Abstract
The aim of this paper is to find a correspondence between the one-loopeffective action WE defined by means of a path integral in Euclideangravity and the free energy F obtained by summation over the modes. Theanalysis is given for quantum fields on stationary spacetimes of a generalform. For such problems a convenient procedure of a `Wick rotation'from Euclidean to Lorentzian theory becomes quite non-trivial, implyingtransition from one real section of a complexified spacetime manifold toanother. We formulate conditions under which F and WE can beconnected and establish an explicit relation of these functionals. Ourresults are based on the Kaluza-Klein method which enables one toreduce the problem on a stationary spacetime to an equivalent problem on astatic spacetime in the presence of a gauge connection. As a by-product,we discover a relation between the asymptotic heat-kernel coefficients ofelliptic operators on a D-dimensional stationary spacetimes and theheat-kernel coefficients of (D-1)-dimensional elliptic operators with anAbelian gauge connection.
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