Markovian representations of current algebra

by R. F. Streater; J. Phys., A10, 261-266, 1977.

ABSTRACT

Independently of the first paper on Euclidean quantum field theory by Osterwalder and Schrader, Hegerfeldt proposed the concept of T-positivity which did not suffer from a difficulty inherent in that paper, and which was published before the more correct second version of reflection positivity (O-S positivity) appeared. We generalize Hegerfeldt's concept of T-positivity in Euclidean random fields to non-commutative probability theory, that is, to Euclidean Fermi fields and to current algebra with possible Schwinger terms. Our axioms imply the Wightman axioms. A non-abelian form of Markovicity is introduced, and is shown to imply T-positivity if a reflection property holds.

The investigation suggests a generalization of Nelson-Symanzik positivity, which might be valid in cases when the extension of the Schwinger functions to coinciding arguments is not expected to maintain both commutativity and positivity (or anti-commutativity and positivity).