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

Riemann sums can be expressed as Cauchy 8sums where or . In fact the Riemann‐complete integral can be defined in terms of suitably chosen finite samples of the elements in the domain of integration, without resort to measurable functions or measurable subsets—or even without explicit mention of subintervals of the domain of integration.

To define ‐complete integration in “rectangular” or Cartesian product domains such as above—no matter how complex their construction—the only requirements are:

Exact specification of the elements or points of the domain, and

A structuring of finite samples of points consistent with Axioms DS1 to DS8 of chapter 4 of [MTRV].

In other words integration requires a domain and a process of selecting samples of points or elements of —without reference to measurable subsets, or even to intervals of at the most basic level.

This skeletal structuring of finite samples of points of the domain provides us with a system of integration (the ‐complete integral) with all the useful properties—limit theorems, Fubini's theorem, a theory of measure, and so on. More than that, it provides criteria for non‐absolute convergence (theorems 62, , 964, and 65 of [MTRV]) wwynman integrals.

1 2 The attachment “‐complete” was introduced by R. Henstock in [70], the first book‐length exposition of this kind of integration theory. A few of copies of this edition were printed in 1962. A replacement edition with different page size was printed and distributed in 1963. Up to that time J. Kurzweil and R. Henstock had worked independently on this subject from around the mid‐1950s, without knowledge of each other.

2 3 Henstock's introduction of the “‐complete” appendage is suggestive of “enhanced integrability of limits” rather than “completeness of a domain with respect to a norm”.

3 4 As part of the College Prize awarded by St. John's College, Cambridge, on the results of the 1943 Mathematics Tripos Part 2 examination, Henstock received a copy of Dienes’ book [23], which includes close analysis of convergence‐divergence issues. In a late, unfinished work [78], c. 1992–1993, Henstock used some notable ideas from Dienes’ book.

4 5 The final chapter of Henstock's 1962‐1963 book [70] has the title Integration in Statistics. It deals mostly with tests of significance, and touches on some questions of probability theory using the Riemann‐complete method. The 1955 paper is concerned strictly with the nature of integration. But ancillary matters such as probability—and, indeed, differentiation—featured consistently in Henstock's subsequent work.

5 6 In the terminology of [MTRV] and this book, Feynman's method consists of substituting cylinder function approximations in the action functional.

6 7 This simplification represents each of the variables , , and as one‐dimensional. The electric field component is essentially vectorial, and one‐dimensional is contrary to the physical nature of the system. A physically more accurate version can be arrived at by a careful reading of chapter 9 of [FH]. And even though it is a bit more complicated, it is not too difficult to adapt the mathematical theory presented in this book.

7 8 A Cauchy sum has . But allowing to be either of or makes a connection with the Riemann sums of ‐complete integration.

8 9 Theorem 63 (page 175 of [MTRV]) is false. See Section 11.2 below; and also [ website ].

The idea or purpose of stochastic integration is to define a random variable

where is a random or unpredictable quantity, depending in a particular manner on unpredictable entities and ; and where

are stochastic processes and depends on time t . In textbooks, the integrand is usually presented as , but is used here in order to emphasise that the integrand is intended to be random.

The integrand (or, when appropriate, ) is to be regarded as a measurable function—as is —with respect to a probability space .