Multipliers

Abstract

Earlier in Chapter III we showed that in any mathematical description of a quantum mechanical system the requirement of covariance introduces a representation of the symmetry group G of the system into the group Aut(ℒ), where ℒ is the set of all states of the system in question. If we assume that the logic of the system is standard, this representation can be replaced by a representation of G into the group of all symmetries of the Hilbert space underlying the logic. If G ⁰ is the subgroup of G consisting of all elements in the same connected component as the identity of G, then it can be shown that when G is a Lie group the symmetry corresponding to each element of G ⁰ is a unitary rather than an antiunitary operator†. However, these unitary operators are not uniquely determined. Each of them can obviously be multiplied by a complex number of modulus 1 (called a phase factor) without changing the induced automorphism of the logic. It is therefore not immediately obvious that we have a unitary representation of G ⁰, i. e., that the phase factors involved are all removable.