{"product_id":"gauge-integral-structures-for-stochastic-calculus-and-quantum-electrodynamics-isbn-9781119595496","title":"Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics","description":"\u003cb\u003eGAUGE INTEGRAL STRUCTURES FOR STOCHASTIC CALCULUS AND QUANTUM ELECTRODYNAMICS\u003c\/b\u003e \u003cp\u003e\u003cb\u003eA stand-alone introduction to specific integration problems in the probabilistic theory of stochastic calculus\u003c\/b\u003e\u003c\/p\u003e\u003cp\u003ePicking up where his previous book, \u003ci\u003eA Modern Theory of Random Variation\u003c\/i\u003e, left off, \u003ci\u003eGauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics\u003c\/i\u003e introduces readers to particular problems of integration in the probability-like theory of quantum mechanics.\u003c\/p\u003e\u003cp\u003eWritten 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 \u003ci\u003eA Modern Theory of Random Variation\u003c\/i\u003e in order to be understandable.\u003c\/p\u003e\u003cp\u003e\u003ci\u003eGauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics\u003c\/i\u003e takes a gradual, relaxed, and discursive approach to the subject in a successful attempt to engage the reader by exploring a narrower range of themes and problems.\u003c\/p\u003e\u003cp\u003eOrganized around examples with accompanying introductions and explanations, the book covers topics such as:\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eStochastic calculus, including discussions of random variation, integration and probability, and stochastic processes\u003c\/li\u003e\n\u003cli\u003eField theory, including discussions of gauges for product spaces and quantum electrodynamics\u003c\/li\u003e\n\u003cli\u003eRobust and thorough appendices, examples, illustrations, and introductions for each of the concepts discussed within\u003c\/li\u003e\n\u003cli\u003eAn introduction to basic gauge integral theory (for those unfamiliar with the author’s previous book)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eThe 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, \u003ci\u003eGauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics\u003c\/i\u003e is an illuminating and insightful exploration of the complex mathematical topics contained within. \u003c\/p\u003e\u003cp\u003e\u003cb\u003eI Stochastic Calculus 23\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Stochastic Integration 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Random Variation 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 What is Random Variation? 37\u003c\/p\u003e \u003cp\u003e2.2 Probability and Riemann Sums 40\u003c\/p\u003e \u003cp\u003e2.3 A Basic Stochastic Integral 42\u003c\/p\u003e \u003cp\u003e2.4 Choosing a Sample Space 50\u003c\/p\u003e \u003cp\u003e2.5 More on Basic Stochastic Integral 52\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Integration and Probability 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 -Complete Integration 55\u003c\/p\u003e \u003cp\u003e3.2 Burkill-complete Stochastic Integral 62\u003c\/p\u003e \u003cp\u003e3.3 The Henstock Integral 63\u003c\/p\u003e \u003cp\u003e3.4 Riemann Approach to Random Variation 67\u003c\/p\u003e \u003cp\u003e3.5 Riemann Approach to Stochastic Integrals 70\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Stochastic Processes 79\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 From R\u003csup\u003en\u003c\/sup\u003e to R\u003csup\u003eª\u003c\/sup\u003e 79\u003c\/p\u003e \u003cp\u003e4.2 Sample Space R\u003csup\u003eT\u003c\/sup\u003e with T Uncountable 87\u003c\/p\u003e \u003cp\u003e4.3 Stochastic Integrals for Example 12 92\u003c\/p\u003e \u003cp\u003e4.4 Example 12 97\u003c\/p\u003e \u003cp\u003e4.5 Review of Integrability Issues 104\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Brownian Motion 107\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction to Brownian Motion 107\u003c\/p\u003e \u003cp\u003e5.2 Brownian Motion Preliminaries 114\u003c\/p\u003e \u003cp\u003e5.3 Review of Brownian Probability 117\u003c\/p\u003e \u003cp\u003e5.4 Brownian Stochastic Integration 120\u003c\/p\u003e \u003cp\u003e5.5 Some Features of Brownian Motion 127\u003c\/p\u003e \u003cp\u003e5.6 Varieties of Stochastic Integral 130\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Stochastic Sums 139\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Review of Random Variability 140\u003c\/p\u003e \u003cp\u003e6.2 Riemann Sums for Stochastic Integrals 142\u003c\/p\u003e \u003cp\u003e6.3 Stochastic Sum as Observable 145\u003c\/p\u003e \u003cp\u003e6.4 Stochastic Sum as Random Variable 146\u003c\/p\u003e \u003cp\u003e6.5 Introduction to RT(dXs)2 = t 149\u003c\/p\u003e \u003cp\u003e6.6 Isometry Preliminaries 151\u003c\/p\u003e \u003cp\u003e6.7 Isometry Property for Stochastic Sums 153\u003c\/p\u003e \u003cp\u003e6.8 Other Stochastic Sums 157\u003c\/p\u003e \u003cp\u003e6.9 Introduction to Itô's Formula 162\u003c\/p\u003e \u003cp\u003e6.10 Itô's Formula for Stochastic Sums 164\u003c\/p\u003e \u003cp\u003e6.11 Proof of Itô's Formula 165\u003c\/p\u003e \u003cp\u003e6.12 Stochastic Sums or Stochastic Integrals? 167\u003c\/p\u003e \u003cp\u003e\u003cb\u003eII Field Theory 173\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Gauges for Product Spaces 175\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction 175\u003c\/p\u003e \u003cp\u003e7.2 Three-dimensional Brownian Motion 175\u003c\/p\u003e \u003cp\u003e7.3 A Structured Cartesian Product Space 178\u003c\/p\u003e \u003cp\u003e7.4 Gauges for Product Spaces 181\u003c\/p\u003e \u003cp\u003e7.5 Gauges for Infinite-dimensional Spaces 184\u003c\/p\u003e \u003cp\u003e7.6 Higher-dimensional Brownian Motion 191\u003c\/p\u003e \u003cp\u003e7.7 Infinite Products of Infinite Products 196\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Quantum Field Theory 203\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Overview of Feynman Integrals 206\u003c\/p\u003e \u003cp\u003e8.2 Path Integral for Particle Motion 210\u003c\/p\u003e \u003cp\u003e8.3 Action Waves 212\u003c\/p\u003e \u003cp\u003e8.4 Interpretation of Action Waves 215\u003c\/p\u003e \u003cp\u003e8.5 Calculus of Variations 217\u003c\/p\u003e \u003cp\u003e8.6 Integration Issues 221\u003c\/p\u003e \u003cp\u003e8.7 Numerical Estimate of Path Integral 228\u003c\/p\u003e \u003cp\u003e8.8 Free Particle in Three Dimensions 236\u003c\/p\u003e \u003cp\u003e8.9 From Particle to Field 240\u003c\/p\u003e \u003cp\u003e8.10 Simple Harmonic Oscillator 245\u003c\/p\u003e \u003cp\u003e8.11 A Finite Number of Particles 251\u003c\/p\u003e \u003cp\u003e8.12 Continuous Mass Field 257\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Quantum Electrodynamics 265\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Electromagnetic Field Interaction 265\u003c\/p\u003e \u003cp\u003e9.2 Constructing the Field Interaction Integral 270\u003c\/p\u003e \u003cp\u003e9.3 -Complete Integral Over Histories 273\u003c\/p\u003e \u003cp\u003e9.4 Review of Point-Cell Structure 278\u003c\/p\u003e \u003cp\u003e9.5 Calculating Integral Over Histories 279\u003c\/p\u003e \u003cp\u003e9.6 Integration of a Step Function 283\u003c\/p\u003e \u003cp\u003e9.7 Regular Partition Calculation 286\u003c\/p\u003e \u003cp\u003e9.8 Integrand for Integral over Histories 288\u003c\/p\u003e \u003cp\u003e9.9 Action Wave Amplitudes 291\u003c\/p\u003e \u003cp\u003e9.10 Probability and Wave Functions 295\u003c\/p\u003e \u003cp\u003e\u003cb\u003eIII Appendices 303\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Appendix 1: Integration 307\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Monstrous Functions 308\u003c\/p\u003e \u003cp\u003e10.2 A Non-monstrous Function 309\u003c\/p\u003e \u003cp\u003e10.3 Riemann-complete Integration 313\u003c\/p\u003e \u003cp\u003e10.4 Convergence Criteria 318\u003c\/p\u003e \u003cp\u003e10.5 \\I would not care to y in that plane\" 324\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Appendix 2: Theorem 63 325\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Fresnel's Integral 325\u003c\/p\u003e \u003cp\u003e11.2 Theorem 188 of [MTRV] 330\u003c\/p\u003e \u003cp\u003e11.3 Some Consequences of Theorem 63 Fallacy 335\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Appendix 3: Option Pricing 337\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 American Options 337\u003c\/p\u003e \u003cp\u003e12.2 Asian Options 344\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Appendix 4: Listings 357\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Theorems 357\u003c\/p\u003e \u003cp\u003e13.2 Examples 358\u003c\/p\u003e \u003cp\u003e13.3 Definitions 360\u003c\/p\u003e \u003cp\u003e13.4 Symbols 360\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePAT MULDOWNEY, PHD\u003c\/b\u003e, was a lecturer at the Magee Business School at Ulster University for more than twenty years. He has published extensively in his areas of expertise: financial mathematics, random variation, Feynman path integrals, and integration theory.\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eA stand-alone introduction to specific integration problems in the probabilistic theory of stochastic calculus\u003c\/b\u003e\u003c\/p\u003e\u003cp\u003ePicking up where his previous book, \u003ci\u003eA Modern Theory of Random Variation\u003c\/i\u003e, left off, \u003ci\u003eGauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics\u003c\/i\u003e introduces readers to particular problems of integration in the probability-like theory of quantum mechanics.\u003c\/p\u003e\u003cp\u003eWritten 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 \u003ci\u003eA Modern Theory of Random Variation\u003c\/i\u003e in order to be understandable.\u003c\/p\u003e\u003cp\u003e\u003ci\u003eGauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics\u003c\/i\u003e takes a gradual, relaxed, and discursive approach to the subject in a successful attempt to engage the reader by exploring a narrower range of themes and problems.\u003c\/p\u003e\u003cp\u003eOrganized around examples with accompanying introductions and explanations, the book covers topics such as:\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eStochastic calculus, including discussions of random variation, integration and probability, and stochastic processes\u003c\/li\u003e\n\u003cli\u003eField theory, including discussions of gauges for product spaces and quantum electrodynamics\u003c\/li\u003e\n\u003cli\u003eRobust and thorough appendices, examples, illustrations, and introductions for each of the concepts discussed within\u003c\/li\u003e\n\u003cli\u003eAn introduction to basic gauge integral theory (for those unfamiliar with the author’s previous book)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eThe 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, \u003ci\u003eGauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics\u003c\/i\u003e is an illuminating and insightful exploration of the complex mathematical topics contained within.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989272576229,"sku":"NP9781119595496","price":128.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119595496.jpg?v=1761783467","url":"https:\/\/k12savings.com\/es\/products\/gauge-integral-structures-for-stochastic-calculus-and-quantum-electrodynamics-isbn-9781119595496","provider":"K12savings","version":"1.0","type":"link"}