Quantum Field Theory for Economics and Finance Belal Ehsan Baaquie Frontmatter More Information

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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 quantum 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

QUANTUM FIELD THEORY FOR ECONOMICS AND FINANCE

BELAL EHSAN BAAQUIE The International Centre for Education in Islamic Finance

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

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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.

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.

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

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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

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

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 gauges 233 9.9 Faddeev–Popov quantization 234 9.10 Ghost state space and Hamiltonian 235 9.11 BRST charge QB 238 9.12 QB and state space 240 9.13 Summary 245 9.14 Appendix: Fermion calculus 246 10 Forward interest rates’ quantum field* 251 10.1 Introduction 251 10.2 Forward interest rates 252 10.3 Action and Lagrangian 254 10.4 Interest rate propagator 257 10.5 Forward interest rate covariance 260 10.6 Empirical forward interest rates 262 10.7 Time-dependent state space Vt 265 10.8 Time-dependent Hamiltonian 268 10.9 Martingale: Path integral 270 10.10 Martingale: Hamiltonian 272 10.11 Zero coupon bond option 275 10.12 Coupon bonds 277 10.13 Zero coupon bonds from coupon bonds 279 10.14 Forward interest rates from zero coupon bonds 281 10.15 Summary 285 11 Risky interest rates’ quantum fields* 286 11.1 Introduction 286 11.2 Risky forward interest rates 288 11.3 Correlation functions 290 11.4 Stiff propagator 292

Contents xi 11.5 Market correlators 293 11.6 Empirical volatility and propagators 296 11.7 Calibration of US and Singapore models 298 11.8 US-Singapore rates cross-term 300 11.9 Summary of calibration results 304 11.10 Risky coupon bond option 305 11.11 Option: Cumulant expansion 308 11.12 Interest rate swaptions 312 11.13 Summary 318 12 Bonds: Index-linked stochastic coupons* 319 12.1 Introduction 319 12.2 Stochastic coupon’s payoff function 321 12.3 Stochastic coupon’s pricing kernel 324 12.4 State space and Hamiltonian 327 12.5 Evolution kernel: Feynman path integral 330 12.6 Price of stochastic coupons 331 12.7 Martingale condition 332 12.8 Option for index-linked coupon bond 336 12.9 Bond put–call parity 341 12.10 Summary 343 Part III Nonlinear quantum fields 345 13 Operator expectation and S matrix 347 13.1 Introduction 347 13.2 Vacuum expectation values 348 13.3 S matrix: Definition 350 13.4 Incoming and outgoing states 351 13.5 Interaction representation 353 13.6 Scattering 357 13.7 LSZ formula and correlation functions 360 14 Nonlinear scalar field: Feynman diagrams 365 14.1 Introduction 365 14.2 ϕ4 theory: Lagrangian 366 14.3 Scale invariance 368 14.4 Wick’s theorem 369 14.5 Partition function 372 14.6 Connected correlation functions 373 14.7 Two-point correlation function 377

xii Contents 14.8 Four-point correlation function 378 14.9 Dimensional regularization 381 14.10 Two-loop regularized propagator 385 14.11 Vertex function 388 14.12 Divergences of Feynman diagrams 390 14.13 Summary 391 15 Renormalization 392 15.1 Introduction 392 15.2 Renormalization schemes 394 15.3 Bare perturbation theory 395 15.4 Mass and field renormalization 396 15.5 Minimal subtraction 397 15.6 Coupling constant renormalization 400 15.7 Change of scale μ 403 15.8 O(N) symmetric scalar field 404 15.9 Renormalization constants of ϕ4 theory 406 15.10 Renormalized perturbation theory 407 15.11 Momentum cutoff regularization 411 15.12 Background field method 414 15.13 Wilson renormalization 418 15.14 Thinning of degrees of freedom 421 15.15 Renormalizability to all orders 427 15.16 Superficial degree of divergence 432 15.17 Summary 433 16 β-function; fixed points 435 16.1 Introduction 435 16.2 Callan–Symanzik equation 436 16.3 Anomalous dimensions 440 16.4 β-function 442 16.5 Renormalization group 444 16.6 Physical mass and coupling constant 446 16.7 Wilson–Fisher infrared fixed point 447 16.8 β-function and fixed points 450 16.9 Fixed point and anomalous dimension 451 16.10 Summary 454 17 Renormalization group and phase transitions 455 17.1 Introduction 455 17.2 Renormalization group 457

Contents xiii 17.3 Wilson space 459 17.4 Fixed points and renormalizability 461 17.5 Classical phase transitions 466 17.6 Second-order phase transition 467 17.7 Landau–Ginzburg–Wilson Hamiltonian 469 17.8 Wilson–Fisher fixed point 471 17.9 Critical exponents 472 17.10 Fixed points and phase transitions 474 17.11 Phase transitions and Wilson space 475 17.12 Linearized RG equation 476 17.13 Wilson–Fisher fixed point: Topology 480 17.14 Summary 483 18 Effective action 485 18.1 Introduction 485 18.2 Effective action Ŵ: Definition 486 18.3 Classical action and Ŵ 487 18.4 Semi-classical expansion of Ŵ[ϕc] 488 18.5 Connected vertex functions 489 18.6 Loop expansion for the effective action 492 18.7 Effective potential Veff 495 18.8 Spontaneous symmetry breaking 497 18.9 ϕ4: One-loop effective action 501 18.10 Massless ϕ4: Effective potential 507 18.11 Massless scalar quantum electrodynamics 512 18.12 Lagrangian and gauge invariance 512 18.13 Effective potential: One loop 513 18.14 β-function; spontaneous symmetry breaking 519 18.15 Renormalization group invariance 521 18.16 Summary 521 19 Nonlinear interest rates’ quantum field* 524 19.1 Introduction 524 19.2 Libor: Simple interest rate 527 19.3 Wilson expansion and Libor drift 530 19.4 Libor Lagrangian 533 19.5 Libor Hamiltonian, martingale and drift 538 19.6 Swaps; swaptions 542 19.7 Black’s model for swaption 545 19.8 Summary 547

xiv Contents 20 Simulation of nonlinear interest rates* 548 20.1 Introduction 548 20.2 Simulating Libor 548 20.3 Lattice quantum fields Am,n and ϕm,n 553 20.4 Caplet 558 20.5 Caplet price: Numeraire invariant 560 20.6 Libor zero coupon bond option 564 20.7 Libor coupon bond option 569 20.8 Swaption 570 20.9 Summary 574 20.10 Appendix: An accurate expansion 576 21 Interest rate range accrual swap* 577 21.1 Introduction 577 21.2 Review of the Libor market model 578 21.3 Range accrual swap 582 21.4 Extension of Libor drift 586 21.5 Approximate price of range accrual swap 590 21.6 Simulation of range accrual swap 592 21.7 Numerical results and errors 594 21.8 Summary 596 Part IV Two-dimensional quantum fields 599 22 Two-dimensional quantum electrodynamics 601 22.1 Introduction 601 22.2 Euclidean action 602 22.3 Point-split regularization 606 22.4 Fermion path integral 608 22.5 Axial-vector current chiral anomaly 613 22.6 Wilson loop integral 615 22.7 Energy eigenvalues 616 22.8 Gauge-invariant states 621 22.9 String tension and fermion confinement 625 22.10 Summary 627 22.11 One-loop diagram 628 23 Bosonic string theory 630 23.1 Logic of superstrings 630 23.2 Introduction to open strings 633 23.3 Nambu–Goto string action 634

Contents xv 23.4 Covariant quantization 637 23.5 Virasoro algebra 642 23.6 BRST invariance 645 23.7 Physical bosonic state space 646 23.8 Summary 650 24 Futures asset prices* 651 24.1 Introduction 651 24.2 Modeling futures asset prices 652 24.3 Gaussian approximation 655 24.4 Propagator 656 24.5 Propagator for spot asset prices 660 24.6 Contour map of G(t, ξ;0,0) 662 24.7 Spot-spot rate G(t, t; t′, t′): Empirical and model 663 24.8 Spot-futures G(t, ξ;0,0): Empirical and model 665 24.9 Algorithm for empirical GE(z+, z−) 666 (k) 24.10 Binning of empirical DE (a, b, c) 670 24.11 Empirical results for GE(z+; z−) 672 24.12 Summary 673 24.13 Integral I(τ, θ) 674 24.14 Algorithm: Binning the propagator 675 25 Epilogue 677

References 680 Index 686

Foreword

I am aware of the author’s work in applying theories of physics to finance since 2003, and the present book is a logical outcome of the author’s line of thinking. The presentation of quantum field theory (QFT) given in this book is based on four strategic decisions. (1) From the very outset it introduces the notion of quantum mathematics. This immediately attracts the attention of readers, with regard two points. First, they realize that in order to feel at home in QFT, they must devote enough time and attention to mastering these techniques. Second, once they have them well in hand, they can also use them outside of physics because they are just mathematical techniques. (2) The book avoids giving applications of QFT to physics as this does not in the least help to understand QFT as a mathematical discipline. (3) Throughout the book the formalism of the Feynman path integral is used, which intuitively is indeed the most appealing formalism of QFT. (4) Last but not least, the book provides applications of QFT to a variety of economic and financial problems. One must realize that this is indeed quite different from calculations tied to high energy physics. Why? Needless to say, the whole machinery of QFT was created for applications to high energy physics; thus, one just follows the track and there is no need to raise any questions. On the contrary, QFT was not created to price options. Thus, instead of just following the track, at each step we have to modify and adapt our understanding of the mathematical tools of QFT. The book has three distinctive features that are worth highlighting. (1) There are many books on QFT, but this is a ground-breaking book that connects QFT with concepts in economics and finance. (2) Almost half the book is devoted to studying models of economics and finance. As the book proceeds with different topics of QFT, chapters on economics and finance are introduced to show the close mathematical connections between these domains of knowledge.

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xviii Foreword (3) Many of the applications to economics and finance are based on models that can be empirically tested. To me, the most remarkable aspect of the book is that empirical tests show that these models are surprisingly accurate. Going through the applications of QFT is a highly rewarding exercise as it tests our degree of understanding and expands our view of QFT. When readers grasp the logic of the applications, it will bolster their self-confidence and make them feel at home with QFT, and empower them to apply the mathematics of QFT to new fields of inquiry. Bertrand M. Roehner Laboratoire de Physique Theorique´ et Hautes Energies (LPTHE) University of Paris 6 Paris, France

Preface

Quantum field theory is undoubtedly one of the most accurate and important scientific theories in the history of science. Relativistic quantum fields are the theoretical backbone of the Standard Model of particles and interactions. Relativistic and non- relativistic quantum fields are extensively used in myriad branches of theoretical physics, from superstring theory, high energy physics and solid state physics to condensed matter, quantum optics, nuclear physics, astrophysics and so on. The mathematics that emerges from the formalism of quantum mechanics and quantum field theory is quite distinct from other branches of mathematics and is termed quantum mathematics. Quantum mathematics is a synthesis of linear algebra, calculus of infinitely many independent variables, functional analysis, operator algebras, infinite-dimensional linear vector spaces, the theory of probability, Lie groups, geometry, topology, functional integration and so on. One of the mathematical bedrocks of quantum mechanics and quantum field theory is the Feynman path integral [Baaquie (2014)]. Unlike functional integration in general, the Feynman path integral is a functional integral with another key feature, which is that the path integral is constructed out of an underlying (infinite- dimensional) linear vector space. Operators are defined on this vector space, including the central operator of theoretical physics, which is the Hamiltonian. The first application of calculus – made by Newton – was in the study of the dynamics of particles; calculus subsequently has gone on to become the universal language of quantitative modeling. Similarly, although quantum mathematics emerges from the study of quantum phenomena that are intrinsically indeterminate, the mathematical structure is not tied to its origins. Examples discussed below show that the mathematics of quantum field theory extends far beyond only quantum systems and can also be applied to a wide variety of subjects that span natural and social sciences. It is my view that quantum mathematics will, in time, supersede calculus and become the universal framework for quantitative modeling and mathematical thinking.

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xx Preface Important applications of quantum mathematics outside quantum physics have been made in vastly different fields, resulting in many ground-breaking results. Quantum mathematics has been applied to many classical problem; two famous examples are (1) the solution of classical phase transitions by Wilson, which led to his Nobel prize in physics in 1982 [Wilson (1983)] and (2) the complete classification of knots and links in three dimensions by Witten, for which he was awarded the Fields medal in 1989 [Witten (1989)]. More recently, superstring theory has led to a plethora of new results in pure mathematics using quantum mathematics. In fact, it would be no exaggeration to state that superstring theory has opened hitherto uncharted domains of pure mathematics of higher dimensions [Polchinski (1998); Zwiebach (2009)]. The formalism of quantum finance has been developed in this spirit and is based on the application of quantum mathematics to finance [Sornette (2003); Baaquie (2004, 2010)]. Two-dimensional quantum fields have been applied by Baaquie (2010) for analyzing interest rates and coupon bonds. Applications to economics has been made by Baaquie (2013a), and Baaquie and Yu (2018) have utilized a two-dimensional quantum field to describe and model futures asset prices. The bedrock of the application of quantum mathematics to both finance and economics is the employment of the Feynman path integral for modeling the behavior of interest rates and of spot as well as futures asset prices The application of ideas from physics to economics and finance has led to the creation of a new field called econophysics, and to which quantum finance belongs [Mantegna and Stanley (1999); Roehner (2002a)]. Applications to psychology [Baaquie and Martin (2005)], to the social sciences [Haven and Khrennikov (2013)] and to decision sciences [Busemeyer and Bruza (2012)], to name a few, show the increasing utility of quantum mathematics in quantitative studies of social phenomena. Many universities, institutes and centers are teaching courses on the applications of quantum mathematics, and researchers are finding new and unexpected applications of quantum mathematics. For instance, the Institute of Quantum Social and Cognitive Science “promotes and develops high level research on the identification of quantum structures in non-physical domains, in particular, in socio-economic and cognitive sciences. The employment of the mathematical formalism of quantum mechanics outside the microscopic world is a growing research field and it has rapidly attracted the interest of the scientific community and the media.”1 Quantum mathematics needs to be made accessible to a wide readership – beyond science, mathematics and engineering – so that students and researchers from all fields of study, including the social sciences, can employ the mathematical

1 www2.le.ac.uk/departments/business/research/units/iqscs.

Preface xxi tools of quantum mathematics. Only with the knowledge of quantum mathematics being widespread can it fulfill its potential and, like calculus, become the sine qua non of all fields of quantitative modeling. The mathematics of quantum mechanics has been discussed by Baaquie (2014) in Path Integrals and Hamiltonians. In contrast to the study of quantum mechanics, this book is an introduction to the mathematics of quantum field theory.What distinguishes quantum field theory from quantum mechanics is the coupling of infinitely many variables, or infinitely many degrees of freedom. The main purpose of this book is to introduce the mathematics of quantum field theory to researchers in finance and economics. The topics chosen are geared toward imparting the mathematical tools of quantum field theory that can facilitate further studies of finance and economics. This book provides a quick and simple primer to quantum field theory and can also be used as an introductory graduate text for readers from science, mathematics and engineering who are not specializing in theoretical physics. A quantum field has quantum indeterminacy, whereas a classical stochastic field has classical randomness. The subtle difference between these is the subject of measurement theory in quantum mechanics [Baaquie (2013b)]. All the applications of quantum fields to economics and finance are in fact the application of stochastic fields; however, since the mathematics of stochastic and quantum fields are identi- cal, the generic term “quantum field” is used for all applications of quantum fields to domains outside quantum physics. How can one introduce quantum mathematics to students, readers and researchers unfamiliar with quantum field theory? Unlike topics in mathematics, such as calculus, that have a well-defined syllabus, quantum fields cannot be so neatly modularized. Given the vast and increasingly complex mathematics of quantum fields, it is virtually impossible for one book to cover the entire terrain of quantum field theory. This book presumes a working knowledge of linear algebra, calculus and probability theory. All the derivations are done from first principles and are comprehensive; there is no need to refer to any material outside this book. In order to introduce quantum field theory to readers from “distant” subjects, some of the leading examples of quantum fields are studied in detail. Starting from simpler examples, the various chapters lay the groundwork for analyzing more advanced topics. These examples encode many of the leading ideas of quantum field theory and are the building blocks of more advanced models. To make the applications of quantum mathematics to economics and finance more tangible, the chapters on economics and finance are interwoven with chapters on quantum fields. In this manner, the reader can directly examine and connect the ideas of quantum field theory with its application, and in particular can see how

xxii Preface these ideas are carried over to economics and finance. About 60% of the material of the book is directly an exposition of quantum field theory, with the remaining chapters being focused on its various applications to economics and finance. The manner of presentation of the two pillars of the book – quantum fields on the one hand and economics and finance on the other– is quite different. Quantum field theory needs no empirical evidence for its utility and validity since the entire domain of particle physics stands as a testament to its empirical success. Hence the focus in the chapters on quantum field theory is on the various mathematical ideas and derivations, and only a fleeting connection is made with other subjects. An integral and pure presentation of quantum field theory is necessary to show that it is free from a bias toward any specific application. In fact, if one skips the Chapters on economics and finance, which are marked by an asterisk, the book then reads as an introductory graduate text on quantum field theory. Unlike mathematics, which has results of great generality, such as theorems and lemmas, one only needs to flip through the pages of a textbook on quantum mechanics or quantum field theory to see that there are no theorems in quantum physics; instead, what one has are leading models and important examples – with the mathematical analysis flowing naturally in interpreting, explaining and deriving the “physics” of these models. Quantum field theory is illustrated and elaborated on by analyzing a number of exemplary models, such as the scalar, vector and spinor fields. Each of these quantum fields is described by a specific Lagrangian and Hamiltonian – and has distinctive properties on which the book focuses. More advanced chapters such as the structure of the renormalization group are presented later, when the reader has a better grasp of the underlying ideas. The methodology of the chapters on economics and finance is quite different from the chapters on quantum field theory. In my view, the only justification for the application of quantum mathematics to empirical disciplines outside quantum physics – including economics and finance – is that it must be supported by empirical evidence. In the absence of such evidence – and there are many papers and books that make conceptual connections between quantum mathematics and classical systems with little or no empirical evidence [Bagarello (2013)] – the application in my view is still not complete, and stands only as an interesting mathematical metaphor. For the metaphor to become a concrete mathematical model, empirical evidence is indispensable. For this reason, topics from economics and finance have been chosen (for inclu- sion in the book) that have empirical support from market data. Furthermore, a detailed analysis is given on how these quantum mathematical models are adapted to the market – and subsequently calibrated and tested. In chapters on economics and finance, very specific and concrete theoretical models are analyzed – all based on path integrals and Hamiltonians. The introductory chapter on nonlinear interest

Preface xxiii rates concentrates on the formalism; the reason is that a quantum finance model of nonlinear interest rates, as realized by the London Interbank Offered Rate (Libor), has been calibrated and exhaustively tested using market data; hence, only certain key features of the formalism are discussed. Two chapters use numerical algorithms and simulations to study nonlinear interest rates; these chapters illustrate a key feature of nonlinear interest rates, which is that in most cases numerical techniques are necessary for obtaining a solution. The models that have been proposed in economics and finance – all of which are based on work done by myself and collaborators – are quite distinct from those that appear in quantum physics. In particular, all the models in economics and finance have an “acceleration” kinetic term – a term forbidden in quantum mechanics (due to the violation of conservation of probability); it is this term that gives a flavor to all the results in economics and finance that is quite different from what one is familiar with in physics. The derivations in this book are not tied down to the application of quantum fields to physics – as this would require concepts that are not necessary for understanding the mathematical formalism of quantum fields. Furthermore, topics that apparently have no connection with finance or economics – but have played a pivotal role in quantum field theory – have been included in the hope that these ideas may lay lead to ground-breaking theories and models in economics and finance. Nonlinearities of quantum fields arise due to self-interactions or because of coupling to other fields – and require the procedure of renormalization for obtaining finite results. The canonical case of a self-interacting nonlinear scalar field is studied in great detail so as to illustrate and analyze the issues that arise in renormalization. The formalism of quantum field theory culminates in the concepts of renormalization, renormalizability and the renormalization group – and which are among the deepest ideas of quantum field theory. It has been shown by Sornette (2003) that ideas from the renormalization group can provide a mathematical framework for understanding, and even predicting, market meltdowns. Many topics, such as fermions, spinors, ghost fields, bosonic strings and gauge- fixing, are discussed that may seem to have no connection with economics and finance. The reason for including these topics is intentional. The broad range of topics covered gives a flavor to the reader of the great variety and complexity of the models that are a part of quantum field theory. A major omission has been the study of Yang–Mills gauge fields and that of spacetime supersymmetry. These topics need a background far in advance to what has been assumed, and hence could not be covered. It is impossible and unwise to try to second guess what future directions economics and finance will turn toward; furthermore, gearing the topics discussed

xxiv Preface toward what is known closes off many future applications. For this reason, the main thrust of this book is to make the reader aware of, and familiar with, a wide array of quantum mathematical models so that a researcher can make leading edge connections and create new pathways between the domains of quantum ﬁelds and economics and ﬁnance.

Acknowledgments

I would like to acknowledge and express my thanks to many outstanding teachers, scholars and researchers whose work motivated me to study quantum field theory and to grapple with its mathematical formalism. I had the singular privilege of doing my PhD thesis under the guidance of Kenneth G. Wilson; his visionary conception of quantum mechanics and of quantum field theory greatly enlightened and inspired me, and continues to do so till today. As an undergraduate I had the honor of meeting and conversing a number of times with Richard P. Feynman, which left a permanent impression on me. I have benefited from discussions with many physicists and field theorists, including but not limited to, in no particular order, Abdus Salam, John Strathdee, M. Ali Namazie, Jamal Nazrul Islam, John Kogut, Leonard Susskind, Michael Fisher, Kurt Gottfried, H. R. Krishnamurthy, Michael Peskin, Steve Shenker, Spenta Wadia, Ashoke Sen, Tan Meng Chwan, Ahmed Zewail, S. G. Rajeev, Cumrun Vafa, Edward Witten, David Gross, Steven Weinberg, Jack Ng, Lay Nam Chang, Frank Wilczek, Nee Pong Chang, Anthony J. Leggett, Sidney Drell, Marvin Weinstein, Phua Kok Khoo, Andrew Strominger, Jayant Narlikar, Juan Maldecena, Bambi Hu, Chen Ning Yang, Bertrand Roehner, Murray Gell-Mann, Gerard ’t Hooft, James D. Bjorken, Gyan Bhanot, Nicolas Sourlas, Faheem Hussain, Avinash Dhar, Asghar Qadir, Robert C. Richardson, Francois Martin, Kerson Huang, Syed Wassim, Jean Philippe Bouchaud, Munawar Karim, Brian Miller, Obiyathulla Ismath Bacha, Oh Choo Hiap, Michael Spalinski, Andrew Strominger, Baharom Abdul Hamid, Lai Choy Heng, Mohamed Eskandar Shah Mohd Rasid, Thomas Osipowicz, Emmanuel Haven, Sandro Sozzo, Claudio Coriano, Abbas Mirakhor and many others. I thank Ashok Das, Yim Kok Kean, Lee Jianwei, Pan Tang, Jufri Setianegara, Cui Liang, Xin Du, L.C. Kwek, Yu Miao and Cao Yang for many helpful discussions. I thank Mahbub Majumdar and Mazhar Kantakji for their input in the preparation of the book.

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xxvi Acknowledgments I owe a special vote of thanks to Wang Qinghai; our enjoyable and varied discussions over many years led to greater clarity about the key concepts of quantum ﬁelds. The presentation of many of the chapters is the result of our discussions. I specially thank Frederick H. Willeboordse for his valuable advice on how to position this book, and for his consistent support. I would like to thank to Simon Capelin of Cambridge University Press for his valuable advice on how to present the material of this book. I thank my wife, Najma, for being an indispensable presence, a wonderful com- panion and for her uplifting approach to family and professional life. I thank my precious family members Arzish, Farah and Tazkiah for their delightful company and warm encouragement. Their love, affection and support over many years has made this book possible.