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We discuss the generalization to curved spacetime of a path-integral formalism of quantum field theory based on the sum over paths first going forward in time in the presence of one external source from an in vacuum to a state defined on a hypersurface of constant time in the future, and then backwards in time in the presence of a different source to the same in vacuum. This closed-time-path formalism which generalizes the conventional method based on in-out vacuum persistence amplitudes yields real and causal effective actions, field equations, and expectation values. We apply this method to two problems in semiclassical cosmology. First we study the back reaction of particle production in a radiation-filled Bianchi type-I universe with a conformal scalar field. Unlike the in-out formalism which yields complex geometries the real and causal effective action here yields equations for real effective geometries, with more readily interpretable results. It also provides a clear identification of particle production as a dissipative process in semiclassical theories. In the second problem we calculate the vacuum expectation value of the stress-energy tensor for a nonconformal massive ^4 theory in a Robertson-Walker universe. This study serves to illustrate the use of Feynman diagrams and higher-loop calculations in this formalism. It also demonstrates the economy of this method in the calculation of expectation values over the mode-sum Bogolubov transformation methods ordinarily applied to matrix elements calculated in the conventional in-out approach. The capability of the closed-time-path formalism of dealing with Feynman, causal, and correlation functions on the same footing makes it a potentially powerful and versatile technique for treating nonequilibrium statistical properties of dynamical systems as in early-Universe quantum processes.
Calzetta et al. (Thu,) studied this question.