A Note On Space Noncommutativity

TL;DR

This paper models an open string in a flat background with a B field using a spatial lattice approximation, revealing how noncommutative space features emerge from string interactions and gauge modifications.

Contribution

It introduces a lattice-based dipole system that captures key noncommutative space properties and demonstrates the emergence of the star product and gauge transformation modifications.

Findings

Reveals the star product structure from string interactions

Shows position operators can commute despite noncommutativity

Demonstrates modified gauge transformations in the noncommutative setting

Abstract

We consider a two-point spatial lattice approximation to an open string moving in a flat background with B field. It gives a constrained dipole system under the influence of a vector potential. Solving and quantizing this system recover all the essential features of a noncommutative space. In particular, open string interactions induce a canonical product structure on the Hilbert space of the dipole system. It coincides with the usual star product, even though the position operators can be thought of as mutually commuting. Modification of gauge transformations in this noncommutative space also naturally emerges.