Failure of microcausality in noncommutative field theories

TL;DR

This paper investigates violations of microcausality in noncommutative quantum field theories, showing that certain commutators do not vanish at spacelike separation, indicating causality violations in these models.

Contribution

It provides a rigorous analysis of the support properties of commutators in noncommutative field theories and demonstrates causality violations even in the commuting directions.

Findings

Commutators do not vanish at spacelike separation in noncommuting directions.

Matrix elements of the commutator decay rapidly along spacelike directions.

Star commutators violate causality even in commuting directions.

Abstract

We revisit the question of microcausality violations in quantum field theory on noncommutative spacetime, taking $O(x)=:\phi\star\phi:(x)$ as a sample observable. Using methods of the theory of distributions, we precisely describe the support properties of the commutator [O(x),O(y)] and prove that, in the case of space-space noncommutativity, it does not vanish at spacelike separation in the noncommuting directions. However, the matrix elements of this commutator exhibit a rapid falloff along an arbitrary spacelike direction irrespective of the type of noncommutativity. We also consider the star commutator for this observable and show that it fails to vanish even at spacelike separation in the commuting directions and completely violates causality. We conclude with a brief discussion about the modified Wightman functions which are vacuum expectation values of the star products of fields at different spacetime points.