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Cambridge University Press978-1-108-42315-1 — Quantum Field Theory for Economics and FinanceBelal Ehsan Baaquie FrontmatterMore Information

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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.

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QUANTUM FIELD THEORY FOR

ECONOMICS AND FINANCE

BELAL EHSAN BAAQUIEThe International Centre for Education in Islamic Finance


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University Printing House, Cambridge CB2 8BS, United Kingdom

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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.


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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.


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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 = 0 69

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


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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

6.17 Appendix: Derivation of D(0)IJ 142

Part II Linear quantum fields 147

7 Scalar quantum field 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


8.2 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


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x Contents

8.12 Casimir force: (Anti-)periodic 210

8.13 Summary 213

9 Photon gauge field 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


10.2 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


11.2 Risky forward interest rates 288

11.3 Correlation functions 290

11.4 Stiff propagator 292


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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


12.2 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


13.2 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


14.2 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


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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


15.2 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


16.2 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


17.2 Renormalization group 457


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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


18.2 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


19.2 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


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xiv Contents

20 Simulation of nonlinear interest rates* 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


21.2 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


22.2 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


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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


24.2 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

24.10 Binning of empirical D(k)E (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


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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 real-

ize 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


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Preface

Quantum field theory is undoubtedly one of the most accurate and important scien-

tific 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 univer-

sal 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 math-

ematical thinking.

xix


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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 classifica-

tion 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.


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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 prob-

ability theory. All the derivations are done from first principles and are compre-

hensive; 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


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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 empiri-

cal evidence. In the absence of such evidence – and there are many papers and books

that make conceptual connections between quantum mathematics and classical sys-

tems 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


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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 mathemat-

ical 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 eco-

nomics and finance will turn toward; furthermore, gearing the topics discussed


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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.


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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.

xxv


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xxvi Acknowledgments

I owe a special vote of thanks to Wang Qinghai; our enjoyable and varied dis-

cussions 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.


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