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Foundations of Mathematical Physics

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  Foundations of Mathematical Physics Paul P. Cook ∗  and Neil Lambert † Department of Mathematics, King’s College London The Strand, London WC2R 2LS, UK  ∗ email: paul.cook@kcl.ac.uk † email: neil.lambert@kcl.ac.uk  2  Contents 1 Classical Mechanics 5 1.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Hamilton’s equations. . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Duality and the Harmonic Oscillator . . . . . . . . . . . . . . . . 151.3.4 Noether’s theorem in the Hamiltonian formulation. . . . . . . . . 16 2 Special Relativity and Component Notation 19 2.1 The Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . 192.1.1 The Lorentz Group and the Minkowski Inner Product. . . . . . . 232.2 Component Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Matrices and Matrix Multiplication. . . . . . . . . . . . . . . . . 282.2.2 Common Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 332.2.4 Maxwell’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.5 Electromagnetic Duality . . . . . . . . . . . . . . . . . . . . . . . 39 3 Quantum Mechanics 41 3.1 Canonical Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.1 The Hilbert Space and Observables. . . . . . . . . . . . . . . . . 433.1.2 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . 453.1.3 A Countable Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . 463.1.4 A Continuous Basis. . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 The Schr¨odinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.1 The Heisenberg and Schr¨odinger Pictures. . . . . . . . . . . . . . 52 4 Group Theory 59 4.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Common Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.1 The Symmetric Group  S  n  . . . . . . . . . . . . . . . . . . . . . . 614.2.2 Back to Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.1 The First Isomomorphism Theorem . . . . . . . . . . . . . . . . 713  4  CONTENTS  4.4 Some Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . 724.4.1 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.2 The Direct Sum and Tensor Product . . . . . . . . . . . . . . . . 764.5 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.6 Lie Algebras: Infinitesimal Generators . . . . . . . . . . . . . . . . . . . 824.7 Everything you wanted to know about  SU  (2) and  SO (3) but were afraidto ask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.7.1  SO (3) =  SU  (2) / Z 2  . . . . . . . . . . . . . . . . . . . . . . . . . . 854.7.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.7.3 Representations Revisited . . . . . . . . . . . . . . . . . . . . . . 914.8 The Invariance of Physical Law . . . . . . . . . . . . . . . . . . . . . . . 934.8.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.8.2 Special Relativity and the Infinitesimal Generators of   SO (1 , 3). . 934.8.3 The Proper Lorentz Group and  SL (2 , C ). . . . . . . . . . . . . . 954.8.4 Representations of the Lorentz Group and Lorentz Tensors. . . . 97

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