Abstract
A general theory of J -quasi-invariant, Markovian, and generalized path space measures on the Schwartz distribution space J ′ is developed. Ergodicity properties of such measures under the action of different transformation groups of J ′ of interest to quantum field theory are studied. As a result an algebraic approach to Markov field theory and a general, probabilistic (Euclidcan) framework for the study of spontaneous symmetry breaking and phase transitions in quantum field models involving fundamental scalar fields such as the ø 4-model (or the generalized Yukawa model in two space-time dimensions) are presented. Applications of these results to the P( ø) 2 quantum field models are given and it is rigorously proven that the short-distance behavior of these models in each pure phase is canonical and the long-distance behavior is determined by the physical mass.
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