Convoluted Generalized White Noise, Schwinger Functions and Their Analytic Continuation to Wightman Functions

We construct Euclidean random fields X over IR, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F . We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo–differential operators (−∆ + m0) −α for α ∈ (0, 1) and m0 > 0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X = Gα ∗ F , obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property. Finally we give some remarks on scattering theory for these models.