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

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A stand-alone introduction to specific integration problems in the probabilistic theory of stochastic calculus Picking up where his previous book,
, left off,
introduces readers to particular problems of integration in the probability-like theory of quantum mechanics. Written as a motivational explanation of the key points of the underlying mathematical theory, and including ample illustrations of the calculus, this book relies heavily on the mathematical theory set out in the author’s previous work. That said, this work stands alone and does not require a reading of
in order to be understandable.
Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics Stochastic calculus, including discussions of random variation, integration and probability, and stochastic processes. Field theory, including discussions of gauges for product spaces and quantum electrodynamics. Robust and thorough appendices, examples, illustrations, and introductions for each of the concepts discussed within. An introduction to basic gauge integral theory. The methods employed in this book show, for instance, that it is no longer necessary to resort to unreliable «Black Box» theory in financial calculus; that full mathematical rigor can now be combined with clarity and simplicity. Perfect for students and academics with even a passing interest in the application of the gauge integral technique pioneered by R. Henstock and J. Kurzweil,
is an illuminating and insightful exploration of the complex mathematical topics contained within.

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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 картинка 35(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 Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 36, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 37, in the theory of probability (including quantum mechanics), where картинка 38is the set of real numbers, and картинка 39is 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 - фото 40, as domains for integration of functions Though the symbol will generally represent time it is also used as - фото 41,

Though the symbol will generally represent time it is also used as an - фото 42

Though the symbol картинка 43will generally represent time, it is also used as an arbitrary finite or infinite set of labels картинка 44, depending on the context.

The notation картинка 45has two components: domain картинка 46, and integrand картинка 47. When картинка 48is an infinite set, such as an interval of time, two different perspectives are present. These are the perspectives indicated in figures 3.1 and 3.2 on page 87 of [MTRV].

Figure 3.1 represents the graph of where is an argument of integrand Figure 32 represents the domain - фото 49where is an argument of integrand Figure 32 represents the domain whose elements - фото 50is an argument of integrand Figure 32 represents the domain whose elements are points not graphs Both - фото 51. Figure 3.2 represents the domain картинка 52whose elements Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 53are points, not graphs. Both of these perspectives should be kept in mind while using this book.

To see the significance of these alternate perspectives 1 , suppose Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 54and Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 55, so Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 56. In this case, for integrands which appear in this book there may be little difference between the values of - фото 57which appear in this book, there may be little difference between the values of integrands in domain and in - фото 58in domain and in if - фото 59, and in if is continuous But domains - фото 60in картинка 61if картинка 62is continuous.

But domains Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 63and Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 64 Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 65are very different; just as Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 66differs geometrically from Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 67, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - изображение 68. The latter difference is the vehicle for the 19th century satire Flatland by Edwin A. Abbott [1], in which two‐dimensional beings struggle with the idea of a three‐dimensional universe.

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