Noise kernel in stochastic gravity and stress energy bitensor of quantum fields in curved spacetimes

The noise kernel is the vacuum expectation value of the (operator-valued) stress-energy bitensor which describes fluctuations of a quantum field in curved spacetimes. It plays the role in stochastic semiclassical gravity based on the Einstein-Langevin equation similar to the expectation value of the stress-energy tensor in semiclassical gravity based on the semiclassical Einstein equation. According to the stochastic gravity program, this two-point function (and by extension the higher-order correlations in a hierarchy) of the stress energy tensor possesses precious statistical-mechanical information of quantum fields in curved spacetime and, by the self-consistency required of Einstein's equation, provides a probe into the coherence properties of the gravity sector (as measured by the higher-order correlation functions of gravitons) and the quantum and/or extended nature of spacetime. It reflects the medium energy (referring to Planck energy as high energy) or mesoscopic behavior of any viable theory of quantum gravity, including string theory. The stress energy bitensor could be the starting point for a new quantum field theory constructed on spacetimes with extended structures. In the coincidence limit we use the method of point separation to derive a regularized noise-kernel for a scalar field in general curved spacetimes. It is useful for calculating quantum fluctuations of fields in modem theories of structure formation and for back reaction problems in the early universe and black holes. One collorary of our finding is that for a massless conformal field the trace of the noise kernel identically vanishes. We outline how the general framework and results derived here can be used for the calculation of noise kernels for specific cases of physical interest such as Robertson-Walker and Schwarzschild spacetimes.