the Weyl quantization on a function of the form gives the average of all possible orderings of factors of and factors of . For example, we have While this result is conceptually natural, it is not convenient for computations when and are large. In such cases, we can use instead McCoy's formula This expression gives an apparently different answer for the case of from the totally symmetric expression above. There is no contradiction, however, since the canonical commutation relations allow for more than one expression for the same operator. (The reader may find it instructive to use the commutation relations to rewrite the totally symmetric formula for the case of in terms of the operators , , and and verify the first expression in McCoy's formula with .) It is widely thought that the Weyl quantization, among all quantization schemes, comes as close as possible to mapping the Poisson bracket on the classical side to the commutator on the quantum side. (An exact correspondence is impossible, in light of Groenewold's theorem.) For example, Moyal showed the Theorem: If is a polynomial of degree at most 2 and is an arbitrary polynomial, then we have . Weyl quantization of general functions If is a real-valued function, then its Weyl-map image is self-adjoint. If is an element of Schwartz space, then is trace-class. More generally, is a densely defined unbounded operator. The map is one-to-one on the Schwartz space (as a subspace of the square-integrable functions). Deformation quantization Intuitively, a deformation of a mathematical object is a family of the same kind of objects that depend on some parameter(s). Here, it provides rules for how to deform the "classical" commutative algebra of observables to a quantum non-commutative algebra of observables. The basic setup in deformation theory is to start with an algebraic structure (say a Lie algebra) and ask: Does there exist a one or more parameter(s) family of similar structures, such that for an initial value of the parameter(s) one has the same structure (Lie algebra) one started with? (The oldest illustration of this may be the realization of Eratosthenes in the ancient world that a flat earth was deformable to a spherical earth, with deformation parameter 1/R⊕.) E.g., one may define a noncommutative torus as a deformation quantization through a ★-product to implicitly address all convergence subtleties (usually not addressed in formal deformation quantization). Insofar as the algebra of functions