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

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I Stochastic Calculus 23

1 Stochastic Integration 25

2 Random Variation 37

2.1 What is Random Variation? 37

2.2 Probability and Riemann Sums 40

2.3 A Basic Stochastic Integral 42

2.4 Choosing a Sample Space 50

2.5 More on Basic Stochastic Integral 52

3 Integration and Probability 55

3.1 -Complete Integration 55

3.2 Burkill-complete Stochastic Integral 62

3.3 The Henstock Integral 63

3.4 Riemann Approach to Random Variation 67

3.5 Riemann Approach to Stochastic Integrals 70

4 Stochastic Processes 79

4.1 From Rn to R? 79

4.2 Sample Space RT with T Uncountable 87

4.3 Stochastic Integrals for Example 12 92

4.4 Example 12 97

4.5 Review of Integrability Issues 104

5 Brownian Motion 107

5.1 Introduction to Brownian Motion 107

5.2 Brownian Motion Preliminaries 114

5.3 Review of Brownian Probability 117

5.4 Brownian Stochastic Integration 120

5.5 Some Features of Brownian Motion 127

5.6 Varieties of Stochastic Integral 130

6 Stochastic Sums 139

6.1 Review of Random Variability 140

6.2 Riemann Sums for Stochastic Integrals 142

6.3 Stochastic Sum as Observable 145

6.4 Stochastic Sum as Random Variable 146

6.5 Introduction to RT(dXs)2 = t 149

6.6 Isometry Preliminaries 151

6.7 Isometry Property for Stochastic Sums 153

6.8 Other Stochastic Sums 157

6.9 Introduction to It?'s Formula 162

6.10 It?'s Formula for Stochastic Sums 164

6.11 Proof of It?'s Formula 165

6.12 Stochastic Sums or Stochastic Integrals? 167

II Field Theory 173

7 Gauges for Product Spaces 175

7.1 Introduction 175

7.2 Three-dimensional Brownian Motion 175

7.3 A Structured Cartesian Product Space 178

7.4 Gauges for Product Spaces 181

7.5 Gauges for Infinite-dimensional Spaces 184

7.6 Higher-dimensional Brownian Motion 191

7.7 Infinite Products of Infinite Products 196

8 Quantum Field Theory 203

8.1 Overview of Feynman Integrals 206

8.2 Path Integral for Particle Motion 210

8.3 Action Waves 212

8.4 Interpretation of Action Waves 215

8.5 Calculus of Variations 217

8.6 Integration Issues 221

8.7 Numerical Estimate of Path Integral 228

8.8 Free Particle in Three Dimensions 236

8.9 From Particle to Field 240

8.10 Simple Harmonic Oscillator 245

8.11 A Finite Number of Particles 251

8.12 Continuous Mass Field 257

9 Quantum Electrodynamics 265

9.1 Electromagnetic Field Interaction 265

9.2 Constructing the Field Interaction Integral 270

9.3 -Complete Integral Over Histories 273

9.4 Review of Point-Cell Structure 278

9.5 Calculating Integral Over Histories 279

9.6 Integration of a Step Function 283

9.7 Regular Partition Calculation 286

9.8 Integrand for Integral over Histories 288

9.9 Action Wave Amplitudes 291

9.10 Probability and Wave Functions 295

III Appendices 303

10 Appendix 1: Integration 307

10.1 Monstrous Functions 308

10.2 A Non-monstrous Function 309

10.3 Riemann-complete Integration 313

10.4 Convergence Criteria 318

10.5 \I would not care to y in that plane 324

11 Appendix 2: Theorem 63 325

11.1 Fresnel's Integral 325

11.2 Theorem 188 of [MTRV] 330

11.3 Some Consequences of Theorem 63 Fallacy 335

12 Appendix 3: Option Pricing 337

12.1 American Options 337

12.2 Asian Options 344