Schrödinger evolution of a scalar field in Riemannian and pseudo Riemannian expanding metrics

We study the quantum field theory (QFT) of a scalar field in the Schrödinger picture in the functional formulation. We derive a formula for the evolution kernel in a flat expanding metric. We discuss a transition between Riemannian and pseudoRiemannian metrics gµν (signature inversion). We express the real time Schrödinger evolution by the Brownian motion. We discuss the Feynman integral for a scalar field in a radiation background. We show that the unitary Schrödinger evolution for positive time can go over for negative time into a dissipative evolution as a consequence of the imaginary value of √- det(gµν). The time evolution remains unitary if √- det(gµν) in the Hamiltonian is replaced by √| det(gµν)|.