A note on broken dilatation symmetry in planar noncommutative theory

TL;DR

This paper explores how noncommutativity in spatial coordinates affects scale symmetry in quantum Hall systems, deriving anomalies and effective descriptions that highlight quantum effects from noncommutative geometry.

Contribution

It establishes a connection between noncommutative geometry and dilatation anomalies in quantum Hall systems using path-integral methods and effective Landau problem models.

Findings

Noncommutativity naturally emerges at high magnetic fields.

Derived exact anomalous Ward identities for scale symmetry.

Identified quantum-induced scale anomalies from noncommutative structure.

Abstract

A study of a riveting connection between noncommutativity and the anomalous dilatation (scale) symmetry is presented for a generalized quantum Hall system due to time dilatation transformations. On using the "Peierls substitution" scheme, it is shown that noncommutativity between spatial coordinates emerges naturally at a large magnetic field limit. Thereafter, we derive a path-integral action for the corresponding noncommutative quantum system and discuss the equivalence between the considered noncommutative system and the generalized Landau problem thus rendering an effective commmutative description. By exploiting the path-integral method due to Fujikawa, we derive an expression for the unintegrated scale or dilatation anomaly for the generalized Landau system, wherein the anomalies are identified with Jacobian factors arising from measure change under scale transformation and is subsequently renormalised. In fact, we derive exact expressions of anomalous Ward identities from which one may point out the existence of scale anomaly which is a purely quantum effect induced from the noncommutative structure between spatial coordinates.