Quantum Spacetime: a Disambiguation

We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the "canonical quantum spacetime". The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance.Comment: special issue of SIGMA "Noncommutative Spaces and Fields

Quantum Spacetime and Algebraic Quantum Field Theory

We review the investigations on the quantum structure of spactime, to be found at the Planck scale if one takes into account the operational limitations to localization of events which result from the concurrence of Quantum Mechanics and General Relativity. We also discuss the different approaches to (perturbative) Quantum Field Theory on Quantum Spacetime, and some of the possible cosmological consequences.Comment: 49 pages, 2 figure

Twisted Covariance and Weyl Quantisation

In this letter we wish to clarify in which sense the tensor nature of the commutation relations [x^mu,x^nu]=i theta ^{mu nu} underlying Minkowski spacetime quantisation cannot be suppressed even in the twisted approach to Lorentz covariance. We then address the vexata quaestio "why theta"?Comment: 6 pages, pdf has active hyperlinks Slight change in title. Appendix added on more general coordinates for symbols. References added. To appear in the Proceedings of the XXV Max Born Symposium, Wroclaw, June 29-July 3, 200

Any compact group is a gauge group

The assignment of local observables in the vacuum sector, fulfilling the standard axioms of local quantum theory, is known to determine uniquely a compact group G of gauge transformations of the first kind together with a central involutive element k of G, and a complete normal algebra of fields carrying the localizable charges, on which k defines the Bose/Fermi grading. We show here that any such pair {G,k}, where G is compact metrizable, does actually appear. The corresponding model can be chosen to fulfill also the split property. This is not a dynamical phenomenon: a given {G,k} arises as the gauge group of a model where the local algebras of observables are a suitable subnet of local algebras of a possibly infinite product of free field theories.Comment: 13 pages, LaTeX. To appear on Reviews in Mathematical Physics. References added; minor changes in styl

Non Local Theories: New Rules for Old Diagrams

We show that a general variant of the Wick theorems can be used to reduce the time ordered products in the Gell-Mann & Low formula for a certain class on non local quantum field theories, including the case where the interaction Lagrangian is defined in terms of twisted products. The only necessary modification is the replacement of the Stueckelberg-Feynman propagator by the general propagator (the ``contractor'' of Denk and Schweda) D(y-y';tau-tau')= - i (Delta_+(y-y')theta(tau-tau')+Delta_+(y'-y)theta(tau'-tau)), where the violations of locality and causality are represented by the dependence of tau,tau' on other points, besides those involved in the contraction. This leads naturally to a diagrammatic expansion of the Gell-Mann & Low formula, in terms of the same diagrams as in the local case, the only necessary modification concerning the Feynman rules. The ordinary local theory is easily recovered as a special case, and there is a one-to-one correspondence between the local and non local contributions corresponding to the same diagrams, which is preserved while performing the large scale limit of the theory.Comment: LaTeX, 14 pages, 1 figure. Uses hyperref. Symmetry factors added; minor changes in the expositio

Poincar\'e Covariant k-Minkowski Spacetime

A fully Poincare' covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincare' group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincare' covariance is realised a` la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of "Poincare covariance".Comment: 9 pages, with active links. Minor changes in presentation. To appear on PL