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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.
Cambridge University Press978-1-108-42315-1 — Quantum Field Theory for Economics and FinanceBelal Ehsan Baaquie FrontmatterMore Information
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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
BELAL EHSAN BAAQUIEThe International Centre for Education in Islamic Finance
Cambridge University Press978-1-108-42315-1 — Quantum Field Theory for Economics and FinanceBelal Ehsan Baaquie FrontmatterMore Information
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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
7.1 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
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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
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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
12.1 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
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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
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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
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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
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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
24.1 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
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.
xvii
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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.