The Geometry of Noncommutative Spacetimes

Michał Eckstein

doi:10.3390/universe3010025

Abstract

We review the concept of ‘noncommutative spacetime’ approached from an operational stand-point and explain how to endow it with suitable geometrical structures. The latter involves i.a. the causal structure, which we illustrate with a simple—‘almost-commutative’—example. Furthermore, we trace the footprints of noncommutive geometry in the foundations of quantum field theory.

Highlights

The idea that spacetime may be quantised was first pondered by Werner Heisenberg in the 1930s
The revival of Heisenberg’s idea came in the late 1990s with the development of noncommutative geometry [4,5,6]. The latter is an advanced mathematical theory sinking its roots in functional analysis and differential geometry
The classical spacetime has an inherent Lorentzian geometry, which determines, in particular, the causal relations between the events. This raises the question: Are noncommutative spacetimes geometric in any suitable mathematical sense? This riddle affects the expected quantum gravity theory, but any quantum field theory, as the latter are deeply rooted in the principles of locality and causality

Summary

Introduction

The idea that spacetime may be quantised was first pondered by Werner Heisenberg in the 1930s (see [1] for a historical review). On the physical side, it became clear that the concept of a point-like event is an idealisation—untenable in the presence of quantum fields. The classical spacetime has an inherent Lorentzian geometry, which determines, in particular, the causal relations between the events. This raises the question: Are noncommutative spacetimes geometric in any suitable mathematical sense? This riddle affects the expected quantum gravity theory, but any quantum field theory, as the latter are deeply rooted in the principles of locality and causality In this short review we advocate a somewhat different approach to noncommutative spacetime (cf [16,17]), based on an operational viewpoint. We explain how the presumed noncommutative structure of spacetime extorts a modification of the axioms of quantum field theory and might yield empirical consequences

Noncommutative Spacetime—An Operational Approach

Noncommutative Geometry à la Connes

Causality in Noncommutative Spacetimes

The Foundations of Quantum Field Theory Revisited