Modular Wedge Localization and the d=1+1 Formfactor Program

Abstract

In this paper I continue the study of the new framework of modular localization and its constructive use in the nonperturbative d=1+1 Karowski-Weisz-Smirnov formfactor program. Particular attention is focussed on the existence of semilocal generators of the wedge-localized algebra without vauum polarization (FWG-operators) which are closely related to objects fulfilling the Zamolodchikov-Faddeev algebraic structure. They generate a ``thermal Hilbert space'' and allow to understand the equivalence of the KMS conditions with the so-called cyclicity equation for formfactors which was known to be closely related to crossing symmetry properties. The modular setting gives rise to interesting new ideas on ``free'' d=2+1 anyons and plektons.Comment: the fourth section has been rewritten in order to remove an error and to gain additional clarity. 32 pages, tcilate

Modular localization and the holistic structure of causal quantum theory, a historical perspective

“…In fact it will become clear in the course of the present work that it indeed can and should be viewed this way. For some time this problem remained out of my range of interest; I did not want to loose time on something which would draw me into opaque historical problems away from my research on new foundational insights into to QFT via "modular localization" 2 [6]. During a two year stay (2002/2003) in Brazil, a CNPq supported research project "The Modular Structure of Causal Quantum Physics" provided the chance to extend this research.…”

“…The study of the relation between modular operator theory and causal localization in LQP started a decade later [19], and its first application consisted in a more profound understanding [20] of the Unruh Gedankenexperiment [21]. The terminology "modular localization" is more recent and marks the beginning of a new constructive strategy in QFT based on the modular aspects of localization of states and algebras [44] [6]. In mathematics the theory was decisive instrument which led to Connes closure of the Murray-von Neumann project of classifying von Neumann factor algebras.…”

“…These on-shell analytic properties find their important manifestation in the particle crossing property. It is also the starting point of the algebraic construction of integrable QFT [6]. The connection between algebraic and analytic properties is much more subtle for on-shell objects as the S-matrix and formfactors than for off-shell correlation function.…”

“…The localization problems of the infinite spin class were finally solved [44] with the help of modular localization which for different problems was already used in [6]. In fact the main theorem in that paper states [44] that all positive energy wave functions are localizable in noncompact spacelike cones and only the first two classes permit the sharper localization in double cones (the causal shadow of a 3-dim.…”

“…In contrast to the approximation of the latter in terms of box-quantized finite volume Gibbs states, the split formalism for open subsystems is a part of the (presently computational rather inaccessible) modular localization theory. It is in particular not clear whether the density matrix from the split property leads to a plain dimensionless area law f area/ε 2 24 as in (6) or to a logarithmically modified area law [47]. For chiral conformal theories on the lightray there is a rigorous derivation of the localization entropy for an interval with vacuum attenuation length ε (surface fuzziness) from the well-known linear length l → ∞ behavior (the "one-dimensional volume factor" l).…”