Zeta Function Regularization and Effective Action in Curved Spacetime

Abstract

In most quantum field theories (QFTs), we encounter irritating problems of divergences. Several techniques have been developed in the past to separate out a finite ‘physically sensible’ part from the divergent mess. The mess is usually absorbed into the theory by a suitable renormalization of the coupling constants. Over the years, a number of elaborate methods have been developed to carry out this procedure. Simple normal ordering which removes a divergent zero point energy in flat spacetime cannot be justified in general relativity. Unlike in the flat spacetime case, the absolute value of energy does have a meaning in general relativity. To be honest, although methods like mode-sum cutoff, De Witt-Schwinger proper time expansion of the Green’s function, adiabatic regularization, covariant point splitting, dimensional regularization, etc., are a lot more than mere psychaedelic sounding expressions, I personally am very biased and committed to the use of the zeta function regularization scheme. This bias is perhaps due to my own subjective notion of elegance and compactness that this scheme offers.