Patrick Muldowney - Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

and where , , , are the particle position co‐ordinates in for each .)

For system the domain and its elements are less obvious. In this book the domain

is proposed. This involves a one‐dimensional simplification (like the simplification instead of for system ), and also other simplifications which are contrary to physical reality but which make the mathematical exposition a bit easier to follow. An element of this domain is

where is particle position at time ; and, at time , elements and correspond to electromagnetic field components 7at a point in space. (An element is called a historyof the interaction.)

The reason for trailing advance notice of details such as these is to provide a sense of the mathematical challenges presented by quantum electrodynamics (system above), further to the challenges already posed by system .

Feynman's theory of system —or interaction of with —posits certain integrands in domain , the integration being carried out over “all degrees of freedom” of the physical system. But how is an integral on , , to be defined? Is there a theory of measurable sets and measurable functions for ? (Even if such a measure‐theoretic integration actually existed it would fail on the requirement for non‐absolute convergence in quantum mechanics.) And if integrands in “ ” involve action functionals of the form , we face the further problem of how to give meaning to “ ” as integrand in domain .

This is reminiscent of the stochastic integrals/stochastic sums issue mentioned above. The resolution in both cases uses the following feature of the ‐complete or gauge system of integration.

A Riemann‐type integral in a one‐dimensional bounded domain is defined by means of Riemann sum approximations where the subintervals of domain are formed from partitions such as