A Survey of Star Product Geometry

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.