Tensor Product of Finite and Infinite Representations in Physics

Abstract

In field theory, particles with gauge freedom, like the photon, are usually not described as unitary irreducible representations of the spacetime symmetry group, but they appear in indecomposable representations with indefinite invariant scalar product. We call such representations Gupta-Bleuler triplets. There is not yet a general representation theory, but some techniques are available to deal with them [1,2]. One—which is very useful in physical applications—stems from the observation, that the tensor product of finite and infinite representations may contain Gupta-Bleuler triplets in the reduction.