Open String Star as a Continuous Moyal Product

TL;DR

This paper shows that the open string star product can be represented as a continuous family of Moyal products, revealing a new mathematical structure and potential for reformulating string field theory as a noncommutative field theory.

Contribution

It introduces a novel continuous tensor product representation of the open string star product using Moyal products parametrized by a continuous variable.

Findings

The noncommutativity parameter varies with ppa as ppa(ppa) = 2 tanh(ppa /4).

The ppa=0 coordinate corresponds to the string's half-momentum.

The work relates to Bars' approach and explores a noncommutative field theory formulation.

Abstract

We establish that the open string star product in the zero momentum sector can be described as a continuous tensor product of mutually commuting two dimensional Moyal star products. Let the continuous variable $\kappa \in [~0,\infty)$ parametrize the eigenvalues of the Neumann matrices; then the noncommutativity parameter is given by $\theta(\kappa) =2\tanh(\pi\kappa/4)$. For each $\kappa$, the Moyal coordinates are a linear combination of even position modes, and the Fourier transform of a linear combination of odd position modes. The commuting coordinate at $\kappa=0$ is identified as the momentum carried by half the string. We discuss the relation to Bars' work, and attempt to write the string field action as a noncommutative field theory.