Causality in noncommutative two-sheeted space-times

Abstract

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2-or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.

Causality for Nonlocal Phenomena

“…In order to obtain the second inequality, notice that by (27) we have ϕ(p 2 ) ≤ ψ(q). By the transitivity of the relation , this inequality holds also if we replace p 2 with any x p 2 .…”

The Geometry of Noncommutative Spacetimes

“…This peculiarity is attested by the following result [22,Theorem 9] and illustrated on Figure 1. In fact, one can extend this result (cf.…”

“…Let M be a globally hyperbolic spacetime and (A M , K M , D / ) the associated Lorentzian spectral triple. We form an almost-commutative spectral triple as follows (see [22] for the full story):…”

“…What is more, the above-sketched procedure generalises in a straightforward way to the case of other, non-gravitational, interactions (cf. [22,26]). The causal structure of almost-commutative spacetimes might thus open a new avenue towards the construction of a nonperturbative algebra of quantum fields.…”

“…Subsequently, we briefly sketch the rudiments of noncommutative geometry à la Connes [4]. Next, we discuss the notion of causality suitable in this context, summarising the outcome of our recent works [18][19][20][21][22][23][24][25][26]. Finally, we explain how the presumed noncommutative structure of spacetime extorts a modification of the axioms of quantum field theory and thus might yield empirical consequences.…”