Noise and fluctuations in semiclassical gravity

Abstract

We continue our earlier investigation of the backreaction problem in semiclassical gravity with the Schwinger-Keldysh or closed-time-path (CTP) functional formalism using the language of the decoherent history formulation of quantum mechanics. Making use of its intimate relation with the Feynman-Vernon influence functional (IF) method, we examine the statistical mechanical meaning and show the interrelation of the many quantum processes involved in the backreaction problem, such as particle creation, decoherence and dissipation. We show how noise and fluctuation arise naturally from the CTP formalism. We derive an expression for the CTP effective action in terms of the Bogolubov coefficients and show how noise is related to the fluctuations in the number of particles created. In so doing we have extended the old framework of semiclassical gravity, based on the mean field theory of Einstein equation with a source given by the expectation value of the energy-momentum tensor, to that based on a Langevin-type equation, where the dynamics of fluctuations of spacetime is driven by the quantum fluctuations of the matter field. This generalized framework is useful for the investigation of quantum processes in the early universe involving fluctuations, vacuum stability and phase transtion phenomena and the non-equilibrium thermodynamics of black holes. It is also essential to an understanding of the transition from any quantum theory of gravity to classical general relativity.

Untitled

“…The CTP effective action, which is a functional of two metric histories, g (+) µν and g (−) µν , may be expressed in terms of the Bogolyubov coefficients that relate the "in" vacuum state to the "out" vacuum state in these two histories. Such a representation has been derived earlier [59,63] for cosmological metrics. For these cases, the Bogolyubov coefficient matrices (denoted by α and β) are diagonal.…”

“…Following [33,52], we can interpret the real part of the influence functional as the characteristic functional of a non-dynamical stochastic variable j µν (x),…”

“…This means that its two-point correlation function, which is given in terms of the noise kernel by [52] as an analog of (2.7). In our case, this leads to…”

“…Studies of the Influence Functional in the context of semiclassical cosmology [60,61,59,49] show that the Influence Functional can be written in terms of the Bogolyubov coefficients that represent particle production in the specific cosmological background. The noise kernel that arises in this Influence functional can be related to the difference between the number of particles produced along alternative histories.…”

Optomechanical Analogy for Toy Cosmology with Quantized Scale Factor

“…For example, reference [7] assumes that the universe's metric is described by a position wavefunction and considers the resulting implications. Another approach is to describe the cosmology using quantum Brownian motion in a parametric bath as in [8,9]. Here we consider a toy cosmology based upon the Friedmann-Robertson-Walker-Lemaître (FRW) model, where the scale factor is quantized, and a wavefunction describing the state of the metric is defined.…”