom, decoherence as environment induced decoherence and quantum cosmology as the theory of the Wheeler-DeWitt equation or else the consistent history formulation thereof, first in mini super spaces and later through its extension to midi super spaces.The overall conclusion is that consideration of decoherence is necessary (and probably sufficient) to sustain an interpretation of quantum cosmology based on the wavefunction of the Universe adopting a Wentzel-Kramers-Brillouin form for large Universes, but a definitive account of the semiclassical transition in classically chaotic cosmological models is not available in the literature yet.
We use the decoherent histories approach to quantum theory to derive the form of an effective theory describing the coupling of classical and quantum variables. The derivation is carried out for a system consisting of a large particle coupled to a small particle with the important additional feature that the large particle is also coupled to a thermal environment producing the decoherence necessary for classicality. The effective theory is obtained by tracing out both the environment and the small particle variables. It consists of a formula for the probabilities of a set of histories of the large particle, and depends on the dynamics and initial quantum state of the small particle. It has the form of an almost classical particle coupled to a stochastic variable whose probabilities are determined by a formula very similar to that given by quantum measurement theory for continuous measurements of the small particle's position. The effective theory gives intuitively sensible answers when the small particle is in a superposition of localized states (unlike the simple mean field approach of coupling to the expectation values of the small system). The derived effective theory suggests a form of the semiclassical theory even when the quantum theory of the large system is not known, as is the case, for example, when a classical gravitational field is coupled to a quantized matter field, thus offering a new approach to the back reaction problem.
The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.
We study, within a toy model, how a thermal bath of cosmological perturbations can decohere the minisuperspace degrees of freedom of the universe, treating the perturbations as an environment in the framework of open quantum systems. Instead of tracing out the perturbations from a semiclassical ansatz for the total wave function, as in earlier approaches, we incorporate their backreaction on the background through an effective second-order coupling and derive a master equation. The model describes a spatially flat semiclassical universe during slow-roll inflation with a dust clock field, while scalar cosmological perturbations in a phenomenologically assumed thermal state constitute the environment. The master equation is non-Markovian, since the bath correlation functions do not decay in time. Within this framework, we find that, for a representative set of parameter estimates, the characteristic decoherence timescale corresponding to the thermal contribution can be shorter than the Hubble timescale. In addition, we address the question of whether the pointer-state structure induced by the decoherence in such a framework could provide a criterion for the choice of semiclassical states in quantum cosmology.