Pure States for General P(φ) 2 Theories: Construction, Regularity and Variational Equality

Abstract

We give a new construction of the P(A)2 Euclidean (quantum) field theory and propose a structure analysis of this theory. Among our results are: (1) For any polynomial P bounded from below, we construct two Euclidean states (expectations) P,?, not necessarily distinct, which satisfy all Osterwalder-Schrader axioms including clustering and obey the DobrushinLanford-Ruelle (DLR) equations for P. (2) Equality p,+ = , - holds if and only if the pressure awe) corresponding to the polynomial P(x) - px is differentiable at e = 0, and in this case the state p,+ is independent of a large class of different (in particular classical) boundary conditions. (3) All P(0)2 expectations thus far constructed are locally absolutely continuous with respect to the free field Gaussian expectations with LI RadonNikodym derivatives, for all p < oo. (4) The strong Gibbs variational equality holds, for all states constructed so far for a given P.