Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

John Wiley and Sons Ltd

06/2021

384

Dura

Inglês

9781119595496

Pré-lançamento - envio 15 a 20 dias após a sua edição

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I Stochastic Calculus 19 1 Stochastic Integration 21 2 Random Variation 33 2.1 What is Random Variation? 33 2.2 Probability and Riemann Sums 36 2.3 A Basic Stochastic Integral 38 2.4 Choosing a Sample Space 46 2.5 More on Basic Stochastic Integral 48 3 Integration and Probability 51 3.1 -Complete Integration 51 3.2 Burkill-complete Stochastic Integral 58 3.3 The Henstock Integral 59 3.4 Riemann Approach to Random Variation 63 3.5 Riemann Approach to Stochastic Integrals 66 4 Stochastic Processes 75 4.1 From Rn to R 75 4.2 Sample Space RT with T Uncountable 83 4.3 Stochastic Integrals for Example 12 88 4.4 Example 12 93 4.5 Review of Integrability Issues 100 5 Brownian Motion 103 5.1 Introduction to Brownian Motion 103 5.2 Brownian Motion Preliminaries 110 5.3 Review of Brownian Probability 113 5.4 Brownian Stochastic Integration 116 5.5 Some Features of Brownian Motion 123 5.6 Varieties of Stochastic Integral 126 6 Stochastic Sums 135 6.1 Review of Random Variability 136 6.2 Riemann Sums for Stochastic Integrals 138 6.3 Stochastic Sum as Observable 141 6.4 Stochastic Sum as Random Variable 142 6.5 Introduction to T(dXs)2 = t 145 6.6 Isometry Preliminaries 147 6.7 Isometry Property for Stochastic Sums 149 6.8 Other Stochastic Sums 153 6.9 Introduction to Ito's Formula 158 6.10 Ito's Formula for Stochastic Sums 160 6.11 Proof of Ito's Formula 161 6.12 Stochastic Sums or Stochastic Integrals? 163 II Field Theory 169 7 Gauges for Product Spaces 171 7.1 Introduction 171 7.2 Three-dimensional Brownian Motion 171 7.3 A Structured Cartesian Product Space 174 7.4 Gauges for Product Spaces 177 7.5 Gauges for Infinite-dimensional Spaces 180 7.6 Higher-dimensional Brownian Motion 187 7.7 Infinite Products of Infinite Products 192 8 Quantum Field Theory 199 8.1 Overview of Feynman Integrals 202 8.2 Path Integral for Particle Motion 206 8.3 Action Waves 208 8.4 Interpretation of Action Waves 211 8.5 Calculus of Variations 213 8.6 Integration Issues 217 8.7 Numerical Estimate of Path Integral 224 8.8 Free Particle in Three Dimensions 232 8.9 From Particle to Field 236 8.10 Simple Harmonic Oscillator 241 8.11 A Finite Number of Particles 247 8.12 Continuous Mass Field 253 9 Quantum Electrodynamics 261 9.1 Electromagnetic Field Interaction 261 9.2 Constructing the Field Interaction Integral 266 9.3 -Complete Integral Over Histories 269 9.4 Review of Point-Cell Structure 274 9.5 Calculating Integral Over Histories 275 9.6 Integration of a Step Function 279 9.7 Regular Partition Calculation 282 9.8 Integrand for Integral over Histories 284 9.9 Action Wave Amplitudes 287 9.10 Probability and Wave Functions 291 III Appendices 299 10 Appendix 1: Integration 303 10.1 Monstrous Functions 304 10.2 A Non-monstrous Function 305 10.3 Riemann-complete Integration 309 10.4 Convergence Criteria 314 10.5 "I would not care to y in that plane" 320 11 Appendix 2: Theorem 63 321 11.1 Fresnel's Integral 321 11.2 Theorem 188 of [MTRV] 326 11.3 Some Consequences of Theorem 63 Fallacy 331 12 Appendix 3: Option Pricing 333 12.1 American Options 333 12.2 Asian Options 340 13 Listings 353 13.1 Theorems 353 13.2 Examples 354 13.3 Definitions 356 13.4 Symbols 356
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