The structure of a class of representations of the unitary group on a Hilbert space
Irving E. Segal · Proceedings of the American Mathematical Society · 1957
for an arbitrary representation. The purpose of the present note is to determine the structure of the most general representation, i.e., one for which the number of particles is always non-negative. It is shown that this structure is essentially the same as in the case of a finite-dimensional Hilbert space. Specifically, an irreducible physical representation is unitarily equivalent to the canonical representation in a class of covariant tensors of maximal symmetry over the one-particle space H. The most general physical representation is a direct sum of these irreducible covariant tensor representations. For a finite-dimensional space, these results are essentially equivalent to well-known ones giving the structure of the general unitary representation of the unitary group on the space. These known results are established by the use of characters, a technique which is not adaptable to the infinite-dimensional case because there is then no trace for unitary operators on the space. The method employed is rather to approximate the space by subspaces of high finite dimension and to make use of what is already known in the finite-dimensional case. 2. Throughout this paper, G will designate the group of unitary operators on a complex Hilbert space H. Unless otherwise specified, the dimension of H may be arbitrary. F will denote a given continuous unitary representation of G on a complex Hilbert space K. The topology of G that is used is the so-called strong operator topology. For any self-adjoint operator A on H, the self-adjoint generator of the one-parameter group {r(eitA): - o < t < oc } will be designated drI(A) and the mapping dF from the self-adjoint operators in H into those