Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

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

Table of Contents

List of Tables

List of Illustrations

Guide

Pages

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

Preface

Reading this Book

Note

Introduction

Notes

Chapter 1 Stochastic Integration

Example 1

Example 2

Notes

Chapter 2 Random Variation

2.1 What is Random Variation?

Example 3

2.2 Probability and Riemann Sums

Example 4

2.3 A Basic Stochastic Integral

Example 5

Example 6

2.4 Choosing a Sample Space

2.5 More on Basic Stochastic Integral

Notes

Chapter 3 Integration and Probability

3.1 ‐Complete Integration

Definition 1

Definition 2

Definition 3

Definition 4

Definition 5

3.2 Burkill‐complete Stochastic Integral

3.3 The Henstock Integral

Definition 6

Example 7

3.4 Riemann Approach to Random Variation

Definition 7

3.5 Riemann Approach to Stochastic Integrals

Definition 8

Example 8

Proof

Example 9

Notes

Chapter 4 Stochastic Processes

4.1 From to

Example 10

Example 11

4.2 Sample Space with Uncountable

Example 12

4.3 Stochastic Integrals for Example 12

Example 13

Proof

Example 14

4.4 Example 12

Example 15

Example 16

4.5 Review of Integrability Issues

Notes

Chapter 5 Brownian Motion. 5.1 Introduction to Brownian Motion

Example 17

5.2 Brownian Motion Preliminaries

5.3 Review of Brownian Probability

5.4 Brownian Stochastic Integration

Example 18

Example 19

Example 20

Example 21

Example 22

5.5 Some Features of Brownian Motion

5.6 Varieties of Stochastic Integral

Notes

Chapter 6 Stochastic Sums

6.1 Review of Random Variability

6.2 Riemann Sums for Stochastic Integrals

Example 23

6.3 Stochastic Sum as Observable

6.4 Stochastic Sum as Random Variable

Example 24

6.5 Introduction to

6.6 Isometry Preliminaries

6.7 Isometry Property for Stochastic Sums

Example 25

6.8 Other Stochastic Sums

Example 26

Example 27

Example 28

Example 29

Example 30

Theorem 1

Proof

Theorem 2

Proof

6.9 Introduction to Itô’s Formula

6.10 Itô’s Formula for Stochastic Sums

6.11 Proof of Itô’s Formula

6.12 Stochastic Sums or Stochastic Integrals?

Notes

Chapter 7 Gauges for Product Spaces. 7.1 Introduction

7.2 Three‐dimensional Brownian Motion

Example 31

7.3 A Structured Cartesian Product Space. Example 32

Theorem 3

Proof

7.4 Gauges for Product Spaces. Example 33

Theorem 4

Proof

7.5 Gauges for Infinite‐dimensional Spaces

Example 34

Example 35

Example 36

Theorem 5

Proof

Example 37

7.6 Higher‐dimensional Brownian Motion

Theorem 6

Proof

Example 38

Example 39

Theorem 7

Proof

Example 40

7.7 Infinite Products of Infinite Products

Example 41

Theorem 8

Proof

Example 42

Notes

Chapter 8 Quantum Field Theory

8.1 Overview of Feynman Integrals

8.2 Path Integral for Particle Motion

8.3 Action Waves

8.4 Interpretation of Action Waves

8.5 Calculus of Variations

8.6 Integration Issues

8.7 Numerical Estimate of Path Integral

Example 43

Example 44

Example 45

Example 46

Example 47

Example 48

Example 49

Example 50

8.8 Free Particle in Three Dimensions

8.9 From Particle to Field

Example 51

Example 52

8.10 Simple Harmonic Oscillator

Example 53

Example 54

8.11 A Finite Number of Particles

8.12 Continuous Mass Field

Notes

Chapter 9 Quantum Electrodynamics

9.1 Electromagnetic Field Interaction

9.2 Constructing the Field Interaction Integral

9.3 ‐Complete Integral Over Histories

Example 55

9.4 Review of Point‐Cell Structure

9.5 Calculating Integral Over Histories

Example 56

9.6 Integration of a Step Function

9.7 Regular Partition Calculation

9.8 Integrand for Integral over Histories

9.9 Action Wave Amplitudes

9.10 Probability and Wave Functions

Example 57

Notes

Chapter 10 Appendix 1: Integration

10.1 Monstrous Functions

10.2 A Non‐monstrous Function

10.3 Riemann‐complete Integration

10.4 Convergence Criteria

Theorem 9

Theorem 10

Theorem 11

Example 58

10.5 “I would not care to fly in that plane”

Notes

Chapter 11 Appendix 2: Theorem 63

11.1 Fresnel's Integral

Example 59

Theorem 142

Proof of theorem 142

11.2 Theorem 188 of [MTRV]

Theorem 188A

Proof

Theorem 188B

Proof

Theorem 188C

Proof

Theorem 188D

Proof

11.3 Some Consequences of Theorem 63 Fallacy

Example 60

Notes

Chapter 12 Appendix 3: Option Pricing

12.1 American Options

12.2 Asian Options

Example 61

Example 62

Notes

Chapter 13 Appendix 4: Listings. 13.1 Theorems

13.2 Examples

13.3 Definitions

13.4 Symbols

Bibliography

Index

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Patrick Muldowney

A link between stochastic calculus and quantum mechanics is provided in a previous book by the author ([121], A Modern Theory of Random Variation, or [MTRV] for short), which establishes a mathematical connection between large scale Brownian motion on the one hand and, on the other, small scale quantum level phenomena of particle motion subject to a conservative external mechanical force. In [MTRV] each of the two subjects is a special case of ‐Brownian motion.

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The 16 values for can be easily calculated, as in Table 2.5 above. In fact, the 16 outcomes for net wealth (shareholding value) gain are

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