Reduction of operator rings and the irreducibility axiom in quantum field theory

Abstract

The mathematical theory of the reduction of operator rings is used to investigate some structures which can occur in quantum field theory when the postulate that the field operators generate an irreducible ring is relaxed. In particular, it is shown that if a quantum field theory has a commutator which commutes with all field operators it is a direct integral of theories in each of which the commutator is a scalar. If in addition it satisfies the postulates of Lorentz covariance, existence of an invariant vacuum, and mass and energy spectra, then it is a direct integral of generalized free field theories whenever the unitary representation of the Lorentz group can be constructed in terms of functions of the field operators and every state can be constructed by applying field operators on the vacuum. It is also shown that the latter two assumptions together with the requirement of a unique invariant vacuum state are sufficient to prove that the ring generated by the field operators is irreducible. In other words, under these conditions the irreducibility postulate is redundant.