A General Theory of Wightman Functions

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.