Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations in four dimensions

Abstract

Stochastic partial differential equations of the form ∂A=ξ are studied in four dimensions, with ξ a quaternionic white noise or Poisson random field. It is shown that solutions are euclidean Markov fields. This gives besides a new construction of free electromagnetic fields, also a large class of non-gaussian euclidean Markov random fields which give rise to solutions of the ultraviolet problem for self-interacting local fields in four space-time dimensions.