Читать книгу Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - Patrick Muldowney - Страница 11

The gauge integral is a version of the Riemann integral, with much improved convergence properties. Convergence properties are conditions which ensure integrability of a function; in particular, integrability of the limit of a convergent sequence of integrable functions, with integral of the limit equal to the limit of the integrals—the limit theorems.

Another notable property of the gauge integral in one dimension is that, if a function possesses a corresponding derivative function, the derivative is integrable, with indefinite integral equal to the original function. Curiously, this “schooldays meaning”—integration as the reverse of differentiation—does not hold universally for the more widely used integration systems. See Section 10.2 of Chapter 10, which provides an overview of this subject.

The gauge integral (called ‐complete integral in [MTRV], and in this book) is non‐absolute. Other kinds of integration, such as Lebesgue's or Riemann's, have restrictive requirements of absolute convergence. But existence of ‐complete integrals requires only that the Riemann sum approximations converge non‐absolutely to the value of the integral; and this is central to the present book.

In [MTRV], instead of the more familiar term gauge integral, the term ‐complete integral (as in Riemann‐complete) is found to be helpful.2 This is because there are a great many different integration techniques—Riemann, Lebesgue, Stieltjes, Burkill, and others—which are used in different situations; most of which can be subsumed or adapted into a gauge integral system. But to assign indiscriminately the blanket designation “gauge integral” to each of the adapted versions is to ignore, firstly, their considerable difference in usages and origins; and, secondly, the fact that “gauge integral” has become practically a synonym for the one‐dimensional generalized Riemann (or Riemann‐complete) integral—also known as the Kurzweil‐Henstock integral.

Also, the general or abstract integral—called Henstock integral in chapter 4 of [MTRV]—has diverged historically from the more mainstream gauge or Kurzweil integration which has “integral‐as‐antiderivative” as its driving force. This aspect of the subject is touched on in Chapter 10 below.

The integral‐as‐antiderivative feature of one‐dimensional Riemann‐complete integration was mentioned in passing in Henstock's 1962–63 exposition [70], which concentrated on other aspects of integration (such as limit theorems3 and Fubini's theorem).

As a student Henstock was attracted to the theory of divergent series. When in 1944 he applied to Paul Dienes to do research in this subject, he was steered towards integration theory [11]; and his subsequent work often focussed on the margins between divergence and convergence.4

The gauge idea made its first appearance in Henstock's published work in [69], in a scenario of extreme divergence in which the gauge method is “tested to destruction” in its first public outing. (This counter‐example is rehearsed in pages 178–181 of [MTRV], section 4.14, Non‐Integrable Functions. See also Example 13 below.) There is no mention in his 1955 paper [69] of the reversal of differentiation which many students of the subject have found so useful. Nor does it touch on the notion of random variation in which theories of integration and measure play a central role, and where integral convergence is much more important than differentiation.5

The emphasis on convergence is maintained in the present book, which can be read as a stand‐alone, self‐contained, or self‐explanatory volume expanding on certain themes in [MTRV]. Like [MTRV] this book aspires to simplicity and transparency. No prior knowledge of the subject matter is assumed, and simple numerical examples set the scene. There is a degree of repetitiveness which may be tedious for experts. But experts can cope with that; more consideration is owed to less experienced readers.

For reasons demonstrated in [MTRV], and amply confirmed in the present volume, non‐absolute convergence is one of the characteristics which, in comparison with other methods, makes the gauge (or ‐complete) integrals suitable for the two main themes of this book: stochastic calculus and Feynman integration.

Stochastic calculus is the branch of the theory of stochastic processes which deals with stochastic integrals, also known as stochastic differential equations. A landmark result is Itô’s lemma, or Itô’s formula.

Stochastic integration is part of the mathematical theory of probability or random variation. Broadly speaking, quantities or variables are random or non‐deterministic if they can assume various unpredictable values; and they are non‐random or deterministic if they can take only definite known values.

Classically, stochastic integrals are constructed by means of a procedure involving weak limits. The purpose of this book is two‐fold:

 To treat stochastic integrals as actual integrals; so that the limit process which defines a stochastic integral is essentially the same as the limit of Riemann sums which defines the more familiar kinds of integral.

 To provide an alternative theory of stochastic sums which achieves the same purposes as stochastic integrals, but in a simpler way.

Mathematically, integration is more complicated and more sophisticated than summation (or addition) of a finite number of terms. It is demonstrated that stochastic sums can achieve the same (or better) results as stochastic integrals do. In the theory of stochastic processes, stochastic sums can replace stochastic integrals.

Examples of concrete nature are used to illustrate aspects of stochastic integration and stochastic summation, starting with relatively elementary ideas about finite numbers of things or events, in which there is no difference between summation and integration. A basic calculation of financial mathematics (growth of portfolio value) is used as a reference concept, as a vehicle, and as an aid to intuition and motivation.

In a review [145] of a book [31], Laurent Schwartz stated:

Each of us [Schwartz and Emery] tried to help the probabilists absorb stochastic infinitesimal calculus of the second order “without tears”; I don't know whether any of us succeeded or will succeed.

This book is a further effort in this direction.

The action functionals of quantum mechanics (see (8.7), page below) are analogous to stochastic integrals. They appear as integrands in the infinite‐dimensional integrals used by R.P. Feynman in his theory of quantum mechanics and quantum electrodynamics.

In comparison with alternative approaches such as those of J. Schwinger ([147–150]) and S. Tomonaga ([88–92, 164, 165]), Feynman's method is said to be physically intuitive. It contrasts with the mathematics‐leaning approach of Paul Dirac [27]:

The present lectures, like those of Eddington, are concerned with unifying relativity and quantum theory, but they approach the question from a different point of view. Eddington's method is first to get the physical ideas clear and then gradually to build up a mathematical scheme. The present method is just the opposite—first to set up a mathematical scheme and then try to get its physical interpretation.

In reading [FH] it can be helpful to bear in mind that “[Feynman was] the outstanding intuitionist of our age …”, (attributed to Schwinger in [32]).

Feynman's first published paper on path integrals was [F1], Space‐time approach to non‐relativistic quantum mechanics [39]. In a long tradition of the relationship between physics and mathematics it entailed problems of a pure mathematical kind:

There are very interesting problems involved in the attempt to avoid the subdivision and limiting processes [in Feynman's construction of path integrals]. Some sort of complex measure is being associated with the space of functions . Finite results can be obtained under unexpected circumstances because the measure is not positive everywhere, but the contributions from most of the paths largely cancel out. These curious mathematical problems are largely side‐stepped by the subdivision process. However, one feels as Cavalieri must have felt calculating the volume of a pyramid before the invention of calculus. [39] (R.P. Feynman [F1]; also page 79 of [10].)

These are problems essentially of mathematics, not physics or quantum mechanics. And the solutions proposed in [MTRV], and here, are intended to be contributions to mathematics, not physics.

 The space of functions (for ) is where is the set of real numbers and . It is likely that Feynman's reference to “measure” above relates to Lebesgue‐type measure on measurable subsets of , which is not available in the form suggested by Feynman. Here are some mathematical issues:

 Instead of measurable sets and measure of sets, [MTRV] provides a solution based on a structure of interval‐type subsets of , with a “natural” volume function for such subsets, and using the ‐complete system of non‐absolute integration described in [MTRV].

 Feynman's statement that “the contributions from most of the paths largely cancel out” suggests a non‐absolute convergence approach, and confirms the unsuitability of methods requiring absolute convergence.

 Stochastic integrals sometimes have the form where is a stochastic process. Feynman's integrals often include expressions involving the integral of kinetic energy . These are action functionals, integrals such aswhere the latter has the form of a stochastic integral . Generally speaking, for , is non‐differentiable. So none of these functionals actually exists as an integral and, in order to give mathematical meaning to them, various devices have to be used, such as the weak integrals of classical stochastic calculus, or Feynman's subdivision and limiting processes.

 Feynman's “subdivision and limiting processes” are described in [F1], and in [FH] (Quantum Mechanics and Path Integrals [46], by R.P. Feynman and A.R. Hibbs). They are also examined in section 7.18 of [MTRV], along with their relationship6 to the ‐complete integral solution.

This book provides an alternative solution to these problems. Instead of integrals , or , sample times are used to form Riemann sums. These are called stochastic sums in the stochastic integral case, and sampling sums in the case of action integrals:

 These functionals are finite sums, not integrals;

 they are sample versions of stochastic integrals (or of action integrals in the case of quantum mechanics);

 they always exist;

 and their expected values and other properties are defined and calculated by a well defined system of ‐complete (or gauge) integration in .

 And, just as it is reasonable to estimate integrals by means of finite Riemann sums, it is equally reasonable to use finite samples to estimate the functional integrands by means of finite samples (or sampling sums).

This book considers mathematical aspects of the Feynman integral theory as it is expounded in [FH], which starts with

 a single particle interacting with a conservative mechanical force,

 and which progresses through to a system consisting of the interaction of a charged particle with an electromagnetic field .

For the latter system, [FH] declares that a certain action functional should be integrated over “all degrees of freedom” of the system—over all possible values of each of the variables.

This highly intuitive mode of expression is physically suggestive and resonant. But in mathematics a domain of integration must be defined and formulated as a definite mathematical set composed of definite mathematical elements or points.

In [FH] as in [MTRV] this is achieved for motion by translating “integration over all degrees of freedom” of the single particle motion into integration on a domain consisting of elements or points ; or, simply, . (This deals only with one‐dimensional particle motion. For physical realism elements of the domain should be points of , where

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 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, ,9 64, and 65 of [MTRV]) wwynman integrals.