Moyal product

In mathematics, the Moyal product (after José Enrique Moyal; also called the star product or Weyl–Groenewold product, after Hermann Weyl and Hilbrand J. Groenewold) is an example of a phase-space star product. It is an associative, non-commutative product, , on the functions on R2n, equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the -product of the "algebra of symbols" of a universal enveloping algebra. The Moyal product is named after José Enrique Moyal, but is also sometimes called the Weyl–Groenewold product as it was introduced by H. J. Groenewold in his 1946 doctoral dissertation, in a trenchant appreciation of the Weyl correspondence. Moyal actually appears not to know about the product in his celebrated article and was crucially lacking it in his legendary correspondence with Dirac, as illustrated in his biography. The popular naming after Moyal appears to have emerged only in the 1970s, in homage to his flat phase-space quantization picture. The product for smooth functions f and g on R2n takes the form where each Cn is a certain bidifferential operator of order n characterized by the following properties (see below for an explicit formula): Deformation of the pointwise product — implicit in the formula above. Deformation of the Poisson bracket, called Moyal bracket. The 1 of the undeformed algebra is also the identity in the new algebra. The complex conjugate is an antilinear antiautomorphism. Note that, if one wishes to take functions valued in the real numbers, then an alternative version eliminates the i in the second condition and eliminates the fourth condition. If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebra An, and the two offer alternative realizations of the Weyl map of the space of polynomials in n variables (or the symmetric algebra of a vector space of dimension 2n). To provide an explicit formula, consider a constant Poisson bivector Π on R2n: where Πij is a real number for each i, j.