A Survey of Star Product Geometry

TL;DR

This paper provides a comprehensive pedagogical overview of the star product, illustrating its geometric and algebraic properties through examples like Landau orbits and comparing different construction methods.

Contribution

It offers a detailed survey of star product geometry, emphasizing its construction via Weyl correspondence and Fourier kernel, and compares it with phase-space polygon methods.

Findings

Star product can be derived from Dirac Brackets of Landau orbits.

The Fourier representation kernel clarifies associativity and symmetries.

Comparison with phase-space polygon construction highlights different geometric approaches.

Abstract

A brief pedagogical survey of the star product is provided, through Groenewold's original construction based on the Weyl correspondence. It is then illustrated how simple Landau orbits in a constant magnetic field, through their Dirac Brackets, define a noncommutative structure since these brackets exponentiate to a star product---a circumstance rarely operative for generic Dirac Brackets. The geometric picture of the star product based on its Fourier representation kernel is utilized in the evaluation of chains of star products. The intuitive appreciation of their associativity and symmetries is thereby enhanced. This construction is compared and contrasted with the remarkable phase-space polygon construction of Almeida.