Masujima M. Applied Mathematical Methods in Theoretical Physics
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Michio Masujima
Applied Mathematical Methods
in Theoretical Physics
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Michio Masujima
Applied Mathematical Methods
in Theoretical Physics
Second, Enlarged and Improved Edition
The Author
Dr. Michio Masujima
Tokyo, Japan
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2009 WILEY-VCH Verlag GmbH & Co.
KGaA, Weinheim
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ISBN: 978-3-527-40936-5
To my granddaughter, Honoka
VII
Contents
Preface XI
Introduction XV
1 Function Spaces, Linear Operators, and Green’s Functions 1
1.1 Function Spaces 1
1.2 Orthonormal System of Functions 3
1.3 Linear Operators 5
1.4 Eigenvalues and Eigenfunctions 7
1.5 The Fredholm Alternative 9
1.6 Self-Adjoint Operators 12
1.7 Green’s Functions for Differential Equations 14
1.8 Review of Complex Analysis 18
1.9 Review of Fourier Transform 25
2 Integral Equations and Green’s Functions 31
2.1 Introduction to Integral Equations 31
2.2 Relationship of Integral Equations with Differential Equations and
Green’s Functions 37
2.3 Sturm–Liouville System 43
2.4 Green’s Function for Time-Dependent Scattering Problem 47
2.5 Lippmann–Schwinger Equation 51
2.6 Scalar Field Interacting with Static Source 62
2.7 Problems for Chapter 2 67
3 Integral Equations of the Volterra Type 105
3.1 Iterative Solution to Volterra Integral Equation of the Second Kind 105
3.2 Solvable Cases of the Volterra Integral Equation 108
3.3 Problems for Chapter 3 112
Applied Mathematical Methods in Theoretical Physics, Second Edition. Michio Masujima
Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-40936-5
VIII Contents
4 Integral Equations of the Fredholm Type 117
4.1 Iterative Solution to the Fredholm Integral Equation of the Second
Kind 117
4.2 Resolvent Kernel 120
4.3 Pincherle–Goursat Kernel 123
4.4 Fredholm Theory for a Bounded Kernel 127
4.5 Solvable Example 134
4.6 Fredholm Integral Equation with a Translation Kernel 136
4.7 System of Fredholm Integral Equations of the Second Kind 143
4.8 Problems for Chapter 4 143
5 Hilbert–Schmidt Theory of Symmetric Kernel 153
5.1 Real and Symmetric Matrix 153
5.2 Real and Symmetric Kernel 155
5.3 Bounds on the Eigenvalues 166
5.4 Rayleigh Quotient 169
5.5 Completeness of Sturm–Liouville Eigenfunctions 172
5.6 Generalization of Hilbert–Schmidt Theory 174
5.7 Generalization of the Sturm–Liouville System 181
5.8 Problems for Chapter 5 187
6 Singular Integral Equations of the Cauchy Type 193
6.1 Hilbert Problem 193
6.2 Cauchy Integral Equation of the First Kind 197
6.3 Cauchy Integral Equation of the Second Kind 201
6.4 Carleman Integral Equation 205
6.5 Dispersion Relations 211
6.6 Problems for Chapter 6 218
7 Wiener–Hopf Method and Wiener–Hopf Integral Equation 223
7.1 The Wiener–Hopf Method for Partial Differential Equations 223
7.2 Homogeneous Wiener–Hopf Integral Equation of the Second
Kind 237
7.3 General Decomposition Problem 252
7.4 Inhomogeneous Wiener–Hopf Integral Equation of the Second
Kind 261
7.5 Toeplitz Matrix and Wiener–Hopf Sum Equation 272
7.6 Wiener–Hopf Integral Equation of the First Kind and Dual Integral
Equations 281
7.7 Problems for Chapter 7 285
8 Nonlinear Integral Equations 295
8.1 Nonlinear Integral Equation of the Volterra Type 295
8.2 Nonlinear Integral Equation of the Fredholm Type 299
8.3 Nonlinear Integral Equation of the Hammerstein Type 303
8.4 Problems for Chapter 8 305
Contents IX
9 Calculus of Variations: Fundamentals 309
9.1 Historical Background 309
9.2 Examples 313
9.3 Euler Equation 314
9.4 Generalization of the Basic Problems 319
9.5 More Examples 323
9.6 Differential Equations, Integral Equations, and Extremization of
Integrals 326
9.7 The Second Variation 330
9.8 Weierstrass–Erdmann Corner Relation 345
9.9 Problems for Chapter 9 349
10 Calculus of Variations: Applications 353
10.1 Hamilton–Jacobi Equation and Quantum Mechanics 353
10.2 Feynman’s Action Principle in Quantum Theory 361
10.3 Schwinger’s Action Principle in Quantum Theory 368
10.4 Schwinger–Dyson Equation in Quantum Field Theory 371
10.5 Schwinger–Dyson Equation in Quantum Statistical Mechanics 385
10.6 Feynman’s Variational Principle 395
10.7 Poincare Transformation and Spin 407
10.8 Conservation Laws and Noether’s Theorem 411
10.9 Weyl’s Gauge Principle 418
10.10 Path Integral Quantization of Gauge Field I 437
10.11 Path Integral Quantization of Gauge Field II 454
10.12 BRST Invariance and Renormalization 468
10.13 Asymptotic Disaster in QED 475
10.14 Asymptotic Freedom in QCD 479
10.15 Renormalization Group Equations 487
10.16 Standard Model 499
10.17 Lattice Gauge Field Theory and Quark Confinement 518
10.18 WKB Approximation in Path Integral Formalism 523
10.19 Hartree–Fock Equation 526
10.20 Problems for Chapter 10 529
References 567
Index 573
XI
Preface
This book on integral equations and the calculus of variations is intended for use
by senior undergraduate students and first-year graduate students in science and
engineering. Basic familiarity with theories of linear algebra, calculus, differential
equations, and complex analysis on the mathematics side, and classical mechanics,
classical electrodynamics, quantum mechanics including the second quantization,
and quantum statistical mechanics on the physics side is assumed. Another
prerequisite on the mathematics side for this book is a sound understanding of
local analysis and global analysis.
This book grew out of the course notes for the last of the three-semester sequence
of Methods of Applied Mathematics I (Local Analysis), II (Global Analysis) and
III (Integral Equations and Calculus of Variations) taught in the Department of
Mathematics at MIT. About two thirds of the course is devoted to integral equations
and the remaining one third to the calculus of variations. Professor Hung Cheng
taught the course on integral equations and the calculus of variations every other
year from the mid-1960s through the mid-1980s at MIT. Since then, younger faculty
have been teaching the course in turn. The course notes evolved in the intervening
years. This book is the culmination of these joint efforts.
There will be a natural question: Why now another book on integral equations and
the calculus of variations? There exist many excellent books on the theory of integral
equations. No existing book, however, discusses the singular integral equations
in detail, in particular, Wiener–Hopf integral equations and Wiener–Hopf sum
equations with the notion of the Wiener–Hopf index. In this book, the notion of
the Wiener–Hopf index is discussed in detail.
This book is organized as follows. In Chapter 1, we discuss the notion of
function space, the linear operator, the Fredholm alternative, and Green’s functions,
preparing the reader for the further development of the material. In Chapter 2, we
discuss a few examples of integral equations and Green’s functions. In Chapter 3,
we discuss integral equations of the Volterra type. In Chapter 4, we discuss integral
equations of the Fredholm type. In Chapter 5, we discuss the Hilbert–Schmidt
theories of symmetric kernel. In Chapter 6, we discuss singular integral equations
of the Cauchy type. In Chapter 7, we discuss the Wiener–Hopf method for
the mixed boundary-value problem in classical electrodynamics, Wiener–Hopf
integral equations, and Wiener–Hopf sum equations, the latter two topics being
Applied Mathematical Methods in Theoretical Physics, Second Edition. Michio Masujima
Copyright
2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-40936-5
XII Preface
discussed in terms of the notion of index. In Chapter 8, we discuss nonlinear
integral equations of the Volterra type, Fredholm type, and Hammerstein type.
In Chapter 9, we discuss calculus of variations, covering the topics on the second
variations, Legendre test, Jacobi test, and relationship with the theory of integral
equations. In Chapter 10, we discuss the Hamilton–Jacobi equation and quantum
mechanics, Feynman’s action principle, Schwinger’s action principle, system of
Schwinger–Dyson equation in quantum theory, Feynman’s variational principle
and polaron, Poincar
´
e transformation and spin, conservation law and Noether’s
theorem, Weyl’s gauge principle, the path integral quantization of non-Abelian
gauge fields, renormalization of non-Abelian gauge fields, asymptotic disaster
(asymptotic freedom) of Abelian gauge field (non-Abelian gauge field) interacting
with fermions with tri- approximation, renormalization group equation, standard
model, lattice gauge field theory, WKB method, and Hartree–Fock equation.
Chapter 10 is taken from my book, titled Path Integrals and Stochastic Processes in
Theoretical Physics, Feshbach Publishing, Minnesota.
Reasonable understanding of Chapter 10 requires the reader to have a basic
understanding of classical mechanics, classical field theory, classical electrody-
namics, quantum mechanics including the second quantization, and quantum
statistical mechanics. For this reason, Chapter 10 can be read as a side reference
on theoretical physics.
The examples are mostly taken from classical mechanics, classical field theory,
classical electrodynamics, quantum mechanics, quantum statistical mechanics,
and quantum field theory. Most of them are worked out in detail to illustrate the
methods of the solutions. Those examples which are not worked out in detail are
either intended to illustrate the general methods of the solutions or left to the
reader to complete the solutions.
At the end of each chapter with the exception of Chapter 1, problem sets are
given for sound understanding of the contents of the main text. The reader is
recommended to solve all the problems at the end of each chapter. Many of the
problems were created by Professor Hung Cheng during the past three decades.
Theproblemsduetohimaredesignatedwiththenote(duetoH.C.).Someof
the problems are those encountered by Professor Hung Cheng in the course of his
own research activity.
Most of the problems can be solved with the direct application of the method
illustrated in the main text. Difficult problems are accompanied with the citation
of the original references. The problems for Chapter 10 are mostly taken from
classical mechanics, classical electrodynamics, quantum mechanics, quantum
statistical mechanics, and quantum field theory.
Bibliography is provided at the end of the book for the in-depth study of the
background materials in physics besides the standard references on the theory of
integral equations and the calculus of variations.
The instructor can cover Chapters 1 through 9 in one semester or two quarters
with a choice of the topics of his or her own taste from Chapter 10.
I would like to express many heart-felt thanks to Professor Hung Cheng at
MIT, who appointed me as his teaching assistant for the course when I was a