A scale-covariant quantum spacetime
type: Item
instance of: scholarly article
title: A scale-covariant quantum spacetime
zbMATH Open document ID: 1305.81106
DOI: 10.1142/S0129055X14500068
author: Claudio Perini
author: Gabriele Nunzio Tornetta
published in: Reviews in Mathematical Physics
publication date: 2014-10-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.7050
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 47L90
Mathematics Subject Classification ID: 81T75
Mathematics Subject Classification ID: 46L60
zbMATH DE Number: 6362143
zbMATH Keywords: algebraic quantum field theory
zbMATH Keywords: quantum spacetime
zbMATH Keywords: DFR model
zbMATH Keywords: scale covariance
MaRDI profile type: Publication
arXiv ID: 1211.7050
cites work: Q5691044
cites work: Q4891080
cites work: Q5702682
cites work: Q4066072
cites work: Q5508878
OpenAlex ID: W3102989379
review text: The paper under review introduces and describes in detail a so-called scale covariant model of quantum spacetime. The starting point is a consideration of 4 unbounded self-adjoint operators \(q_{\mu}\), representing the coordinates of the system, and whose commutators are central elements, satisfying constraints related to the formal scaling limit. Specifically the set of commutation relations is the following: \([q_{\mu}, q_{\nu}] \subset i R_{\mu \nu}\), \([q_{\lambda}, R_{\mu\nu}]=0\), \(R_{\mu\nu} R^{\mu\nu} = 0\), \(R_{\mu\nu} (*R)^{\mu\nu} = 0\). In irreducible representations of these relations \(R_{\mu \nu}\) naturally become scalars, and the corresponding joint spectrum of the \(R\) operators, after deleting the `classical' point 0, is denoted \(\Sigma_0\). The authors analyse in detail the action of (the proper and ortochronous part of) the Lorentz group on \(\Sigma_0\), define the \(C^*\)-algebra of the model arising as the universal completion of a certain convolution type Banach algebra of functions on \(\Sigma_0\) with values in \(L^1(\mathbb{R}^4)\) and show that both the Poincaré group and the `scaling' group act on this algebra by automorphisms. In the second part of the article a massless scalar neutral field, understood as a family of operators in the tensor product of the model algebra and the algebra of operators on the Fock space, is constructed. This turns out to have similar covariance properties as those mentioned above (Poincaré group and `scaling' group), but lacks the locality property, which the authors connect to the removal of the `classical point', resulting in the non-existence of the states of absolute optimal localisation. Next a net of field algebras associated to certain particular (essentially two-dimensional) regions in the classical spacetime is studied and shown to be covariant with respect to the symmetry group discussed above. Finally a theorem concerning the irreducibility of a particular field algebra is proved and physic…
reviewed by: Adam Skalski
description: scientific article; zbMATH DE number 6362143