Group Algebra, Convolution Algebra, and Applications to Quantum Mechanics

The symmetry group of the Hamiltonian plays a fundamental role in quantum theory in the classification of stationary states and in studying transition probabilities and selection rules. It is here shown that the properties of the group may be given a condensed and transparent description in terms of the convolution algebra, and that Schur's lemma immediately leads to the construction of the fundamental set of projection and shift operators. The projection operators form a resolution of the identity which may be used to split the Hilbert space into orthogonal and noninteracting subspaces of infinite order. The question of the splitting of the conventional secular equations is discussed, and the explicit form of the decomposed equation is derived in terms of the convolution algebra and the characters. The theory is here discussed only for finite groups, but the results may be generalized to the compact infinite groups having a well-defined "invariant mean."