Time-ordered Perturbation Theory on Noncommutative Spacetime: Basic Rules

TL;DR

This paper develops a consistent time-ordered perturbation theory framework for noncommutative spacetime, revealing significant differences from naive approaches and highlighting the impact of nonlocal interactions on Green functions.

Contribution

It introduces a rigorous extension of time-ordered perturbation theory to noncommutative field theory, correcting previous naive methods and clarifying the role of nonlocal interactions.

Findings

Different Green functions from naive approaches

A natural framework for NC perturbation theory

Modified analytic properties due to nonlocal interactions

Abstract

Assuming the S-matrix on noncommutative (NC) spacetime can still be developped perturbatively in terms of the time-ordered exponential of the interaction Lagrangian, we investigate the perturbation theory of NC field theory. We first work out with care some typical Green functions starting from the usual concepts of time-ordering and commutation relations for free fields. The results are found to be very different from those in the naive approach pursued in the literature. A simple framework then appears naturally which can incorporate the new features of our results and which turns out to be the usual time-ordered perturbation theory extended to the NC context. We provide the prescriptions for computing S-matrix elements and Green functions in this framework. We also emphasize that the naive seemingly covariant approach cannot be reproduced from the current one, in contrast to the field theory on ordinary spacetime. We attribute this to the phase-like nonlocal interaction intrinsic in NC field theory which modifies the analytic properties of Green functions significantly.