Spectral Representation and Dispersion Relations in Field Theory on Noncommutative Space

TL;DR

This paper investigates spectral representations and dispersion relations in noncommutative field theory, highlighting the role of NC parameters, weaker causality conditions, and challenges in formulating dispersion relations due to unphysical regions.

Contribution

It introduces a spectral analysis framework for NC field theory, explores the implications of reduced symmetries, and discusses the difficulties in establishing dispersion relations in this context.

Findings

Spectral variables naturally incorporate NC parameters.

Subtractions for divergences are made at fixed NC variable values.

Dispersion relations face obstacles due to unphysical regions in NC space.

Abstract

We study the spectral representation and dispersion relations that follow from some basic assumptions and the reduced spacetime symmetries on noncommutative (NC) space. Kinematic variables involving the NC parameter appear naturally as parametric variables in this analysis. When subtractions are necessary to remove ultraviolet divergences, they are always made at the fixed values of these NC variables. This point is also illustrated by a perturbative analysis of self-energies. Our analysis of the reduced spacetime symmetries suggests a weaker microcausality requirement. Starting from it, we make a first attempt at dispersion relations for forward scattering. It turns out that the attempt is hampered by a new unphysical region specified by a given motion in the NC plane which does not seem to be surmountable using the usual tricks. Implications for a possible subtraction and renormalization scheme for NC field theory in which the ultraviolet-infrared (UV/IR) mixing is removed are also briefly commented on.