Abstract:

Let F(A) and G(A) be functions of a gauge connection A on R³. We will assume that all functions of gauge connections are “rapidly vanishing at infinity” in the sense that F(A) goes to zero as the coordinates A^a_i(x) of the gauge connection go to infinity, and that the same is true for all functional derivatives of the function. Let DF denote a functional derivative with respect to a gauge coordinate at some point x in R³. I say that F is functionally equivalent to G if

F - G = DH

for some rapidly vanishing H, a function of gauge fields and some functional derivative D. Let $\int F$ denote the equivalence class of F. I intend $\int F$ as a mathematical substitute for the non-existent functional integral of F over all gauge fields. This function class satisfies integration by parts and change of variables formulas for functions with power series expansions. We can use the functional classes to retrieve large amounts of functional integral heuristics and turn them into mathematics.

The goal is to put the physical end of topological quantum field theory on rigorous mathematical ground. This talk is still heuristic in many respects, and it will be basically elementary (at the advanced calculus level). At the very least we get a new look at the construction of Vassiliev invariants in the context of this revised version of Witten Theory. By eliminating the functional integral and making the above definitions, the heuristic domain is transformed into a mathematical domain. The most interesting problems then lie in understanding the asymptotics for large k, since stationary phase heuristics appear to depend upon more then just these functional classes.

Web Site Information and Return Links

Contact person: Larry Peterson

E-mail: lawrence_peterson@und.nodak.edu

Phone: (701) 777-4609

Date of most recent update: August 13, 2002

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