Three remarks on Powers’ theorem about irreducible fields fulfilling CAR

Abstract

First it is shown that within a relativistic Fermi field theory, a bound ∥Ψk( f,t)∥ ≤C∥ f ∥2 already implies canonical anticommutation relations (CAR). Then under Powers’ assumptions a linear, first-order differential equation for the fields ψk(x,t) is derived. This shows that in the set of generalized free fields fulfilling CAR only the free fields are irreducible at time zero. Finally Fermi fields in two space-time dimensions are considered. It is shown that only four-fermion interaction might be compatible with CAR and a bound on the coupling strength is derived.