The Structure of a Class of Representations of the Unitary Group on a Hilbert Space

1. In the mathematical theory of fields of elementary particles one has to do with a representation of the unitary group on a Hilbert space 77. With any such representation T and subspace M of 77 there is associated a self-adjoint operator called the "number of particles in M." The particle interpretation of general wave fields is made possible in the first instance by the fact that these operators have integral proper values. The direct physical approach suggests that these proper values should be non-negative, which however is not the case for an arbitrary representation. The purpose of the present note is to determine the structure of the most general "physical" 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 77. 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.