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 functionsf and g on ℝ2n takes the form where each Cn is a certain bidifferential operator of order n characterized by the following properties :
> 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 condition 2 and eliminates condition 4. If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebraAn, and the two offer alternative realizations of the Weyl map of the space of polynomials in variables. To provide an explicit formula, consider a constant Poisson bivector Π on ℝ2n: where Πij is a complex number for each i, j. The star product of two functions and can then be defined as where ħ 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. The closed form can be obtained by using the exponential: where is the multiplication map,, and the exponential is treated as a power series: That is, the formula for is As indicated, often one eliminates all occurrences of above, and the formulas then restrict naturally to real numbers. Note that if the functions f and g are polynomials, the above infinite sums become finite. The relationship of the Moyal product to the generalized ★-product used in the definition of the "algebra of symbols" of a universal enveloping algebrafollows from the fact that the Weyl algebra is the universal enveloping algebra of the Heisenberg algebra.
On manifolds
On any symplectic manifold, one can, at least locally, choose coordinates so as to make the symplectic structureconstant, by Darboux's theorem; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally, as a function on the whole manifold, one must equip the symplectic manifold with a torsion-free symplectic connection. This makes it a Fedosov manifold. More general results for arbitrary Poisson manifolds are given by the Kontsevich quantization formula.
Examples
A simple explicit example of the construction and utility of the ★-product is given in the article on the Wigner–Weyl transform: two Gaussians compose with this ★-product according to a hyperbolic tangent law: Every correspondence prescription between phase space and Hilbert space, however, induces its ownproper ★-product. Similar results are seen in the Segal–Bargmann space and in the theta representation of the Heisenberg group, where the creation and annihilation operators and are understood to act on the complex plane, so that the position and momenta operators are given by and. This situation is clearly different from the case where the positions are taken to be real-valued, but does offer insights into the overall algebraic structure of the Heisenberg algebra and its envelope, the Weyl algebra.
