Adiabatic regularization of the quantum stress-energy tensor in curved spacetimes: Stochastic quantization method

It is found that the adiabatic regularized quantum stress-energy tensor for matter in any background spacetime can be obtained in the framework of stochastic quantization. As an illustration, we investigate the modes of a massive scalar field with arbitrary curvature coupling to the inhomogeneous conformally flat spacetime. The difficulty of the mode-mixing behavior arising from the spacetime inhomogeneity is overcome and a simple algorithm is presented to evaluate the adiabatic regularized quantum stress-energy tensor. The expressions so obtained are useful in formulating the semiclassical theory of gravity and in the numerical study of the back-reaction effects of quantized fields in inhomogeneous spacetimes, such as the problems of homogenization in the early universe and the evaporation of a quantum black hole. It can be seen that our prescriptions may be extended to investigate more realistic models with fermions and gauge fields. This will be studied in future papers.