Massless Asymptotic Fields and Haag–Ruelle Theory

Abstract

Abstract.We revisit the problem of the existence of asymptotic massless boson fields in quantum field theory. The well-known construction of such fields by Buchholz (Commun. Math. Phys. 52:147-173, 1977; Commun. Math. Phys. 85:49-71, 1982) is based on locality and the existence of vacuum vector, at least in regions spacelike to spacelike cones. Our analysis does not depend on these assumptions and supplies a more general framework for fields only very weakly decaying in spacelike directions. In this setting, the existence of appropriate null asymptotes of fields is linked with their spectral properties in the neighborhood of the lightcone. The main technical tool is one of the results of a recent analysis by one of us (Herdegen in Lett. Math. Phys. 104:1263-1280. doi:10.1007/ s11005-014-0710-5, 2014), which allows application of the null asymptotic limit separately to creation/annihilation parts of a wide class of nonlocal fields. In vacuum representation the scheme allows application of the methods of the Haag-Ruelle theory closely analogous to those of the massive case. In local case this Haag-Ruelle procedure may be combined with the Buchholz method, which leads to significant simplification.Mathematics Subject Classification. 81T05, 81U99.

The Bisognano–Wichmann Property for Asymptotically Complete Massless QFT

“…Namely, we extract the creation and annihilation parts of the asymptotic fields (2.13) in order to facilitate the construction of scattering states. We still follow [AD17] which in turn relied here on [DH15]. Let θ ∈ C ∞ (R), 0 ≤ θ ≤ 1, be supported in (0, ∞) and equal to one on (1, ∞).…”

Compton scattering in the Buchholz–Roberts framework of relativistic QED

“…A coordinate independent treatment of Fourier transforms on the sphere can be found in [DH15]. We give here an elementary coordinate dependent computation.…”

“…Instead, we take the opportunity to indicate another simplification. Namely, in the analysis of commutators of asymptotic fields in Proposition 5.7 below, the clustering estimates from [Bu77,DH15] can be avoided due to Theorem 4.6.…”

“…in theories satisfying some kind of asymptotic abelianess in spacelike directions. For example, it should help to remove Assumption 4 of [He14] and Assumption 3 of [DH15]. It might also find applications in scattering theory of quantum spin systems satisfying the Lieb-Robinson bounds [BDN14].…”

On Energy-Momentum Transfer of Quantum Fields

“…Appendices contain a few lemmas which are needed in the main text, but which may also be of a more general technical interest. The implications of Theorem 5 will also be used (in a future publication [7]) for the extension to the case of massless fields of the methods used in [12] for the analysis of scattering and particle structure in quantum field theory.…”