Applied Mathematical Methods in Theoretical Physics

All there is to know about functional analysis, integral equations and calculus of variations in one handy volume, written for the specific needs of physicists and applied mathematicians.

The new edition of this handbook starts with a short introduction to functional analysis, including a review of complex analysis, before continuing a systematic discussion of different types of integral equations. After a few remarks on the historical development, the second part provides an introduction to the calculus of variations and the relationship between integral equations and applications of the calculus of variations. It further covers applications of the calculus of variations developed in the second half of the 20th century in the fields of quantum mechanics, quantum statistical mechanics and quantum field theory.

Throughout the book, the author presents a wealth of problems and examples often with a physical background. He provides outlines of the solutions for each problem, while detailed solutions are also given, supplementing the materials discussed in the main text. The problems can be solved by directly applying the method illustrated in the main text, and difficult problems are accompanied by a citation of the original references.

Highly recommended as a textbook for senior undergraduates and first-year graduates in science and engineering, this is equally useful as a reference or self-study guide.

Der Schwerpunkt dieses Lehrbuchs der angewandten Mathematik liegt auf den Bedürfnissen in der Praxis tätiger Physiker und Mathematiker. An die ausführliche Erläuterung der Grundlagen von Funktionalanalysis, Integralgleichungen und Variationsrechnung schließen sich 300 Seiten konkreter Anwendungsbeispiele aus dem physikalisch-technischen Umfeld an.

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

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

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

Michio Masujima, born in 1947, studied physics and mathematics at the Massachusetts Institute of Technology and Stanford University. He received his PhD in mathematics from the MIT in 1983. Dr. Masujima worked for many years at the NEC Fundamental Research Laboratory in Japan, where he was in charge of computational physics, and later as a lecturer at the NEC Junior Technical College, where he was responsible for the subjects mathematics and physics. Dr. Masujima works currently in private enterprise. All there is to know about functional analysis, integral equations and calculus of variations in one handy volume, written for the specific needs of physicists and applied mathematicians.

The new edition of this handbook starts with a short introduction to functional analysis, including a review of complex analysis, before continuing a systematic discussion of different types of integral equations. After a few remarks on the historical development, the second part provides an introduction to the calculus of variations and the relationship between integral equations and applications of the calculus of variations. It further covers applications of the calculus of variations developed in the second half of the 20th century in the fields of quantum mechanics, quantum statistical mechanics and quantum field theory.

Throughout the book, the author presents a wealth of problems and examples, often with a physical background. He provides outlines of the solutions for each problem, while detailed solutions are also given, supplementing the materials discussed in the main text. The problems can be solved by directly applying the method illustrated in the main text, and difficult problems are accompanied by a citation of the original references.

Highly recommended as a textbook for senior undergraduates and first-year graduates in science and engineering, this is equally useful as a reference or self-study guide.

From the contents:

• Function Spaces, Linear Operators and Green's Functions
  
• Integral Equations and Green's Functions
  
• Integral Equations of the Volterra Type
  
• Integral Equations of the Fredholm Type
  
• Hilbert-Schmidt Theory of Symmetric Kernel
  
• Singular Integral Equations of the Cauchy Type
  
• Wiener-Hopf Method and the Wiener-Hopf Integral Equation
  
• Non-linear Integral Equations
  
• Calculus of Variations: Fundamentals
  
• Calculus of Variations: Applications