Canonical Quantization of Open String and Noncommutative Geometry

TL;DR

This paper develops a canonical quantization framework for open strings in a D-brane background with a B-field, revealing the noncommutative geometry through explicit solutions and constraints.

Contribution

It introduces a novel approach to quantize open strings with mixed boundary conditions, linking them to orbifold conditions and clarifying the emergence of noncommutativity.

Findings

Derived a simple Hamiltonian for open strings in B-field backgrounds.

Showed the equivalence of boundary constraints to orbifold conditions.

Explicitly demonstrated the noncommutative structure of the string coordinates.

Abstract

We perform canonical quantization of open strings in the $D$-brane background with a $B$-field. Treating the mixed boundary condition as a primary constraint, we get a set of secondary constraints. Then these constraints are shown to be equivalent to orbifold conditions to be imposed on normal string modes. These orbifold conditions are a generalization of the familiar orbifold conditions which arise when we describe open strings in terms of closed strings. Solving the constraints explicitly, we obtain a simple Hamiltonian for the open string, which reveals the nature of noncommutativity transparently.