Group theory in solid state physics

Abstract

Part 1. The concept of a vector space irreducible under a set of operators is developed from first principles, and then introduced into the many-particle formalism of quantum mechanics by means of an explicit postulate of irreducibility.The calculus of the irreducible matrix representations of finite groups is developed ab initio, including the theory of characters and projection operators. The use of this calculus to simplify eigenvalue calculations is explained in detail.Part 2. The structure of the general crystal space group, including glide-planes and screw-axes, is discussed briefly. The theory is developed of the symmetry group of a many-electron system with spin-orbit coupling, using the Dirac formalism. A detailed discussion is given of Wigner's time-reversal theorems for a many-electron system, including the character tests for time-reversal degeneracy. A general theory of the permutation symmetry of a many-electron system is developed, and shown to contain the Dirac vector model as a special case.A new treatment is given of the theory of the irreducible representations of space groups, including the double space groups and Herring's time-reversal theorems.