Open String Star as a Continuous Moyal Product

TL;DR

The paper demonstrates that the open string star product in the zero-momentum sector can be recast as a continuous tensor product of mutually commuting two-dimensional Moyal products labeled by a continuous parameter $\kappa\in[0,\infty)$. The noncommutativity parameter is $\theta(\kappa)=2\tanh(\pi\kappa/4)$, with Moyal coordinates $(x(\kappa), y(\kappa))$ defined from even-position modes and Fourier-transformed odd-position modes; at $\kappa>0$ the sectors are decoupled and satisfy $[x(\kappa), y(\kappa')]_* = i\,\theta(\kappa)\,\delta(\kappa-\kappa')$, while the $\kappa=0$ mode commutes and corresponds to half-string momentum. The analysis uses diagonalization of the Neumann matrices to construct continuous oscillators and a functional Moyal product, and it clarifies the relation to Bars’ half-string approach while highlighting issues in formulating the full string field theory as a noncommutative field theory, particularly regarding the kinetic term and regularization. The work provides a structurally transparent view of the string field star algebra and points toward a path for its K-theoretic and NCFT interpretations, albeit with open questions about ghosts and rigorous operator definitions.

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 $κ\in [~0,\infty)$ parametrize the eigenvalues of the Neumann matrices; then the noncommutativity parameter is given by $θ(κ) =2\tanh(πκ/4)$. For each $κ$, 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 $κ=0$ 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.