A fresh look at midpoint singularities in the algebra of string fields

TL;DR

This paper investigates the midpoint structure of open string algebra using the operator/Moyal formalism, revealing new (anti)commutative coordinates, associativity issues, and obstructions in the Moyal product formulation.

Contribution

It introduces a split string description, identifies new coordinates, and analyzes associativity breakdown and poles in the non(anti)commutativity parameter.

Findings

Identified new (anti)commutative coordinates in the string star algebra.

Showed how poles relate to null operators affecting the star product.

Highlighted obstructions to a well-defined Moyal product formulation.

Abstract

In this paper we study the midpoint structure of the algebra of open strings from the standpoint of the operator/Moyal formalism. We construct a split string description for the continuous Moyal product of hep-th/0202087, study the breakdown of associativity in the star algebra, and identify in infinite sequence of new (anti)commutative coordinates for the star product in in the complex plane. We also explain how poles in the open string non(anti)commutativity parameter correspond to certain ``null'' operators which annihilate the vertex, implying that states proportional to such operators tend to have vanishing star product with other string fields. The existence of such poles, we argue, presents an obstruction to realizing a well-defined formulation of the theory in terms of a Moyal product. We also comment on the interesting, but singular, representation $L_0$ which has appeared prominently in the recent studies of Bars {\it et al}.