The disappearance of causality at small scale in almost-commutative manifolds

TL;DR

This paper explores how causality, modeled by isocones in almost-commutative algebras, vanishes at small scales, suggesting a fundamental limit to causal structure in noncommutative geometric models of spacetime.

Contribution

It introduces a family of isocones in almost-commutative algebras and conjectures this family provides a complete classification, linking noncommutative geometry to causality breakdown.

Findings

Every isocone in noncommutative algebra induces an order with incomparable points.

Causality disappears at small scales in models with noncommutative algebraic structures.

A proposed classification of isocones relates to the noncommutative Stone-Weierstrass conjecture.

Abstract

This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by so-called \emph{isocones} which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form ${\cal C}(M)\otimes A_f$, with $M$ a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in ${\cal C}(M)\otimes A_f$, with $A_f$ noncommutative, induces an order relation on $M$ with the property that every point in $M$ lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale.