The chief aim of this chapter is to give brief accounts of topics in nonrelativistic quantum mechanics that are not always treated in elementary texts. We begin with the Hilbert space formulation of quantum mechanics as set down by von Neumann in his book Mathematical Foundations of Quantum Mechanics (von Neumann, 1955), which will henceforth be referred to as von Neumann's book. Much of our concern will be with continuous spectra, which cannot be discussed adequately in the Dirac formalism. The density matrix, which will play a key role in Chapters 8 and 10, will be treated in some detail.

The section on formalism is followed by one on the probability interpretation; the latter is included because Sewell's theory of measurement suggests a subtle reformulation of a part of it. These are followed by sections on superselection rules and the Galilei group, which is the relativity group of quantum mechanics. They are based on the pioneering works of Wigner and Bargmann. The last section is devoted to the fundamental theorems of von Neumann and Stone, and to Reeh's observation on the physical significance of the failure of the Stone–von Neumann uniqueness theorem at the Lie algebra level.

The formalism of quantum mechanics

By quantum mechanics we shall mean the nonrelativistic quantum theory of a system with a finite number, N, of particles. The number N is assumed fixed. It will be convenient, for later recall, to divide the material into subsections.