Polarization-Free Generators and the S-Matrix

Abstract

Polarization-free generators, i.e. ``interacting'' Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories.Comment: Dedicated to the memory of Harry Lehmann, 19 pages; revised version (proof of Lemma 3.4 corrected

Deformations of Quantum Field Theories and Integrable Models

“…If the velocity supports of f 1 , g 1 lie in a suitable relative position to the wedge W 0 , namely Γ(f 1 ) − Γ(g 1 ) ⊂ W 0 , these regions are spacelike for t > 0. As t → ∞, we therefore find two-particle outgoing scattering states as the limits [BBS01] …”

“…Using the algebraic framework of quantum field theory [Haa96], it is also in principle possible [BL04] to extract all observables localized in bounded spacetime regions. Moreover, the localization in wedges is sharp enough to consistently compute the two-particle scattering matrix [BBS01], and decide if the constructed model exhibits non-trivial interaction.…”

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

“…This means that for a vector state created by applying a wedge-local operator from A in (W ) to the vacuum there will be a corresponding uniquely defined operator in A(W ) operator which, applied to the vacuum creates the same vector. Existence and uniqueness is secured by modular theory applied to the wedge region [53]. We refer to this bijection between wedge local operators as: emulation of wedge localized free fields within the interacting wedge algebra [10][26] and denote the emulated image by a subscript A(W )…”

“…Similarly the action of emulated incoming fields on an incoming state is an infinite superposition of incoming particle states even though the emulated momenta are on-shell. 32 In earlier publications the special case of an emulated incoming field was referred to as a vacuum polarization free generators (PFG) [53]. transforms of the Zamolodchikov operators which obey the Zamolodchikov-Faddeev algebra.…”

“…This simplicity has its mathematical origin in restrictive domain properties of emulats which characterize integrability [53]. Emulats in general QFT only inherit the invariance property of their domains under the wedge-preserving subgroup.…”