The Geometry of Noncommutative Spacetimes

The Geometry of Noncommutative Spacetimes The idea that spacetime may be quantised was first pondered by Werner Heisenberg in the 1930s (see [1] for a historical review). His proposal was motivated by the urgency of providing a suitable regularisation for quantum electrodynamics. The first concrete model of a quantum spacetime, based on a noncommutative algebra of ‘coordinates’, was constructed by Hartland Snyder in 1949 [2] and extended by Chen-Ning Yang shortly afterwards [3]. With the development of the renormalisation theory, the concept of quantum spacetime became, however, less popular.
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. It permits one to equip noncommutative algebras with differential calculi, compatible with their inherent topology [7,8,9].
Meanwhile, 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. This is because particles can never be strictly localised [10,11,12] and, more generally, quantum states cannot be distinguished by means of observables in a very small region of spacetime (cf. [13], p. 131).
Nowadays, there exists a plethora of models of noncommutative (i.e., quantum) spacetimes. Most of them are connected with some quantum gravity theory and founded on the postulate that there exists a fundamental length-scale in Nature, which is of the order of Planck length λP∼(Gℏc−3)1/2≈1.6×10−35 m (see, for instance, [14] for a comprehensive review).
The ‘hypothesis of noncommutative spacetime’ is, however, plagued with serious conceptual problems (cf. for instance, [15]). Firstly, one needs to adjust the very notions of space and time. This is not only a philosophical problem, but also a practical one: we need a reasonable physical quantity to parametrise the observed evolution of various phenomena. Secondly, 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 also geometric in any suitable mathematical sense? This riddle not only affects the expected quantum gravity theory, but in fact 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 argue that the latter provides a conceptually transparent framework, although this comes at the price of involving rather abstract mathematical structures. In the next section we introduce the language of C∗-algebras and provide a short survey of the operational viewpoint on noncommutative spacetime. 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.