| Formatted contents note |
Review of linear algebra -- Symmetry and groups -- Finite groups -- Rotation and the notion of Lie algebra -- Representation theory -- Schur's lemma and the great orthogonality theorem -- Character is a function of class -- Real, pseudoreal, complex, and the number of square roots -- Crystals -- Euler, Fermat, and Wilson -- Frobenius groups -- Quantum mechanics and group theory: degeneracy -- Group theory and harmonic motion -- Symmetry in the laws of physics: Lagrangian and Hamiltonian -- Tensors and representations of the rotation groups SO(N) -- Lie algebra of SO(3) and ladder operators: creation and annihilation -- Angular momentum and Clebsch-Gordan decomposition -- Tensors and representations of the unitary groups SU(N) -- SU(2): double covering and the spinor -- The electron spin and Kramer's degeneracy -- Integration over continuous groups, topology, and coset manifolds -- Symplectic groups and their algebras -- From Lagrangian mechanics to quantum field theory: it's but a skip and a hop -- Multiplying irreducible representations of finite groups: return to the tetrahedral group -- Crystal field splitting -- Group theory and special functions -- Covering the tetrahedron -- Isospin and and the discovery of a vast internal space -- The Eightfold Way of SU(3) -- The Lie algebra of SU(3) and its root vectors -- Group theory guides us into the microscopic world -- The poor man finds his roots -- Roots and weights for orthogonal, unitary, and symplectic algebras -- Lie algebras in general -- Killing-Cartan classification -- Dynkin diagrams -- SO(2N) and its spinors -- Galileo, Lorentz, and Poincaré -- SL(2,C) double covers SO(3,1): group theory leads us to the Weyl equation -- From the Weyl equation to the Dirac equation -- Dirac and Majorana spinors: antimatter and pseudoreality -- The hydrogen atom and SO(4) -- The unexpected emergence of the Dirac equation in condensed matter physics -- The even more unexpected emergence of the Majorana equation in condensed matter physics -- Contraction and extension -- Conformal algebra -- The expanding universe and group theory -- The gauged universe -- Grand unification and SU(5) -- From SU(5) to SO(10) -- The family mystery. |
