Normal ordering and boundary conditions in open bosonic strings

TL;DR

This paper develops a method for constructing normal ordered products of bosonic string operators that satisfy boundary conditions and equations of motion, revealing non-commutative structures in string theory with background fields.

Contribution

It introduces a systematic way to build normal ordered products for open bosonic strings that respect boundary conditions at the quantum level, including in backgrounds with antisymmetric tensors.

Findings

Normal ordered products satisfy boundary conditions and equations of motion.

Equal time commutators reveal non-commutativity induced by background fields.

Method applies to various boundary conditions, including mixed types.

Abstract

Boundary conditions play a non trivial role in string theory. For instance the rich structure of D-branes is generated by choosing appropriate combinations of Dirichlet and Neumann boundary conditions. Furthermore, when an antisymmetric background is present at the string end-points (corresponding to mixed boundary conditions) space time becomes non-commutative there. We show here how to build up normal ordered products for bosonic string position operators that satisfy both equations of motion and open string boundary conditions at quantum level. We also calculate the equal time commutator of these normal ordered products in the presence of antisymmetric tensor background.