On Unitary Ray Representations of Continuous Groups

Abstract

1. This paper, although mathematical in content, is motivated by quantumtheoretical considerations. The states of a quantum-mechanical system are usually described by vectors f of norm 1 in some Hilbert space A, and we assume explicitly that to every unit vector f corresponds a state of the system. This correspondence, however, is not one-to-one. In fact, the vectors which describe the same state form a ray f (in Weyl's terminology, cf. [13], p. 4 and p. 20),1 i.e. a set consisting of all vectors f = Tfo where fo is a fixed unit vector in & and r any complex number of modulus 1. (Every vector f in f will be called a representative of the ray f.) We have therefore a one-to-one correspondence between quantum states and rays, and every significant statement in Quantum Theory is a statement about rays. The transition probability from a state f to a state g equals (f, I)'2 where f, g are representatives of the rays f, g respectively. This suggests the introduction of the inner product of two rays by the definition