Springer Online Journal Archives 1860-2000
Abstract Quantum field theory in curved spacetime is examined from the Euclidean approach, where one seeks to define the theory for metrics of positive (rather than Lorentzian) signature. Methods of functional analysis are used to give a proof of the heat kernel expansion for the Laplacian, which extends the well known result for compact manifolds to all non-compact manifolds for which the Laplacian and its powers are essentially self-adjoint on the initial domain of smooth functions of compact support. Using this result, precise prescriptions of the zeta-function, dimensional, and point-splitting type are given for renormalizing the action of a Klein-Gordon scalar field. These prescriptions are shown to be equivalent up to local curvature terms. It is also shown that for static spacetimes, the Euclidean prescription for defining the Feynman propagator agrees with the definition of Feyman propagator obtained by working directly on the spacetime.
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