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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 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 is defined by means of Riemann sum approximations where the subintervals of domain are formed from partitions such as
Riemann sums can be expressed as Cauchy8 sums 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,
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