Commutative/Non-Commutative Dualities

TL;DR

This paper presents a systematic method to construct dualities between commutative and non-commutative theories using a generalized ERG, enabling the development of consistent non-commutative models that preserve Lorentz symmetry and unitarity.

Contribution

It introduces a novel systematic approach to establish dualities between commutative and non-commutative theories via a generalized ERG, applicable to various models.

Findings

Constructed dualities for the Landau problem and scalar field theories.

Demonstrated Lorentz symmetry can be implemented in non-commutative duals.

Provided a framework for consistent non-commutative, non-local theories.

Abstract

We show that it is in principle possible to construct dualities between commutative and non-commutative theories in a systematic way. This construction exploits a generalization of the exact renormalization group equation (ERG). We apply this to the simple case of the Landau problem and then generalize it to the free and interacting non-canonical scalar field theory. This constructive approach offers the advantage of tracking the implementation of the Lorentz symmetry in the non-commutative dual theory. In principle, it allows for the construction of completely consistent non-commutative and non-local theories where the Lorentz symmetry and unitarity are still respected, but may be implemented in a highly non-trivial and non-local manner.