Talk:Weyl algebra

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Seems to me, that this is a special representation of the more general concept, mentioned here under quotient algebra. Let <,> be a non-degenerate bilinear form on a (real) vector space V. Let ten(V) be the (infinite-dimensional) tensor algebra. For two elements x and y of V, consider the two-sided ideal in this algebra, generated by xy-yx - <x,y> 1 if <,> is skew i.e. (V,<,>) a symplectic vectorspace, resp. xy+yx - <x,y> 1 if <,> is symmetric i.e. (V,<,>) a pseudo-orthogonal vector space. Let ideal(<,>) be this two-sided ideal. Then the factor algebra

         ten(V)/ideal(<,>)

is another associative algebra with unit element. There are four special cases: If <,> is skew than we get the infinite-dimensional Weyl Algebra weyl(V,<,>). If <,> is trivial, i.e. not non-degenerate, we get the Symmetric Algebra on V. If <,> is symmetric we get the Clifford algebra, and for vanishing bilinear form as a special case the Exterior Algebra. These two are finite dimensional. All four of them are universal algebras, in which the linear ,tensors' fulfill the canonical commutation relations resp. the defining relations of the Clifford algebra (Dirac is the special case of a Minkowski bilinear form in four dimensions). In the symplectic resp. pseudo-orthogonal case, the two-tensors fulfill the commutation relations of the symplectic resp. the pseuo-orthogonal Lie algebras. These therefore can be written basis free, like the canonical commutation resp. anticommutation relations. I have studied these algebras in several papers. They even may combined into one Z2-,graded' concept. Hannes Tilgner

There even are doubts on the mathematical contents of this articles. It starts with a certain representation on a function space of an algebraic concept ,Weyl algebra', then proceeding to the universal definition. But is it really true, that this functional representation is a representation of the Weyl algebra in the sence of (infinite-dimensional) Lie algebras? Sure, we have in both cases the canonical commutation relations in the embedded symplectic vector space. This means that the nilpotent Heisenberg Lie algebra is represented by Poisson brackets as a Lie algebra. With the help of the symplectic bilinear form they are written basis free. The same applies to the subspace of tensor grade two in the Weyl algebra, resulting in a nice basis free formulation of the commutation relations of the (semi-simple) symplectic Lie algebra. So up to the second tensor grade there is isomophy of Lie subalgebras, since Poisson brackets represent (in the sense of Lie algebraic isomophisms) the symplectic Lie algebra as well. But does that really imply isomorphy of the whole infinite-dimensional Weyl algebra, seen as a Lie algebra with respect to the commutator of its associative multiplication? Note that taking the subspaces of tensor grade 0, 1, 2, 3, ... a generating system for each vector space of grade i is given by the totally symmetrized elements of the form x1 times x2 times ... times xi. Already in the case of grade 3 I found deviations between Poisson brackets and commutators in the Weyl algebra. The calculations are tedious. So does anybody know how to computerize this? In which language? LISP? If there is no Lie algebraic isomophism between the Weyl algebra and the classical mechanical realization by means of Poisson brackets, classical and quantum realizations are nonisomorphic in a Lie algebra sense. All that can be excluded, if one can prove, that a Lie algebra representation of the defining two sided ideal already gives rise to a representation of the whole infinite Lie algebra. This proof is not available to me. If there is no such proof, than classical and quantum mechanics are not isomorphic. And there is no concept of quantization, but one of dequantization. Hannes Tilgner