Wigner particle theory and local quantum physics

Abstract

Wigner's irreducible positive energy representations of the Poincaré group are often used to give additional justifications for the Lagrangian quantization formalism of standard QFT. Here we study another more recent aspect. We explain in this paper modular concepts by which we are able to construct the local operator algebras for all standard positive energy representations directly i.e. without going through field coordinatizations. In this way the artificial emphasis on Lagrangian field coordinates is avoided from the very beginning. These new concepts allow to treat also those cases of "exceptional" Wigner representations associated with anyons and the famous Wigner "spin tower"which have remained inaccessible to Lagrangian quantization. Together with the d=1+1 factorizing models (whose modular construction has been studied previously), they form an interesting family of theories with a rich vacuum-polarization structure (but no on shell real particle creation) to which the modular methods can be applied for their explicit construction. We explain and illustrate the algebraic strategy of this construction.We also comment on possibilities of formulating the Wigner theory in a setting of a noncommutative spacetime substrate. This is potentially interesting in connection with recent unitarity-and Lorentz invariancepreserving results of the special nonlocality caused by this kind of noncommutativity. * work supported by CNPq

Constructive proposals for QFT based on the crossing property and on lightfront holography

“…These representations come with two localization concepts: the Newton-Wigner localization [15] and the more recent modular localization [16][17] [18].…”

“…In the absence of interactions they turn out to consist precisely of those O-localized operators which are linear in the Wigner creation/annihilation operators. In that case a denumerable covariant pointlike basis of PFGs is conveniently described in terms of the well-known set of interwining functions 13 To maintain simplicity we limit our presentation to the bosonic situation and refer to [17] [18] for the general treatment.…”

“…This raises the interesting question whether there is some common feature to interaction-induced vacuum polarization clouds which permits a finer classification of interactions; this is a problem which certainly must be solved if one wants to use this intrinsic characterization 17 The only exception is the case of massless theories in d=1+1. 18 The J-S theorem can easily be generalized to subwedge-localized PFGs [3].…”

“…The representation of the generators (28) in terms of on-shell Z-F creation/annihilation operators simplifies the calculation of the lightray limit x − = 0. The method of lightfront restriction works exactly as in the case of d=1+1 free fields (18) except that corresponding formulas in terms of the Z-F operators only serve as generators of half-line algebras. The algebras of finite intervals have to be calculated as relative commutants by the modular inclusion formalism; the resulting infinite series in the Z-F operators are completely analogous to (35) in section 4.…”

Massless Relativistic Wave Equations and Quantum Field Theory

“…[53]). We conclude this paper commenting on the relations of the free net construction in Section 3 with the construction given in [20] which is based on the notion of modular localisation (see also [27]). …”