Effective Actions for Spin 0,1/2,1 in Curved Spacetimes

Abstract

We calculate the effective potentials for scalar, Dirac and Yang-Mills fields in curved backgrounds using a new method for the determination of the heat kernel involving a re-summation of the Schwinger-DeWitt series. Self-interactions are treated both to one loop order as usual and slightly beyond one-loop order by means of a mean-field approximation. The new approach gives the familiar result for scalar fields, the Coleman-Weinberg potential plus corrections such as the leading-log terms, but the actual calculation is much faster. We furthermore show how to go systematically to higher loop order. The Schwarzschild space-time is used to exemplify the procedure. Next we consider phase transitions and we show that for a classical critical point to be a critical point of the effective potential too, certain restrictions must be imposed on as well its value as on the form of the classical potential and the background geometry. We derive this extra condition for scalar fields with arbitrary self couplings and comment on the case of fermions and gauge bosons too. Critical points of the effective action which are not there classically are also discussed. This has implications for inflation. The renormalised energy-momentum tensor for a scalar field with arbitrary self-interaction and non-minimal coupling to the gravitational background is also calculated to this improved one-loop order as is the resulting conformal anomaly. Conditions for the violation of energy conditions are described. Finally we discuss metric fluctuations and a self-consistency condition for such fluctuations is written down for spin 0,1/2,1 quantum fields. This is of importance for the study of cosmic density fluctuations. All calculations are performed in the physically relevant case of d=4 dimensions.