The Reduced Phase Space of An Open String in The Background B-Field

TL;DR

This paper analyzes the phase space structure of an open string in a background B-field, revealing non-commutative boundary coordinates and proposing a method to compute Dirac brackets without expanding in equations of motion solutions.

Contribution

It introduces a discretized model to identify the reduced phase space influenced by infinite second class constraints and clarifies how to compute Dirac brackets effectively.

Findings

String coordinates are non-commutative at boundaries

A discretized approach simplifies Dirac bracket calculations

Constraints should be imposed in suitable coordinates, not via equations of motion solutions

Abstract

The problem of an open string in background $B$-field is discussed. Using the discretized model in details we show that the system is influenced by infinite number of second class constraints. We interpret the allowed Fourier modes as the coordinates of the reduced phase space. This enables us to compute the Dirac brackets more easily. We prove that the coordinates of the string are non-commutative at the boundaries. We argue that in order to find the Dirac bracket or commutator algebra of the physical variables, one should not expand the fields in terms of the solutions of the equations of motion. Instead, one should impose the set of constraints in suitable coordinates.