Quantized equations of motion in non-commutative theories

TL;DR

This paper proposes a new light-cone based time ordering in non-commutative quantum field theories, resolving issues with equations of motion, conserved currents, and spacetime symmetries, while maintaining unitarity.

Contribution

It introduces a novel time ordering method with respect to a light-cone variable that preserves symmetries and unitarity in non-commutative quantum field theories.

Findings

Time ordering with light-cone variable resolves equations of motion issues.

Spacetime symmetries SO(1,1) x SO(2) are preserved.

Unitarity of the theory is maintained.

Abstract

Quantum field theories based on interactions which contain the Moyal star product suffer, in the general case when time does not commute with space, from several diseases: quantum equation of motions contain unusual terms, conserved currents can not be defined and the residual spacetime symmetry is not maintained. All these problems have the same origin: time ordering does not commute with taking the star product. Here we show that these difficulties can be circumvented by a new definition of time ordering: namely with respect to a light-cone variable. In particular the original spacetime symmetries SO(1,1) x SO(2) and translation invariance turn out to be respected. Unitarity is guaranteed as well.