Space/Time Non-Commutativity and Causality

TL;DR

This paper compares non-commutative field theories and string theories, showing that while non-commutative fields exhibit acausality and nonlocal effects, string theories with similar non-commutativity do not, due to stringy effects.

Contribution

It demonstrates that string theory cancels acausal effects seen in non-commutative field theories with space/time non-commutativity.

Findings

Non-commutative field theories with space/time non-commutativity are acausal and violate unitarity.

String theory in a background electric field does not exhibit acausal behavior despite similar non-commutativity.

Stringy effects cancel acausal effects in string theory, unlike in non-commutative field theories.

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.