Regularization and renormalization of quantum field theory in curved space-time

Abstract

It is proposed that field theories quantized in a curved space-time manifold can be conveniently regularized and renormalized with the aid of Pauli-Villars regulator fields. The method avoids the conceptual difficulties of covariant point-separation approaches, by always starting from a manifestly generally covariant action, and the technical limitations of the dimensional regularization approach, which requires solution of the theory in arbitrary dimension in order to go beyond a weak-field expansion. An action is constructed which renormalizes the weak-field perturbation theory of a massive scalar field in two space-time dimensions, it is shown that the trace anomaly previously found in dimensional regularization and some point-separation calculations also arises in perturbation theory when the theory is Pauli-Villars regulated. We then study a specific solvable two-dimensional model of a massive scalar field in a Robertson-Walker asymptotically flat universe. It is shown that the action previously considered leads, in this model, to a well-defined finite expectation value for the stress-energy tensor. The particle production (<0 in | θ μν ( x, t)| 0 in> for t → +∞) is computed explicitly. Finally, the validity of weak-field perturbation theory (in the appropriate range of parameters) is checked directly in the solvable model, and the trace anomaly computed in the asymptotic regions t → ±∞ independently of any weak-field approximation. The extension of our model to higher dimensions and the renormalization of interacting (scalar) field theories are briefly discussed.