On Unitary Ray Representations of Continuous Groups

TLDR

Quantum states are represented by unit vectors in a Hilbert space, but equivalent states form rays, establishing a one‑to‑one correspondence between states and rays. The paper aims to introduce an inner product for rays, motivated by quantum‑theoretical considerations. The authors define the transition probability between rays as the squared inner product of their representative vectors. Reference [13] cited on pages 4 and 20.

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