Asymptotic completeness in quantum field theory

• Applicant: Professor Dr. Wojciech Dybalski
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields ; Mathematics
• Term: from 2013 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 243517318

• Final Report Year: 2021

Final Report Abstract

The research work performed within my Emmy Noether project brought numerous new insights into scattering theory of quantum systems with infinitely many degrees of freedom. In particular: • Construction of scattering states of one electron and photons in the Buchholz-Roberts setting of relativistic QED. This work helped to clarify the distinction between the infraparticle picture of the electron as a composite object with fuzzy mass, consisting of a bare electron and a soft-photon cloud correlated with its velocity, and the infravacuum picture of the electron as an elementary object, possibly with a sharp mass, moving in a highly fluctuating background radiation, not correlated with its velocity. The infravacuum picture of the electron has been investigated, where some evidence for the sharp mass has been found (e.g. absence of velocity superselection). • Development of a new method, based on non-commutative recurrence relations, for controlling localization of electrons in space which led to a mathematical control of atom-electron scattering and deeper understanding of the Faddeev-Kulish approach in the massless Nelson model. These works bring us closer to a mathematical control of collisions of several infraparticles in this model. • Non-triviality of infraparticle scattering in 2d massless free field in an irregular vacuum representation. • Construction of scattering states for ‘spin waves’ in a class of gapped quantum spin systems. This work received the AHP Prize for the most remarkable paper in Annales Henri Poincaré in 2016 and provided a basis for model-independent investigations of the problem of asymptotic completeness in spin systems. General criteria for asymptotic completeness, in terms of asymptotic observables, were obtained. • A surprise in the course of the project was the construction of N-particle scattering states in wedgelocal relativistic QFT due to my PhD student Mr. Duell. This is a remarkable result, given the long-standing belief that only two-particle scattering states can be constructed in wedgelocal theories. This latter conviction, based on the geometric fact that at most two particles can be localized in two spacelike-separated wedges, misses certain swapping symmetry between the opposite wedge-algebras, which makes the construction of N-particle scattering states possible. In a follow up work of Dr. Duell and myself asymptotic completeness for interacting wedge-local QFT is shown. • Insight that asymptotically complete theories of massless Wigner particles satisfy the Bisognano-Wichmann property.

Publications

• Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems. Ann. Henri Poincaré 17, (2016) 1737–1791 [AHP Prize 2016]
  S. Bachmann, W. Dybalski and P. Naaijkens
  (See online at https://doi.org/10.1007/s00023-015-0440-y)
• Compton scattering in the Buchholz-Roberts framework of relativistic QED. Lett. Math. Phys. 107, (2017) 81–106
  S. Alazzawi and W. Dybalski
  (See online at https://doi.org/10.1007/s11005-016-0889-8)
• From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons. Nuclear Physics B 925, (2017) 455–469
  W. Dybalski
  (See online at https://doi.org/10.1016/j.nuclphysb.2017.10.018)
• Strengthened Reeh-Schlieder property and scattering in quantum field theories without mass-gaps. Commun. Math. Phys. 352, (2017) 935–966
  M. Duell
  (See online at https://doi.org/10.1007/s00220-017-2841-x)
• Asymptotic observables in gapped quantum spin systems. Commun. Math. Phys. 357, (2018) 231–248
  W. Dybalski
  (See online at https://doi.org/10.1007/s00220-017-2954-2)
• Coulomb scattering in the massless Nelson model III. Ground state wave functions and non-commutative recurrence relations. Ann. Henri Poincaré 19, (2018) 463–514
  W. Dybalski and A. Pizzo
  (See online at https://doi.org/10.1007/s00023-017-0642-6)
• Coulomb scattering in the massless Nelson model IV. Atomelectron scattering
  W. Dybalski and A. Pizzo
  (See online at https://doi.org/10.48550/arXiv.1902.08799)
• N-particle scattering in relativistic wedge-local QFT. Commun. Math. Phys. 364, (2018) 203–232
  M. Duell
  (See online at https://doi.org/10.1007/s00220-018-3183-z)
• The Bisognano-Wichmann property for asymptotically complete massless QFT. Commun. Math. Phys. 380, (2020) 1267–1294
  W. Dybalski and V. Morinelli
  (See online at https://doi.org/10.1007/s00220-020-03755-8)
• Asymptotic completeness in a class of massive wedge-local quantum field theories in any dimension
  M. Duell and W. Dybalski
  (See online at https://doi.org/10.48550/arXiv.2111.04831)
• Infraparticle states in the massless Nelson model – revisited
  V. Beaud, W. Dybalski and G.M. Graf
  (See online at https://doi.org/10.48550/arXiv.2105.05723)
• Interacting massless infraparticles in 1+1 dimensions
  W. Dybalski and J. Mund
  (See online at https://doi.org/10.1007/s00220-022-04451-5)