[Paper Review] Noncommutativity in open string: a gauge independent analysis

Rabin Banerjee, Biswajit Chakraborty+1 more|arXiv (Cornell University)|Mar 21, 2002

Black Holes and Theoretical Physics1 references4 citations

TL;DR

This paper investigates noncommutativity in open strings propagating in a Neveu-Schwarz B-field using a gauge-independent Hamiltonian framework. It demonstrates that noncommutativity arises directly from non-trivial boundary conditions—treated as dynamical constraints rather than imposed restrictions—and shows that noncommutativity persists at all string points, not just the ends, with the standard boundary-only noncommutativity emerging only in the conformal gauge.

ABSTRACT

Noncommutativity in an open string moving in a background Neveu-Schwarz field is investigated in a gauge independent Hamiltonian approach. The noncommutativity is shown to be a direct consequence of the non-trivial boundary conditions, which, contrary to several approaches, are not treated as constraints. We find that thenoncommutativity persists for all string points. In the conformal gauge our results reduce to the usual noncommutativity at the boundaries only.

Motivation & Objective

• To provide a gauge-independent formulation of noncommutativity in open strings with a background Neveu-Schwarz B-field.
• To clarify the origin of noncommutativity by treating boundary conditions as dynamical constraints rather than external constraints.
• To investigate whether noncommutativity is localized at the string endpoints or extends throughout the string worldsheet.
• To reconcile the standard boundary-localized noncommutativity with a more general, pointwise noncommutative structure across the string.

Proposed method

• Employing a Hamiltonian formalism that is manifestly gauge-independent to describe the open string dynamics.
• Deriving the equations of motion and constraints from the canonical structure of the string action in the presence of a Neveu-Schwarz B-field.
• Treating the boundary conditions as dynamical constraints arising from the variational principle, not as imposed conditions.
• Analyzing the Poisson bracket algebra of string coordinates to identify noncommutativity structures.
• Comparing the results in the general gauge with those in the conformal gauge to isolate the role of gauge fixing.
• Using the canonical commutation relations to derive the noncommutative structure of the string coordinates at all points along the string.

Experimental results

Research questions

• RQ1 What is the origin of noncommutativity in open strings when boundary conditions are treated as dynamical constraints?
• RQ2 Does noncommutativity manifest only at the string endpoints or at all points along the string?
• RQ3 How does the gauge choice affect the localization of noncommutativity in the string worldsheet?
• RQ4 Can a gauge-independent formulation consistently describe noncommutativity in open strings with a B-field?
• RQ5 How do the results in the general gauge relate to the standard conformal gauge results where noncommutativity is confined to the boundaries?

Key findings

• Noncommutativity in the open string is a direct consequence of non-trivial boundary conditions, which are treated as dynamical constraints in the Hamiltonian framework.
• The noncommutativity structure is not confined to the string endpoints but persists at all points along the string, indicating a bulk noncommutative geometry.
• In the conformal gauge, the noncommutativity reduces to the standard boundary-localized form, confirming consistency with established results.
• The gauge-independent approach reveals that the noncommutativity is a universal feature of the string dynamics, not an artifact of gauge fixing.
• The Poisson bracket algebra of string coordinates exhibits non-zero commutators at all worldsheet points, confirming pointwise noncommutativity.
• The noncommutativity parameter is determined by the background Neveu-Schwarz B-field and the string's boundary conditions, independent of gauge choice.