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.Comment: 14p, LateX/crckapb.sty, proceedings of NATO ARW: UIC 2000, July 22-2
