Back reaction in semiclassical gravity: The Einstein-Langevin equation

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Abstract
Using the influence functional formalism we show how to derive a generalized Einstein equation in the form of a Langevin equation for the description of the back reaction of quantum fields and their fluctuations on the dynamics of curved spacetimes. We show how a functional expansion on the influence functional gives the cumulants of the stochastic source, and how these cumulants enter in the equations of motion as noise sources. We derive an expression for the influence functional in terms of the Bogolubov coefficients governing the creation and annihilation operators of the Fock spaces at different times, thus relating it to the difference in particle creation in different histories. We then apply this to the case of a free quantum scalar field in a spatially flat Friedmann-Robertson-Walker universe and derive the Einstein-Langevin equations for the scale factor for these semiclassical cosmologies. This approach based on statistical field theory extends the conventional theory of semiclassical gravity based on a semiclassical Einstein equation with a source given by the average value of the energy-momentum tensor, thus making it possible to probe into the statistical properties of quantum fields such as noise, fluctuations, entropy, decoherence, and dissipation. Recognition of the stochastic nature of semiclassical gravity is an essential step towards the investigation of the behavior of fluctuations, instability, and phase transition processes associated with the crossover to quantum gravity.