Functional measure for quantum field theory in curved spacetime

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Abstract
An examination of the functional measure for quantum field theory defined on a general curved background spacetime is presented. It is shown how to define the measure in field space to be invariant under general coordinate transformations based upon the simpler problem of defining an invariant inner product. The weight chosen for the variables of integration is seen not to matter in contrast with the claim of Fujikawa that they are uniquely specified. It is shown how the weight -1/2 variables advocated by Fujikawa are equivalent to working in a local orthonormal frame. In view of this, the interpretation of conformal anomalies as arising from the measure is reexamined. It is also shown how to define the invariant measure in phase space for a scalar field, which turns out not to be the naive generalization of the finite-dimensional result. The extension to complex and anticommuting fields is discussed. It is also shown how the choice of field variables does not alter the effective field equations.

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