Generalized Dirichlet normal ordering in open bosonic strings

TL;DR

This paper develops a generalized Dirichlet normal ordering for open bosonic strings in the presence of an antisymmetric background field, ensuring consistency with boundary conditions and equations of motion at the quantum level.

Contribution

It introduces a novel generalized Dirichlet normal ordered product for open bosonic strings with antisymmetric backgrounds, extending previous work on Neumann conditions.

Findings

Constructed a generalized Dirichlet normal ordered product.

Ensured the product satisfies equations of motion and boundary conditions.

Provided a consistency check for the new normal ordering.

Abstract

Generally, open string boundary conditions play a nontrivial role in string theory. For example, in the presence of an antisymmetric tensor background field, they will lead the spacetime coordinates noncommutative. In this paper, we mainly discuss how to build up a generalized Dirichlet normal ordered product of open bosonic string embedding operators that satisfies both the equations of motion and the generalized Dirichlet boundary conditions at the quantum level in the presence of an antisymmetric background field, as the generalized Neumann case has already been discussed in the literature. Further, we also give a brief check of the consistency of the theory under the newly introduced normal ordering.