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      The examples, explanations, and illustrations of this book, though prompted by issues in physics and finance, are primarily about underlying mathematical problems. A reader whose primary purpose is to investigate finance or physics as such should seek out other, more relevant sources.

      Also, it is inadvisable to approach the gauge or Riemann sum theory of integration as if it were unchallenging or easy compared with, say, Lebesgue's integration theory. That said, the general idea of gauge integration is more accessible (initially, at least) than the mathematical theory of measure which underpins other approaches to integration. (Gauge integration theory includes measure and measurability—see section A.1 of [MTRV]—but these are outcomes rather than prerequisites of the theory.)

      Another central theme of this book is to devise new and better versions of stochastic integrals. Functionals of the form

      Such integral functionals either do not actually exist, or are not easy to define. In their place, this book proposes replacement or equivalent functionals which involve Riemann sums rather than integrals. In the case of stochastic processes such sums are called stochastic sums. In quantum theory they are designated sampling sums. (The former are a special case of the latter.)

      The general idea of infinite‐dimensional integration on a Cartesian product domain can be found in chapter 3 of [MTRV]. The technical foundations of the subject are in chapter 4 of that book.

      The core of this book consists of Chapters 6 and 9. Without tackling the companion book [MTRV], a reader who wants a quick introduction to basic gauge integral theory can get it in Chapter 10 below. This includes a re‐definition of the standard Lebesgue integral (on an abstract measure space) as a Riemann‐Stieltjes integral

      There is a website for commentary on technical issues: https://sites.google.com/site/StieltjesComplete/

       [email protected]

      Pat Muldowney

Reading this Book

      The term “gauge” in the title relates to gauge integration in mathematics (a generalized form of Riemann integration). It is not about gauge symmetry or gauge transformations in theoretical physics.

      The following abbreviations are used because of frequent references to these sources:

       [F1] for [39], Space‐time approach to non‐relativistic quantum mechanics, by R.P. Feynman;

       [FH] for [46], Quantum Mechanics and Path Integrals, by R.P. Feynman and A.R. Hibbs;

       [MTRV] for [121], A Modern Theory of Random Variation, by P. Muldowney.

       [ website ] for [122], https://sites.google.com/site/StieltjesComplete/ This is the website for this book, and for [MTRV].

      References to chapters, sections, and figures in this book use a capital letter, “Chapter x”; but for material from other sources lower‐case is used: “figure y”.

      This book develops themes in probability and quantum mechanics which were introduced in [MTRV]. The range of topics is smaller, and the range of notation is correspondingly smaller, with only a few new symbols. One such is the notation

(denoting stochastic sum) on page below. Section 13.4 has a list of the main symbols used.

      The subject of the predecessor book [MTRV] is the role of Cartesian product spaces

,

, in the theory of probability (including quantum mechanics), where

is the set of real numbers, and

is an interval of time.

      The present book examines in more detail different kinds of Cartesian products of

, as domains for integration of functions

,

      Though the symbol

will generally represent time, it is also used as an arbitrary finite or infinite set of labels

, depending on the context.

The notation Скачать книгу