Detectors, Coincidence Arrangements, Cross Sections

Abstract

In section 1 of Chapter III we claimed that the theory is fully characterized once the abstract net of observable algebras 𝔄 (𝒪) is known. This means that all physical consequences must be derivable from this net. For comparison with experiments in high energy physics the most interesting ones concern the types of particles which occur and their collision cross sections. What is a particle? In section 3 of Chapter I we argued that the state vectors in an irreducible representation of ${\bar {\frak P}}$describe a single particle, a particle alone in the world. This was taken as the starting point for the discussion of the particle aspects of the theory in Chapter II, sections 3 and 4. Within the Hilbert space setting provided by the Wightman axioms the single particle subspace ℋ⁽¹⁾ was defined as the part belonging to the discrete spectrum of the mass operator M = (P _μ P^μ)^1/2. Prom ℋ⁽¹⁾ we constructed the states of asymptotic particle configurations by introducing a product composition ⊗^t between state vectors which becomes unambiguous and has an obvious physical interpretation when it is applied to states which at time t are localized in far separated regions. It was then shown that this condition is satisfied at asymptotic times for any subset of single particle states and that ⊗^t converges for t → ±∞ to a composition ⊗^out, ⊗ⁱⁿ, respectively of the state vectors of ℋ⁽¹⁾. The proof depended on the assumption of a fast decrease of correlation functions with increasing space-like separation. This assumption in turn was shown to be a consequence of locality and positivity of energy provided that the vacuum was clearly separated from all other states by a mass gap. When compared to the claim at the beginning of this section the results and methods¹ of Chapter II fall short on several counts.