Functional Formulation of Quantum Theory of a Scalar Field in a Metric with Lorentzian and Euclidean Signatures

We study the Schr\"odinger equation in quantum field theory (QFT) in its functional formulation. In this approach quantum correlation functions can be expressed as classical expectation values over (complex) stochastic processes. We obtain a stochastic representation of the Schr\"odinger time evolution on Wentzel-Kramers-Brillouin (WKB) states by means of the Wiener integral. We discuss QFT in a flat expanding metric and in de Sitter space-time. We calculate the evolution kernel in an expanding flat metric in the real time formulation. We discuss a field interaction in pseudoRiemannian and Riemannian metrics showing that an inversion of the signature leads to some substantial simplifications of the singularity problems in QFT.