FUB HEP May 74/8 Euclidean Fermi Fields with a Hermitean Feynman-Kac-Nelson Formula, II∗

We present a second heuristic FKN formula for fermions which is more suitable for constructing relativistic fields in the interacting case than our first attempt. The FKN formula remains Hermitean, and the Euclidean Dirac fields remain undoubled. This version is based on a local extension of Osterwalder-Schrader positivity to overlapping, Euclidean time arguments, which is not quite so immediate for us as it was for them. We propose a modified set of axioms for Euclidean Dirac fields, abstracted from the FKN formula. We show from Osterwalder-Schrader positivity that the Schwinger functions relative to the physical Hilbert space are at least well-defined as distributions, and they rigorously correspond to Wightman fields if one admits their existence as analytic functions with the appropriate continuations from Euclidean to Minkowski points. We do not check in this paper the natural conjecture that Nelson’s Axiom (A) implies the continuation. Work partially supported by NSF Grant GP-17523. Copies of this document may be distributed without restriction as long as the content, including attributions, is unchanged. Permanent address: The Harrison M. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, 48109. March 17, 2008: This LTEX version has only cosmetic changes from the original.