Mathematical Physics: A Modern Introduction to its Foundations

Description

This book is for physics students interested in the mathematics they use and for mathematics students interested in seeing how some of the ideas of their discipline find realization in an applied setting. The presentation tries to strike a balance between formalism and application, between abstract and concrete. The interconnections among the various topics are clarified both by the use of vector spaces as a central unifying theme, recurring throughout the book, and by putting ideas into their historical context. Enough of the essential formalism is included to make the presentation self-contained. The book is divided into eight parts: The first covers finite- dimensional vector spaces and the linear operators defined on them. The second is devoted to infinite-dimensional vector spaces, and includes discussions of the classical orthogonal polynomials and of Fourier series and transforms. The third part deals with complex analysis, including complex series and their convergence, the calculus of residues, multivalued functions, and analytic continuation. Part IV treats ordinary differential equations, concentrating on second-order equations and discussing both analytical and numerical methods of solution. The next part deals with operator theory, focusing on integral and Sturm--Liouville operators. Part VI is devoted to Green's functions, both for ordinary differential equations and in multidimensional spaces. Parts VII and VIII contain a thorough discussion of differential geometry and Lie groups and their applications, concluding with Noether's theorem on the relationship between symmetries and conservation laws. Intended for advanced undergraduates or beginning graduate students, this comprehensive guide should also prove useful as a refresher or reference for physicists and applied mathematicians. Over 300 worked-out examples and more than 800 problems provide valuable learning aids. Numerous enhancements and revision are incorporated into this new edition. For example, fiber bundle techniques are used to introduce differential geometry. This more elegant and intuitive approach naturally connects differential geometry with not only the general theory of relativity, but also gauge theories of fundamental forces. Some praise for the previous edition: PAGEOPH [Pure and Applied Geophysics] Review by Daniel Wojcik, University of Maryland "This volume should be a welcome addition to any collection. The book is well written and explanations are usually clear. Lives of famous mathematicians and physicists are scattered within the book. They are quite extended, often amusing, making nice interludes. Numerous exercises help the student practice the methods introduced. I have recently been using this book for an extended time and acquired a liking for it. Among all the available books treating mathematical methods of physics this one certainly stands out and assuredly it would suit the needs of many physics readers." ZENTRALBLATT MATH Review by G.Roepstorff, University of Aachen, Germany " Unlike most existing texts with the same emphasis and audience, which are merely collections of facts and formulas, the present book is more systematic, self-contained, with a level of presentation that tends to be more formal and abstract. This entails proving a large number of theorems, lemmas, and corollaries, deferring most of the applications that physics students might be interested in to the example sections in small print. Indeed, there are 350 worked-out examples and about 850 problems. A very nice feature is the way the author intertwines the formalism with the life stories and anecdotes of some mathematicians and physicists, leading at their times. As is often the case, the historical view point helps to understand and appreciate the ideas presented in the text. For the physics student in the middle of his training, it will certainly prove to be extremely useful." THE PHYSICIST Review by Paul Davies, Orion Productions, Adelaide, Australia "I am pleased to have so many topics collected in a single volume. All the tricks are there of course, but supported by sufficient rigour and substantiation to make the dedicated mathematical physicist sigh with delight." EMS [EUROPEAN MATHEMATICAL SOCIETY] NEWSLETTER "This book is a condensed exposition of the mathematics that is met in most parts of physics. The presentation attains a very good balance between the formal introduction of concepts, theorems and proofs on one hand, and the applied approach on the other, with many examples, fully or partially solved problems, and historical remarks. An impressive amount of mathematics is covered. This book can be warmly recommended as a basic source for the study of mathematics for advanced undergraduates or beginning graduate students in physics and applied mathematics, and also as a reference book for all working mathematicians and physicists." Mathematical Physics Preface to Second Edition Preface to First Edition Level and Philosophy of Presentation Features Organization and Topical Coverage Acknowledgments Note to the Reader Contents List of Symbols Chapter 1: Mathematical Preliminaries 1.1 Sets 1.1.1 Equivalence Relations 1.2 Maps 1.3 Metric Spaces 1.4 Cardinality 1.5 Mathematical Induction 1.6 Problems Part I: Finite-Dimensional Vector Spaces Chapter 2: Vectors and Linear Maps 2.1 Vector Spaces 2.1.1 Subspaces 2.1.2 Factor Space 2.1.3 Direct Sums 2.1.4 Tensor Product of Vector Spaces 2.2 Inner Product 2.2.1 Orthogonality 2.2.2 The Gram-Schmidt Process 2.2.3 The Schwarz Inequality 2.2.4 Length of a Vector 2.3 Linear Maps 2.3.1 Kernel of a Linear Map 2.3.2 Linear Isomorphism 2.4 Complex Structures 2.5 Linear Functionals 2.6 Multilinear Maps 2.6.1 Determinant of a Linear Operator 2.6.2 Classical Adjoint 2.7 Problems Chapter 3: Algebras 3.1 From Vector Space to Algebra 3.1.1 General Properties 3.1.2 Homomorphisms 3.2 Ideals 3.2.1 Factor Algebras 3.3 Total Matrix Algebra 3.4 Derivation of an Algebra 3.5 Decomposition of Algebras 3.5.1 The Radical 3.5.2 Semi-simple Algebras 3.5.3 Classification of Simple Algebras 3.6 Polynomial Algebra 3.7 Problems Chapter 4: Operator Algebra 4.1 Algebra of `39`42`"613A``45`47`"603AEnd(V) 4.1.1 Polynomials of Operators 4.1.2 Functions of Operators 4.1.3 Commutators 4.2 Derivatives of Operators 4.3 Conjugation of Operators 4.3.1 Hermitian Operators 4.3.2 Unitary Operators 4.4 Idempotents 4.4.1 Projection Operators 4.5 Representation of Algebras 4.6 Problems Chapter 5: Matrices 5.1 Representing Vectors and Operators 5.2 Operations on Matrices 5.3 Orthonormal Bases 5.4 Change of Basis 5.5 Determinant of a Matrix 5.5.1 Matrix of the Classical Adjoint 5.5.2 Inverse of a Matrix Algorithm for Calculating the Inverse of a Matrix Rank of a Matrix 5.5.3 Dual Determinant Function 5.6 The Trace 5.7 Problems Chapter 6: Spectral Decomposition 6.1 Invariant Subspaces 6.2 Eigenvalues and Eigenvectors 6.3 Upper-Triangular Representations 6.4 Complex Spectral Decomposition 6.4.1 Simultaneous Diagonalization 6.5 Functions of Operators 6.6 Real Spectral Decomposition 6.6.1 The Case of Symmetric Operators 6.6.2 The Case of Real Normal Operators 6.7 Polar Decomposition 6.8 Problems Part II: Infinite-Dimensional Vector Spaces Chapter 7: Hilbert Spaces 7.1 The Question of Convergence 7.2 The Space of Square-Integrable Functions 7.2.1 Orthogonal Polynomials 7.2.2 Orthogonal Polynomials and Least Squares 7.3 Continuous Index 7.4 Generalized Functions 7.5 Problems Chapter 8: Classical Orthogonal Polynomials 8.1 General Properties 8.2 Classification 8.3 Recurrence Relations 8.4 Details of Specific Examples 8.4.1 Hermite Polynomials 8.4.2 Laguerre Polynomials 8.4.3 Legendre Polynomials 8.4.4 Other Classical Orthogonal Polynomials Jacobi Polynomials, Pnµ,nu(x) Gegenbauer Polynomials, Cnlambda(x) Chebyshev Polynomials of the First Kind, Tn(x) Chebyshev Polynomials of the Second Kind, Un(x) 8.5 Expansion in Terms of Orthogonal Polynomials 8.6 Generating Functions 8.7 Problems Chapter 9: Fourier Analysis 9.1 Fourier Series 9.1.1 The Gibbs Phenomenon 9.1.2 Fourier Series in Higher Dimensions 9.2 Fourier Transform 9.2.1 Fourier Transforms and Derivatives 9.2.2 The Discrete Fourier Transform 9.2.3 Fourier Transform of a Distribution 9.3 Problems Part III: Complex Analysis Chapter 10: Complex Calculus 10.1 Complex Functions 10.2 Analytic Functions 10.3 Conformal Maps 10.4 Integration of Complex Functions 10.5 Derivatives as Integrals 10.6 Infinite Complex Series 10.6.1 Properties of Series 10.6.2 Taylor and Laurent Series 10.7 Problems Chapter 11: Calculus of Residues 11.1 Residues 11.2 Classification of Isolated Singularities 11.3 Evaluation of Definite Integrals 11.3.1 Integrals of Rational Functions 11.3.2 Products of Rational and Trigonometric Functions 11.3.3 Functions of Trigonometric Functions 11.3.4 Some Other Integrals 11.3.5 Principal Value of an Integral 11.4 Problems Chapter 12: Advanced Topics 12.1 Meromorphic Functions 12.2 Multivalued Functions 12.2.1 Riemann Surfaces 12.3 Analytic Continuation 12.3.1 The Schwarz Reflection Principle 12.3.2 Dispersion Relations 12.4 The Gamma and Beta Functions 12.5 Method of Steepest Descent 12.6 Problems Part IV: Differential Equations Chapter 13: Separation of Variables in Spherical Coordinates 13.1 PDEs of Mathematical Physics 13.2 Separation of the Angular Part 13.3 Construction of Eigenvalues of L2 13.4 Eigenvectors of L2: Spherical Harmonics 13.4.1 Expansion of Angular Functions 13.4.2 Addition Theorem for Spherical Harmonics 13.5 Problems Chapter 14: Second-Order Linear Differential Equations 14.1 General Properties of ODEs 14.2 Existence/Uniqueness for First-Order DEs 14.3 General Properties of SOLDEs 14.4 The Wronskian 14.4.1 A Second Solution to the HSOLDE 14.4.2 The General Solution to an ISOLDE 14.4.3 Separation and Comparison Theorems 14.5 Adjoint Differential Operators 14.6 Power-Series Solutions of SOLDEs 14.6.1 Frobenius Method of Undetermined Coefficients 14.6.2 Quantum Harmonic Oscillator 14.7 SOLDEs with Constant Coefficients 14.8 The WKB Method 14.8.1 Classical Limit of the Schrödinger Equation 14.9 Problems Chapter 15: Complex Analysis of SOLDEs 15.1 Analytic Properties of Complex DEs 15.1.1 Complex FOLDEs 15.1.2 The Circuit Matrix 15.2 Complex SOLDEs 15.3 Fuchsian Differential Equations 15.4 The Hypergeometric Function 15.5 Confluent Hypergeometric Functions 15.5.1 Hydrogen-Like Atoms 15.5.2 Bessel Functions 15.6 Problems Chapter 16: Integral Transforms and Differential Equations 16.1 Integral Representation of the Hypergeometric Function 16.1.1 Integral Representation of the Confluent Hypergeometric Function 16.2 Integral Representation of Bessel Functions 16.2.1 Asymptotic Behavior of Bessel Functions 16.3 Problems Part V: Operators on Hilbert Spaces Chapter 17: Introductory Operator Theory 17.1 From Abstract to Integral and Differential Operators 17.2 Bounded Operators in Hilbert Spaces 17.2.1 Adjoints of Bounded Operators 17.3 Spectra of Linear Operators 17.4 Compact Sets 17.4.1 Compactness and Infinite Sequences 17.5 Compact Operators 17.5.1 Spectrum of Compact Operators 17.6 Spectral Theorem for Compact Operators 17.6.1 Compact Hermitian Operator 17.6.2 Compact Normal Operator 17.7 Resolvents 17.8 Problems Chapter 18: Integral Equations 18.1 Classification 18.2 Fredholm Integral Equations 18.2.1 Hermitian Kernel 18.2.2 Degenerate Kernels 18.3 Problems Chapter 19: Sturm-Liouville Systems 19.1 Compact-Resolvent Unbounded Operators 19.2 Sturm-Liouville Systems and SOLDEs 19.3 Asymptotic Behavior 19.3.1 Large Eigenvalues 19.3.2 Large Argument 19.4 Expansions in Terms of Eigenfunctions 19.5 Separation in Cartesian Coordinates 19.5.1 Rectangular Conducting Box 19.5.2 Heat Conduction in a Rectangular Plate 19.5.3 Quantum Particle in a Box 19.5.4 Wave Guides 19.6 Separation in Cylindrical Coordinates 19.6.1 Conducting Cylindrical Can 19.6.2 Cylindrical Wave Guide 19.6.3 Current Distribution in a Circular Wire 19.7 Separation in Spherical Coordinates 19.7.1 Radial Part of Laplace's Equation 19.7.2 Helmholtz Equation in Spherical Coordinates 19.7.3 Quantum Particle in a Hard Sphere 19.7.4 Plane Wave Expansion 19.8 Problems Part VI: Green's Functions Chapter 20: Green's Functions in One Dimension 20.1 Calculation of Some Green's Functions 20.2 Formal Considerations 20.2.1 Second-Order Linear DOs 20.2.2 Self-adjoint SOLDOs 20.3 Green's Functions for SOLDOs 20.3.1 Properties of Green's Functions 20.3.2 Construction and Uniqueness of Green's Functions 20.3.3 Inhomogeneous BCs 20.4 Eigenfunction Expansion 20.5 Problems Chapter 21: Multidimensional Green's Functions: Formalism 21.1 Properties of Partial Differential Equations 21.1.1 Characteristic Hypersurfaces 21.1.2 Second-Order PDEs in m Dimensions 21.2 Multidimensional GFs and Delta Functions 21.2.1 Spherical Coordinates in m Dimensions 21.2.2 Green's Function for the Laplacian 21.3 Formal Development 21.3.1 General Properties 21.3.2 Fundamental (Singular) Solutions 21.4 Integral Equations and GFs 21.5 Perturbation Theory 21.5.1 The Nondegenerate Case 21.5.2 The Degenerate Case 21.6 Problems Chapter 22: Multidimensional Green's Functions: Applications 22.1 Elliptic Equations 22.1.1 The Dirichlet Boundary Value Problem 22.1.2 The Neumann Boundary Value Problem 22.2 Parabolic Equations 22.3 Hyperbolic Equations 22.4 The Fourier Transform Technique 22.4.1 GF for the m-Dimensional Laplacian 22.4.2 GF for the m-Dimensional Helmholtz Operator 22.4.3 GF for the m-Dimensional Diffusion Operator 22.4.4 GF for the m-Dimensional Wave Equation 22.5 The Eigenfunction Expansion Technique 22.6 Problems Part VII: Groups and Their Representations Chapter 23: Group Theory 23.1 Groups 23.2 Subgroups 23.2.1 Direct Products 23.3 Group Action 23.4 The Symmetric Group Sn 23.5 Problems Chapter 24: Representation of Groups 24.1 Definitions and Examples 24.2 Irreducible Representations 24.3 Orthogonality Properties 24.4 Analysis of Representations 24.5 Group Algebra 24.5.1 Group Algebra and Representations 24.6 Relationship of Characters to Those of a Subgroup 24.7 Irreducible Basis Functions 24.8 Tensor Product of Representations 24.8.1 Clebsch-Gordan Decomposition 24.8.2 Irreducible Tensor Operators 24.9 Problems Chapter 25: Representations of the Symmetric Group 25.1 Analytic Construction 25.2 Graphical Construction 25.3 Graphical Construction of Characters 25.4 Young Operators 25.5 Products of Representations of Sn 25.6 Problems Part VIII: Tensors and Manifolds Chapter 26: Tensors 26.1 Tensors as Multilinear Maps 26.2 Symmetries of Tensors 26.3 Exterior Algebra 26.3.1 Orientation 26.4 Symplectic Vector Spaces 26.5 Inner Product Revisited 26.5.1 Subspaces 26.5.2 Orthonormal Basis 26.5.3 Inner Product on Lambdap(V,U) 26.6 The Hodge Star Operator 26.7 Problems Chapter 27: Clifford Algebras 27.1 Construction of Clifford Algebras 27.1.1 The Dirac Equation 27.2 General Properties of the Clifford Algebra 27.2.1 Homomorphism with Other Algebras 27.2.2 The Canonical Element 27.2.3 Center and Anticenter 27.2.4 Isomorphisms 27.3 General Classification of Clifford Algebras 27.4 The Clifford Algebras Cµnu(R) 27.4.1 Classification of Cn0(R) and C0n(R) 27.4.2 Classification of Cµnu(R) 27.4.3 The Algebra C31(R) 27.5 Problems Chapter 28: Analysis of Tensors 28.1 Differentiable Manifolds 28.2 Curves and Tangent Vectors 28.3 Differential of a Map 28.4 Tensor Fields on Manifolds 28.4.1 Vector Fields 28.4.2 Tensor Fields 28.5 Exterior Calculus 28.6 Integration on Manifolds 28.7 Symplectic Geometry 28.8 Problems Part IX: Lie Groups and Their Applications Chapter 29: Lie Groups and Lie Algebras 29.1 Lie Groups and Their Algebras 29.1.1 Group Action 29.1.2 Lie Algebra of a Lie Group 29.1.3 Invariant Forms 29.1.4 Infinitesimal Action 29.1.5 Integration on Lie Groups 29.2 An Outline of Lie Algebra Theory 29.2.1 The Lie Algebras o(p,n-p) and p(p,n-p) 29.2.2 Operations on Lie Algebras 29.3 Problems Chapter 30: Representation of Lie Groups and Lie Algebras 30.1 Representation of Compact Lie Groups 30.2 Representation of the General Linear Group 30.3 Representation of Lie Algebras 30.3.1 Representation of Subgroups of GL(V) 30.3.2 Casimir Operators 30.3.3 Representation of so(3) and so(3,1) 30.3.4 Representation of the Poincaré Algebra 30.4 Problems Chapter 31: Representation of Clifford Algebras 31.1 The Clifford Group 31.2 Spinors 31.2.1 Pauli Spin Matrices and Spinors 31.2.2 Spinors for Cµnu(R) 31.2.3 C31(R) Revisited 31.3 Problems Chapter 32: Lie Groups and Differential Equations 32.1 Symmetries of Algebraic Equations 32.2 Symmetry Groups of Differential Equations 32.2.1 Prolongation of Functions 32.2.2 Prolongation of Groups 32.2.3 Prolongation of Vector Fields 32.3 The Central Theorems 32.4 Application to Some Known PDEs 32.4.1 The Heat Equation 32.4.2 The Wave Equation 32.5 Application to ODEs 32.5.1 First-Order ODEs 32.5.2 Higher-Order ODEs 32.5.3 DEs with Multiparameter Symmetries 32.6 Problems Chapter 33: Calculus of Variations, Symmetries, and Conservation Laws 33.1 The Calculus of Variations 33.1.1 Derivative for Hilbert Spaces 33.1.2 Functional Derivative 33.1.3 Variational Problems 33.1.4 Divergence and Null Lagrangians 33.2 Symmetry Groups of Variational Problems 33.3 Conservation Laws and Noether's Theorem 33.4 Application to Classical Field Theory 33.5 Problems Part X: Fiber Bundles Chapter 34: Fiber Bundles and Connections 34.1 Principal Fiber Bundles 34.1.1 Associated Bundles 34.2 Connections in a PFB 34.2.1 Local Expression for a Connection 34.2.2 Parallelism 34.3 Curvature Form 34.3.1 Flat Connections 34.3.2 Matrix Structure Group 34.4 Problems Chapter 35: Gauge Theories 35.1 Gauge Potentials and Fields 35.1.1 Particle Fields 35.1.2 Gauge Transformation 35.2 Gauge-Invariant Lagrangians 35.3 Construction of Gauge-Invariant Lagrangians 35.4 Local Equations 35.5 Problems Chapter 36: Differential Geometry 36.1 Connections in a Vector Bundle 36.2 Linear Connections 36.2.1 Covariant Derivative of Tensor Fields 36.2.2 From Forms on P to Tensor Fields on M 36.2.3 Component Expressions 36.2.4 General Basis 36.3 Geodesics 36.3.1 Riemann Normal Coordinates 36.4 Problems Chapter 37: Riemannian Geometry 37.1 The Metric Connection 37.1.1 Orthogonal Bases 37.2 Isometries and Killing Vector Fields 37.3 Geodesic Deviation and Curvature 37.3.1 Newtonian Gravity 37.4 General Theory of Relativity 37.4.1 Einstein's Equation 37.4.2 Static Spherically Symmetric Solutions 37.4.3 Schwarzschild Geodesics Massive Particle Massless Particle 37.5 Problems References Index