Some aspects of Markov and Euclidean field theories

Abstract

Various aspects of Markov field theory are treated. We give a Fock space description of the scalar free field with Nelson's Markov property formulated in terms of projections. We consider conditions imposed on analytically continued Wightman distributions at Euclidean points so that a Euclidean Markov field theory will result. Euclidean theories in higher dimensional imaginary times are considered. We show how the generalized free field theory can be interpreted as a Markov Euclidean field theory. The spatially cutoff linear perturbation model is solved in arbitrary space-time dimensions and the Wightman distributions are obtained explicitly in the limit as the cutoff is removed. The appendices contain a discussion and derivation of the Segal isomorphism and we give some generalizations of Feynman-Kac formulas inRn and in the Fock space of Euclidean field theory.