In this chapter we look at some aspects of group representation theory specifically pertaining to the ‘needs’ of quantum mechanics, and involving quite subtle features. We begin by recalling the basic mathematical – or perhaps better, the kinematical framework of quantum mechanics. With this preparation, we define the concept of a symmetry operation in quantum mechanics in the manner of Wigner, followed by a description of his celebrated unitary–antiunitary theorem. A proof of this theorem, well known for its elegance, was given by V. Bargmann in 1964.We indicate the structure of this proof, and then present two other recent proofs which afford considerable insight into the conceptualisation of symmetry in quantum mechanics.

Wigner's theorem leads us to examine ray representations, or representations up to phases, for Lie groups. Such representations have important consequences for the generator commutation relations, bringing in the concept of neutral elements and a certain degree of freedom or flexibility in the choice of generators.We study the extent to which this flexibility can be used to simplify, or possibly completely eliminate, neutral elements in the commutation relations associated with a given Lie group.

Neutral elements appear also in the context of realisations of Lie groups via canonical transformations in the phase space formalism of classical mechanics. Comparison of the situations in the two cases, classical and quantum mechanics helps us appreciate that whereas the origins of neutral elements are different, their algebraic properties are common.

Hilbert and Ray Space Descriptions of Pure Quantum States

Quantum mechanics uses vectors in a suitable complex Hilbert space H to describe the pure states – states of maximum possible information – of a quantum system. Depending on the system, the dimension of H may be finite or infinite. Each unit vector ψ ∈ H determines uniquely a certain pure physical state. However, this is a many-to-one rather than a one-to-one relationship, since for any phase α , the vector eiαψ determines the same pure state as ψ . Overall phases are unobservable and unphysical. In spite of this, the use of H is very convenient as one can express the Superposition Principle very simply. If ψ 1,ψ 2, · · · , are any vectors in H , each determining a corresponding pure state, and c 1, c 2, · · · are any complex numbers such that

is nonzero, then ψ also determines, in general after normalisation, a certain pure state.