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 and are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl-algebra case). The relationship of the Moyal product to the generalized ★-product used in the definition of the "algebra of symbols" of a universal enveloping algebra follows from the fact that the Weyl algebra is the universal enveloping algebra of the Heisenberg algebra (modulo that the center equals the unit). On manifolds On any symplectic manifold, one can, at least locally, choose coordinates so as to make the symplectic structure constant, 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 (and not just a local formula), 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 (where the Darboux theorem does not apply) are given by the Kontsevich quantization formula. Examples A simple explicit example of the construction and utility of the ★-product (for the simplest case of a two-dimensional euclidean phase space) is given in the article on the Wigner–Weyl transform: two Gaussians compose with this ★-product according to a hyperbolic tangent law: (Note the classical limit, .) Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ★-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 (respectively, the upper half-plane for the Heisenberg group), 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. Inside phase-space integrals Inside a phase-space integral, just one star product of the Moyal type may be dropped, resulting in plain multiplication, as evident by integration by parts, making the cyclicity of the phase-space trace manifest. This is a unique property of the above