Space/Time Non-Commutativity and Causality

TL;DR

The paper investigates whether space/time noncommutativity induces acausality and unitarity violations by comparing scattering in NC field theory with open string theory in a background electric field. In NC field theory, space/space NC yields spatial displacements consistent with rod-like nonlocality, while space/time NC in 1+1 dimensions produces advanced wave components that appear to precede the collision, signaling acausal behavior. By contrast, open string scattering with an electric field exhibits stringy oscillator effects that cancel these acausal phases, leading only to causal time delays that scale with momentum; the delays grow as the field approaches its critical value, though the amplitudes are suppressed by the effective coupling. The results show that acausal signatures are not a universal feature of space/time NC and that string dynamics play a crucial role in preserving causality, highlighting a fundamental distinction between NC field theories and their string-theoretic realizations.

Abstract

Field theories based on non-commutative spacetimes exhibit very distinctive nonlocal effects which mix the ultraviolet with the infrared in bizarre ways. In particular if the time coordinate is involved in the non-commutativity the theory seems to be seriously acausal and inconsistent with conventional Hamiltonian evolution. To illustrate these effects we study the scattering of wave packets in a field theory with space/time non-commutativity. In this theory we find effects which seem to precede their causes and rigid rods which grow instead of Lorentz contract as they are boosted. These field theories are evidently inconsistent and violate causality and unitarity. On the other hand open string theory in a background electric field is expected to exhibit space/time non-commutativity. This raises the question of whether they also lead to acausal behavior. We show that this is not the case. Stringy effects conspire to cancel the acausal effects that are present for the non-commutative field theory.