Normal ordering and noncommutativity in open bosonic strings

TL;DR

This paper investigates noncommutativity in open bosonic strings with background B-fields using a novel normal ordering approach that satisfies both equations of motion and boundary conditions, clarifying the origin of noncommutativity.

Contribution

It introduces a new normal ordering method that simplifies the derivation of mode algebra and boundary noncommutativity in open bosonic strings.

Findings

Normal ordering satisfies both equations of motion and boundary conditions.

The mode algebra matches previous results, confirming noncommutativity at boundaries.

The approach provides a clearer understanding of the origin of noncommutativity.

Abstract

Noncommutativity in an open bosonic string moving in the presence of a background Neveu-Schwarz two-form field $B_{\mu \nu}$ is investigated in a conformal field theory approach, leading to noncommutativity at the boundaries. In contrast to several discussions, in which boundary conditions are taken as Dirac constraints, we first obtain the mode algebra by using the newly proposed normal ordering, which satisfies both equations of motion and boundary conditions. Using these the commutator among the string coordinates is obtained. Interestingly, this new normal ordering yields the same algebra between the modes as the one satisfying only the equations of motion. In this approach, we find that noncommutativity originates more transparently and our results match with the existing results in the literature.