Renormalization group in curved space and the problem of conformal anomaly

Abstract

Renormalization Group (RG) is a powerful method for investigating quantum effects of matter fields on curved background. The formalism of RG in curved space is well known since 1984, but its applications to cosmology and black hole physics require more knowledge and opens a new interesting field of study. We review recent results about the derivation of renormalization group in a mass-dependent scheme and also consider ambiguities of conformal anomaly using dimensional and covariant Pauli-Villars regularizations.The Effective Action of vacuum (EA) is a most useful object to parametrize the quantum effects within quantum field theory in curved space-time [1,2]. Sometimes this approach is refereed to as semiclassical approximation, in order to emphasize that the metric is not quantized. The EA is a metricdependent functional, at one-loop level it consists of the classical action of vacuum and the contributions of closed loops of free matter fields. At higher loops the fields interactions become relevant. However, the practical derivations of the EA are mainly restricted by the one-loop order and here we shall also work in the framework of this approximation. In d = 4 even the contributions of free matter fields are very complicated to calculate, and this task has been successful only for some particular cases. The word "particular" corresponds to the choice of both the model for the quantum field and the one for the background classical metric. The most remarkable example is the case of massless conformal invariant fields on the cosmological (homogeneous and isotropic) metric background, where the EA can be derived exactly [3,4]. This result has been obtained through integration of conformal anomaly [5,6], showing the importance of anomaly and the related renormalization group for the analysis of the EA.The investigation of EA in more complicated situations, in particular for massive fields, requires a choice of the approximation and of the calculational scheme. In this respect the seminal role belongs to the Renormalization Group (RG), which enables one to reduce the information about complicated non-local quantum contributions to the relatively simple dependence on a single parameter µ. The most simple and most formal version of RG, based on the Minimal Subtraction MS renormalization scheme, has been successfully applied to the quantum field theory in curved space (see, e.g. [2] for the * On leave from Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine † Also at Tomsk State Pedagogical University, Russia review and introduction). The weak side of the MS-based RG is the problem of interpretation of µ and related difficulties with applying RG, e.g. in cosmology.An alternative way is to use some more sophisticated scheme of renormalization, which would not be universal one as the MS-based RG is, but which should be designed especially for the given application. Unfortunately, there is no covariant renormalization scheme except the MS one. Furthermore, the known renormalization schemes (e.g. the momen...