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We study the quantum field theory (QFT) of a scalar field in theSchr\"odinger picture in the functional formulation.  We derive a formula for the evolution kernel in a flat expanding metric. Wediscuss a transition between Riemannian and pseudoRiemannian metrics (signatureinversion). We express the real time Schr\"odinger evolution by the Brownianmotion (Feynman-Kac formula). We discuss the Feynman integral for a scalarfield in a radiation background. We show that the unitary Schr\"odingerevolution for positive time can go over for negative time into a dissipativeevolution described by diffusive paths.

Schr\"odinger evolution of a scalar field in Riemannian and pseudoRiemannian expanding metrics