Exploring the Causal Structures of Almost Commutative Geometries

Abstract

Abstract. We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra S(R 1,1 ) ⊗ M 2 (C), which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.

Causality for Nonlocal Phenomena

“…The concept of causality in the space of states was explored [26,27] in the framework of 'almost commutative spacetimes', i.e. for C * -algebras of the form C 0 (M) ⊗ A F , with A F being a finite dimensional matrix algebra.…”

The Geometry of Noncommutative Spacetimes

“…In particular, the causal structure not only forbids a superluminal motion in M, but imposes explicit bounds on the speed of change in the models' internal space. This effect was visible in the example presented in the previous section, where the internal space consisted of just two points, and it would manifest itself in any almost-commutative spacetime (see e.g., [19]). …”

“…Example 3) is simply a Cartesian product of a spacetime M and an internal space F = P(A F ) [19,21], as all of the pure states on the algebra A M ⊗ A F are separable [30,Theorem 11.3.7]. In other words, there is no entanglement between spacetime and the internal space of an almost-commutative model.…”

“…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].…”

“…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.…”