The foundational origin of integrability in quantum field theory

Details The foundational origin of integrability in quantum field theory The foundational origin of integrability in quantum field theory

2011-09-06

arxiv.org/abs/1109.1212v3

Physics

Mathematical Physics

52 pages, more detailed explanations and some corrections. arXiv admin note: text overlap with arXiv:0912.2874

Scientific paper

There are two foundational model-independent ways of thinking about integrability in QFT. One is "dynamical" and generalizes the solvability in closed form known from the Kepler two-body problem and its quantum mechanical counterpart. The other, referred to as kinematical integrability, has no classical nor quantum mechanical counterpart; it describes the relation between observable algebras and their inequivalent representation classes in form of a discrete algebra describing the a group dual or a braid group representation. The dynamical integrability is defined in terms of properties of wedge localization and uses the fact that modular localization theory allows to "emulate" the application of operators of the incoming wedge-localized interaction-free algebra inside the corresponding interacting algebra. Emulation can be viewed as a kind of generalization of the intrinsic (no Lagrangian quantization) Wigner particle representation theory to interacting systems. It also leads to a profound understanding of crossing in particle physics and places it into sharp difference to the kind of crossing used in the dual model and string theory. Discrete integrability has a particular interesting realization in higher dimensional conformal theories where the spectrum of anomalous scale dimensions turns out to be a special aspect of this integrability. In its formulation the Huygens principle (timelike commutation) of observables play an important role. Since QFT is the theory which arises from the principle of causal localization, dynamical as well as discrete integrability have their foundational origin in localization.

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