Dual families of non-commutative quantum systems

TL;DR

This paper constructs a family of physically equivalent non-commutative Hamiltonians in two dimensions, demonstrating an approximate duality between interacting commutative and non-commutative systems, especially relevant in strong magnetic fields.

Contribution

It provides an exact analytical construction of a one-parameter family of dual Hamiltonians in non-commutative quantum mechanics, highlighting the role of the Seiberg-Witten map and low-energy approximations.

Findings

Exact construction of dual Hamiltonians in non-commutative quantum mechanics

Approximate duality between interacting commutative and non-commutative systems at low energy

Duality validity extends to strong magnetic fields and weak Landau-level mixing

Abstract

We demonstrate how a one parameter family of interacting non-commuting Hamiltonians, which are physically equivalent, can be constructed in non-commutative quantum mechanics. This construction is carried out exactly (to all orders in the non-commutative parameter) and analytically in two dimensions for a free particle and a harmonic oscillator moving in a constant magnetic field. We discuss the significance of the Seiberg-Witten map in this context. It is shown for the harmonic oscillator potential that an approximate duality, valid in the low energy sector, can be constructed between the interacting commutative and a non-interacting non-commutative Hamiltonian. This approximation holds to order 1/B and is therefore valid in the case of strong magnetic fields and weak Landau-level mixing.