A scale-covariant quantum spacetime
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 physical interpretations together with further perspectives presented.NEWLINENEWLINEThe main inspiration and many specific techniques in the paper originate from the article [Commun. Math. Phys. 172, No. 1, 187--220 (1995; Zbl 0847.53051)] of \textit{S. Doplicher} et al., where a (non-scale-covariant) model of quantum Minkowski spacetime was introduced.
scientific article; zbMATH DE number 6362143