Unitarity in space-time noncommutative field theories

TL;DR

This paper constructs a manifestly unitary S-matrix for space-time noncommutative scalar field theories by redefining time-ordering, resolving previous unitarity issues in perturbative analyses.

Contribution

It introduces a proper functional formalism for the S-matrix in noncommutative field theories, ensuring unitarity even when time is involved in noncommutativity.

Findings

The S-matrix is explicitly shown to be unitary.

Perturbation theory respects unitarity with the new formalism.

Exact two-particle scattering amplitude derived for 1+1D noncommutative nonlinear Schrödinger model.

Abstract

In non-commutative field theories conventional wisdom is that the unitarity is non-compatible with the perturbation analysis when time is involved in the non-commutative coordinates. However, as suggested by Bahns et.al. recently, the root of the problem lies in the improper definition of the time-ordered product. In this article, functional formalism of S-matrix is explicitly constructed for the non-commutative $\phi^p$ scalar field theory using the field equation in the Heisenberg picture and proper definition of time-ordering. This S-matrix is manifestly unitary. Using the free spectral (Wightmann) function as the free field propagator, we demonstrate the perturbation obeys the unitarity, and present the exact two particle scattering amplitude for 1+1 dimensional non-commutative nonlinear Schr\"odinger model.