Towards Asymptotic Completeness of Two-Particle Scattering in Local Relativistic QFT

Abstract

We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Letp 1 andp 2 be two energy-momentum vectors of a massive particle and let ∆ be a small neighbourhood ofp 1 +p 2 . We construct asymptotic observables (two-particle Araki-Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods ofp 1 andp 2 . We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of ∆. Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag-Ruelle scattering states with total energy-momenta in ∆. The result holds under very general conditions which are satisfied, for example, in λφ 4 2 . The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.1991 Mathematics Subject Classification. 81T05, 81U99. Key words and phrases. local quantum field theory, Haag-Ruelle scattering theory, Araki-Haag detectors, asymptotic completeness. 1 We consider only the limit t → +∞ and outgoing scattering states in this paper as the case t → −∞ is completely analogous.

Asymptotic observables, propagation estimates and the problem of asymptotic completeness in algebraic QFT

“…The above construction of generalized creation and annihilation operators is well known since early days of AQFT. We conclude this section with a more recent concept of 'improper' annihilation operators, introduced in our recent work [4]. They are defined as maps…”

“…In the case of n = 2, using relation (5.3) and the standard phase space propagation estimates (see e.g. [3]) one can show the convergence of t → f (t) [4]. In this case functions χ appearing in condition (b) above are not needed and one can prove the convergence of a product of two conventional Araki-Haag detectors (4.6).…”

“…In an ongoing project with C. Gérard we gave a solution of this problem for a certain class of detectors [4,5]. Moreover, we demonstrated that the linear span of the ranges of these detectors coincides with the subspace of scattering states.…”

“…(uniqueness and cyclicity of the vacuum) 1l {0} (U ) = |Ω Ω| and AΩ = H. We adopt a restrictive variant of the positivity of energy assumption (4), suitable for massive theories, which requires that Sp U consists of {0}, corresponding to the vacuum vector Ω, a mass hyperboloid H m := { (E, p) ∈ R d+1 | E = p 2 + m 2 } carrying single-particle states and the multiparticle spectrum G 2m := { (E, p) ∈ R d+1 | E ≥ p 2 + (2m) 2 }. We recall that there exist interacting quantum field theories which satisfy the above assumptions, for example λφ 4 2 at small λ [6].…”

Asymptotic Observables in Gapped Quantum Spin Systems

“…This analysis is a step towards asymptotic completeness for lattice systems, which appears to be an open problem beyond the two body scattering [GS97,AB01]. We build on recent advances in algebraic QFT [DG12,DG13] but in contrast to these two references we will not use the method of propagation estimates [SiSo87] to prove the existence of asymptotic observables. We introduce a different technique, explained in more detailed below, which is based on compactness of the relevant propagation observables at any fixed time.…”