A General Theory of Wightman Functions

TL;DR

This paper generalizes the Wightman reconstruction theorem to include gauge fields and noncommutative spaces, providing a framework for quantum gauge theories and matrix models.

Contribution

It introduces a Wightman construction for fields valued in topological *-algebras, extending the theorem to gauge and noncommutative quantum field theories.

Findings

Large class of quantum theories as limits of matrix models

Framework applicable to gauge theories and matrix models

Conjectured link between deformation quantization and quantum field theories

Abstract

One of the main open problems of mathematical physics is to consistently quantize Yang-Mills gauge theory. If such a consistent quantization were to exist, it is reasonable to expect a ``Wightman reconstruction theorem,'' by which a Hilbert space and quantum field operators are recovered from n-point functions. However, the original version of the Wightman theorem is not equipped to deal with gauge fields or fields taking values in a noncommutative space. This paper explores a generalization of the Wightman construction which allows the fundamental fields to take values in an arbitrary topological *-algebra. In particular, the construction applies to fields valued in a Lie algebra representation, of the type required by Yang-Mills theory. This appears to be the correct framework for a generalized reconstruction theorem amenable to modern quantum theories such as gauge theories and matrix models. We obtain the interesting result that a large class of quantum theories are expected to arise as limits of matrix models, which may be related to the well-known conjecture of Kazakov. Further, by considering deformations of the associative algebra structure in the noncommutative target space, we define certain one-parameter families of quantum field theories and conjecture a relationship with deformation quantization.