  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.
