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Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information QUANTUM FIELD THEORY FOR ECONOMICS AND FINANCE An introduction to how the mathematical tools from quantum field theory can be applied to economics and finance, this book provides a wide range of quan- tum mathematical techniques for designing financial instruments. The ideas of Lagrangians, Hamiltonians, state spaces, operators and Feynman path integrals are demonstrated to be the mathematical underpinning of quantum field theory and are employed to formulate a comprehensive mathematical theory of asset pricing as well as of interest rates, which are validated by empirical evidence. Numerical algorithms and simulations are applied to the study of asset pricing models as well as of nonlinear interest rates. A range of economic and financial topics is shown to have quantum mechanical formulations, including options, coupon bonds, nonlinear interest rates, risky bonds and the microeconomic action functional. This is an invaluable resource for experts in quantitative finance and in mathematics who have no specialist knowledge of quantum field theory. belal ehsan baaquie is a professor at the International Centre for Education in Islamic Finance. He received his training in theoretical physics at Caltech and Cornell University, specializing in quantum field theory. He later developed an interest in finance and economics, and started applying quantum mathematics to these fields. He has written two books on quantum finance: Quantum Finance (Cambridge University Press, 2007) and Interest Rates and Coupon Bonds in Quantum Finance (Cambridge University Press, 2009), in addition to several other books focusing on topics from quantum mechanics and mathematics to books on leading ideas in science. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information QUANTUM FIELD THEORY FOR ECONOMICS AND FINANCE BELAL EHSAN BAAQUIE The International Centre for Education in Islamic Finance © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108423151 DOI: 10.1017/9781108399685 © Cambridge University Press 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Baaquie, B. E., author. Title: Quantum field theory for economics and finance / Belal Ehsan Baaquie, The International Centre for Education in Islamic Finance, Singapore. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2017061436 | ISBN 9781108423151 (Hardback : alk. paper) Subjects: LCSH: Economics–Mathematical models. | Finance–Mathematical models. | Quantum field theory. Classification: LCC HB135 .B28 2018 | DDC 330.01/530143–dc23 LC record available at https://lccn.loc.gov/2017061436 ISBN 978-1-108-42315-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information This book is dedicated to all the scholars, thinkers and visionaries who have been striving and contributing to enhance the knowledge and wisdom of humanity. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information Contents Foreword page xvii Preface xix Acknowledgments xxv 1 Synopsis 1 1.1 Organization of the book 2 1.2 What is a quantum field? 8 Part I Introduction 11 2 Quantum mechanics 13 2.1 Introduction 13 2.2 Quantum principles 14 2.3 Theory of measurement 16 2.4 Dirac delta function 17 2.5 Schrodinger¨ and Heisenberg formalism 19 2.6 Feynman path integral 20 2.7 Hamiltonian and path integral 23 2.8 Hamiltonian from Lagrangian 24 2.9 Summary 27 2.10 Appendix: Dirac bracket and vector notation 28 2.11 Appendix: Gaussian integration 30 3 Classical field theory 35 3.1 Introduction 35 3.2 Lagrangian mechanics 36 3.3 Classical field equation 38 3.4 Free scalar field 40 3.5 Symmetries 41 3.6 Noether’s theorem 43 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information viii Contents 3.7 Stress tensor 44 3.8 Spontaneous symmetry breaking 46 3.9 Landau–Ginzburg Lagrangian 49 3.10 Higgs mechanism 54 3.11 Lorentz group 56 3.12 Relativistic fields 58 3.13 Summary 62 4 Acceleration action 63 4.1 Action and Hamiltonian 63 4.2 Transition amplitude: Hamiltonian 64 4.3 Limiting case: τ = 069 4.4 Transition amplitude: Path integral derivation 70 4.5 Summary 76 5 Option theory* 77 5.1 Introduction 77 5.2 Options on a security 77 5.3 European call and put option 78 5.4 Quantum mechanical pricing of options 80 5.5 Martingale condition: Hamiltonian 82 5.6 Hamiltonian and option pricing 82 5.7 Black–Scholes Hamiltonian: Pricing kernel 85 5.8 Black–Scholes option price 87 5.9 Option pricing 88 5.10 Option price: Baaquie–Yang (BY) model 90 5.11 Martingale: Conditional probability 91 5.12 Market time 92 5.13 Empirical results 93 5.14 FX options and market instability 97 5.15 Summary 100 6 Path integral of asset prices* 101 6.1 Introduction 101 6.2 Microeconomic potential 103 6.3 Microeconomic action functional 105 6.4 Equilibrium asset prices 108 6.5 Feynman perturbation expansion 111 6.6 Nonlinear terms: Feynman diagrams 116 6.7 Normalization 118 6.8 Path integral: Monte Carlo simulation 121 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information Contents ix 6.9 Mapping model to market data 125 6.10 Equity and FX rates 127 6.11 Single and multiple commodities fit 130 6.12 Empirical microeconomic potential 131 6.13 Multiple commodities and market data 133 6.14 Commodity coupling coefficient ij 134 6.15 Fits for GII, GIJ 135 6.16 Summary 139 (0) 6.17 Appendix: Derivation of DIJ 142 Part II Linear quantum fields 147 7 Scalar quantum field 149 7.1 Introduction 149 7.2 Two-dimensional quantum field 150 7.3 Fock space 151 7.4 Scalar field: Hamiltonian 160 7.5 Heisenberg operators 161 7.6 Fourier expansion 163 7.7 Creation and destruction operators 165 7.8 Energy eigenstates 167 7.9 Schrodinger wave functional 168 7.10 Unequal time commutation equations 169 7.11 The Feynman propagator 172 7.12 Complex scalar field 176 7.13 Free scalar field: Path integral 179 7.14 Continuation to Euclidean time 181 7.15 Summary 185 8 Dirac spinor field 187 8.1 Introduction 187 8.2 Dirac equation 188 8.3 Dirac Lagrangian and Hamiltonian 190 8.4 Euclidean Dirac Lagrangian 192 8.5 Plane wave solutions 192 8.6 Quantization 195 8.7 Positrons: Hole theory 198 8.8 Antiparticles 199 8.9 Hamiltonian, charge and momentum 200 8.10 Charge conjugation 202 8.11 Dirac field: Casimir force 205 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-42315-1 — Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information x Contents 8.12 Casimir force: (Anti-)periodic 210 8.13 Summary 213 9 Photon gauge field 215 9.1 Introduction 215 9.2 Hamiltonian and state space 216 9.3 Hamiltonian gauge-fixing: Coulomb gauge 219 9.4 Coulomb gauge: Normal modes 223 9.5 Gauge symmetry and path integral 227 9.6 Gauge-fixing the action 228 9.7 Ghost field: Finite time path integral 231 9.8 Feynman and Landau
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