Asymptotic behaviour of spin-momentum distribution observables for fermion fields with cut-off self-coupling

Abstract

In a previous paper asymptotic creation and annhilation operatorsa ^#_± have been constructed by the Kato-Mugibayashi method from the creation and annihilation operatorsa ^# for spin 1/2 fields with an interaction Hamiltonian density which is an evendegree polynomial in the field with ultra-violet cut-off and its derivatives. For any eigenvector φ of the total HamiltonianH=H ₀+H _I partial isometries Ω_± have been defined so thata ^#_± equal Ω_± a ^# Ω*_± on the ranges ℱ_± of Ω_±. Since the existence of a groundstate ofH has been proved, the existence of at least one pair Ω_± follows. The purpose of this paper is to show that for any ψ ∈ ℱ_± orthogonal to φ the distribution of spins and momenta of the interacting Schrödinger states exp[−itH]Ω_±ψ approaches fort→∓∞ the distributions of spins and momenta of the free state exp[−itH ₀] ψ if a wave-amplitude renormalization is carried out in ℱ_±. This is achieved by studying the expectation values of the operators in themaximally abelian W*-algebra

generated by operators of the form ∫ρa*a, in terms of whichany information about spins and momenta can be expressed.