Star Product, on flat space and torus

Groenewold's noncommutative star product [1] (adumbrated in [2]) of phase-space functions f(x,p) and g(x,p) is the unique associative pseudodifferential deformation of ordinary products:

It is the cornerstone of Moyal deformation (phase-space) quantization [3], as well as noncommutative geometry .

In practice, since the star product involves exponentials of derivative operators, it may be evaluated through translation of function arguments,

Baker [4] has utilized the more practical Fourier representation of this product as an integral kernel

The cyclic determinantal expression multiplying in the exponent is twice the area of the phase-space triangle (r′′,r′,r), where r≡ (x,p). Associativity is evident in this formulation, which further suggests efficient geometrical evaluation [5].

Groenewold's original definition of this product was through enforcement of the homomorphisms beetween Weyl-ordered operator multiplication and their classical kernel (c-number) functions. Weyl's association...