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

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

bold upper R"/> correspond to electromagnetic field components7 at a point

in space. (An element

is called a history of 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 left-bracket a comma b right-bracket

is defined by means of Riemann sum approximations sigma-summation f left-parenthesis x right-parenthesis StartAbsoluteValue upper I EndAbsoluteValue

sigma-summation f left-parenthesis x right-parenthesis StartAbsoluteValue upper I EndAbsoluteValue

where the subintervals StartSet upper I EndSet

of domain

are formed from partitions such as

script upper P equals StartSet x 1 comma x 2 comma ellipsis comma x Subscript n minus 1 Baseline EndSet comma x 0 equals a comma x Subscript n Baseline equals b comma x 0 less-than x 1 less-than midline-horizontal-ellipsis less-than x Subscript n Baseline period

Riemann sums can be expressed as Cauchy8 sums sigma-summation Underscript j equals 1 Overscript n Endscripts f left-parenthesis x overbar Subscript j Baseline right-parenthesis left-parenthesis x Subscript j Baseline minus x Subscript j minus 1 Baseline right-parenthesis where or . In fact the Riemann‐complete integral can be defined in terms of suitably chosen finite samples StartSet x 1 comma ellipsis comma x Subscript n minus 1 Baseline EndSet 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 normal upper Omega 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 normal upper Omega 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, left-parenthesis asterisk right-parenthesis ,9 64, and 65 of [MTRV]) wwynman integrals.

Notes

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

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