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

      

      Table of Contents

      1  Cover

      2  Title Page

      3  Copyright

      4  Preface

      5  Reading this BooknotesSet Note

      6  Introduction Notes

      7  Part I: Stochastic Calculus Chapter 1: Stochastic Integration Notes Chapter 2: Random Variation 2.1 What is Random Variation? 2.2 Probability and Riemann Sums 2.3 A Basic Stochastic Integral 2.4 Choosing a Sample Space 2.5 More on Basic Stochastic Integral Notes Chapter 3: Integration and Probability 3.1 ‐Complete Integration 3.2 Burkill‐complete Stochastic Integral 3.3 The Henstock Integral 3.4 Riemann Approach to Random Variation 3.5 Riemann Approach to Stochastic Integrals Notes Chapter 4: Stochastic Processes 4.1 From

to

4.2 Sample Space with Uncountable 4.3 Stochastic Integrals for Example 12 4.4 Example 12 4.5 Review of Integrability Issues Notes Chapter 5: Brownian Motion 5.1 Introduction to Brownian Motion 5.2 Brownian Motion Preliminaries 5.3 Review of Brownian Probability 5.4 Brownian Stochastic Integration 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 6.3 Stochastic Sum as Observable 6.4 Stochastic Sum as Random Variable 6.5 Introduction to

6.6 Isometry Preliminaries 6.7 Isometry Property for Stochastic Sums 6.8 Other Stochastic Sums 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

      8  Part II: Field Theory Chapter 7: Gauges for Product Spaces 7.1 Introduction 7.2 Three‐dimensional Brownian Motion 7.3 A Structured Cartesian Product Space 7.4 Gauges for Product Spaces 7.5 Gauges for Infinite‐dimensional Spaces 7.6 Higher‐dimensional Brownian Motion 7.7 Infinite Products of Infinite Products 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 8.8