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. Historical comments 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. Definition The product for smooth functions and on ℝ2n takes the form where each Cn is a certain bidifferential operator of order 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 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 variables (or the symmetric algebra of a vector space of dimension 2 ). To provide an explicit formula, consider a constant Poisson bivector on ℝ2n: where Πij is a real number for each i, j. The star product of two functions and can then be defined as the pseudo-differential operator acting on both of them, where is the reduced Planck constant, treated as a formal parameter here. This is a special case of what is known as the Berezin formula on the algebra of symbols and can be given a closed form (which follows from the Baker–Campbell–Hausdorff formula). The closed form can