Metrics and causality on Moyal planes

Abstract

Abstract. Metrics structures stemming from the Connes distance promote Moyal planes to the status of quantum metric spaces. We discuss this aspect in the light of recent developments, emphasizing the role of Moyal planes as representative examples of a recently introduced notion of quantum (noncommutative) locally compact space. We move then to the framework of Lorentzian noncommutative geometry and we examine the possibility of defining a notion of causality on Moyal plane, which is somewhat controversial in the area of mathematical physics. We show the actual existence of causal relations between the elements of a particular class of pure (coherent) states on Moyal plane with related causal structure similar to the one of the usual Minkowski space, up to the notion of locality.

Single extra dimension from κ-Poincaré and gauge invariance

“…Lorentzian noncommutative geometry involves a natural notion of causality rooted to the Dirac operator playing in some sense the role of a metric and underlies the notion of noncommutative metric geometry 5 . This noncommutative causality, which coincides to the usual one at the commutative limit, has been applied for almost-commutative manifolds [66] and for Moyal plane [71], giving in that latter case the explicit characterization of the relevant causal structure and by the way closing a controversy of the physics literature about the existence of some causality on the Moyal plane at the Planck scale.…”

The Lorentzian distance formula in noncommutative geometry

“…For almost commutative manifolds, all pure states are product states between well-defined states on the based spacetime (evaluation maps) and vector states on the discrete algebra. For deformation spaces, the elements of the initial pre-C * -algebra of bounded continuous functions are compact operators so all states correspond to vector states which can be easily and uniquely extended to unbounded functions as long as their evaluation remain finite, as used in [29]. Once more, this problem is an abstract one and does not prevent the application of the Lorentzian distance formula to particular models of noncommutative spacetimes.…”

The Geometry of Noncommutative Spacetimes

“…The concept of almost-commutative geometry naturally carries over to the Lorentzian setting [19,57] and the noncommutative examples include the Moyal-Minkowski spectral triple [53,63].…”

“…Now, the standard approach assumes that the fields are initially free, i.e., [Ψ L (x),Ψ R (y)] = 0 for all x, y ∈ M and one takes into account the interaction by the standard perturbative techniques. 3 A concrete model of a noncommutative spacetime with nonlocal events, but a rigid causal structure was developed in [63]. In the example presented above it is likely that the resulting quantum field theories would be equivalent, i.e., they would lead to the same S-matrix.…”