Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces

In the theory of a quantized scalar field interacting with the classical Einstein gravitational field, the formal expression for the energy-momentum tensor has infinite expectation values. We propose a procedure for defining, in certain cosmological models, suitable finite expectation values of this tensor, when the mass of the scalar matter field does not vanish. Our method uses the decomposition of the scalar field into modes permitted by the symmetry of the models. The identification of the divergent terms, which are to be subtracted mode by mode from the formal tensor, follows in a natural manner from the identification of physically relevant creation and annihilation operators under conditions of arbitrarily slow (adiabatic) time dependence of the metric. The extension of the results to periods of strong time dependence is accomplished with the aid of the requirement that the four-divergence of the regularized energy-momentum tensor remain zero at all times. The energy-momentum tensor obtained by adiabatic regularization is the same as that obtained by the $n$-wave regularization procedure of Zel'dovich and Starobinsky, although the two methods are conceptually quite different. In this paper we apply the adiabatic-regularization method to the minimally coupled scalar field with positive mass in the Robertson-Walker universes. Later papers will concern extensions to conformal coupling, anisotropic metrics, and massless fields, as well as a possible physical interpretation of the regularization procedure in terms of renormalization of coupling constants in Einstein's equation.