Mathematical Physics

  1. Group Theory and Symmetries in Physics
    1. Abstract Group Theory
      1. Group Axioms and Basic Properties
        1. Closure and Associativity
          1. Identity and Inverse Elements
            1. Order of Groups and Elements
            2. Subgroups and Quotient Groups
              1. Lagrange's Theorem
                1. Normal Subgroups
                  1. Factor Groups
                  2. Group Homomorphisms
                    1. Kernel and Image
                      1. Isomorphism Theorems
                        1. Automorphisms
                        2. Group Actions
                          1. Definition and Examples
                            1. Orbits and Stabilizers
                              1. Burnside's Lemma
                            2. Lie Groups
                              1. Matrix Lie Groups
                                1. General Linear Groups
                                  1. Orthogonal Groups
                                    1. Unitary Groups
                                      1. Special Linear Groups
                                      2. Manifold Structure
                                        1. Smooth Manifolds
                                          1. Lie Group as Manifold
                                            1. Left and Right Translations
                                            2. One-Parameter Subgroups
                                              1. Exponential Map
                                                1. Generators of Lie Groups
                                              2. Lie Algebras
                                                1. Definition and Structure
                                                  1. Lie Bracket
                                                    1. Jacobi Identity
                                                      1. Structure Constants
                                                      2. Relationship to Lie Groups
                                                        1. Tangent Space at Identity
                                                          1. Exponential Map
                                                            1. Baker-Campbell-Hausdorff Formula
                                                            2. Classification of Lie Algebras
                                                              1. Simple and Semisimple Algebras
                                                                1. Root Systems
                                                                  1. Cartan Subalgebras
                                                                2. Representation Theory
                                                                  1. Linear Representations
                                                                    1. Definition and Examples
                                                                      1. Faithful Representations
                                                                        1. Equivalent Representations
                                                                        2. Reducible and Irreducible Representations
                                                                          1. Schur's Lemmas
                                                                            1. Complete Reducibility
                                                                              1. Maschke's Theorem
                                                                              2. Character Theory
                                                                                1. Characters of Representations
                                                                                  1. Character Tables
                                                                                    1. Orthogonality Relations
                                                                                    2. Representations of Lie Groups
                                                                                      1. Representations of SU(2)
                                                                                        1. Representations of SO(3)
                                                                                          1. Weight Spaces and Highest Weight Theory
                                                                                        2. Physical Applications
                                                                                          1. Rotational Symmetry
                                                                                            1. Angular Momentum Algebra
                                                                                              1. Spherical Harmonics
                                                                                                1. Addition of Angular Momenta
                                                                                                2. Spin Systems
                                                                                                  1. SU(2) and Spinor Representations
                                                                                                    1. Pauli Matrices
                                                                                                      1. Spin-Orbit Coupling
                                                                                                      2. Relativistic Symmetries
                                                                                                        1. Lorentz Group
                                                                                                          1. Poincaré Group
                                                                                                            1. Representations in Particle Physics
                                                                                                            2. Gauge Symmetries
                                                                                                              1. Abelian Gauge Groups
                                                                                                                1. Non-Abelian Gauge Groups
                                                                                                                  1. Spontaneous Symmetry Breaking
                                                                                                                  2. Crystallographic Groups
                                                                                                                    1. Point Groups
                                                                                                                      1. Space Groups
                                                                                                                        1. Applications in Solid State Physics

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                                                                                                                    3. Mathematical Formulation of Quantum Mechanics

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                                                                                                                    5. Differential Geometry and General Relativity