Talk:Weyl algebra

Generalization

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. Hans Tilgner — Preceding unsigned comment added by 160.45.16.132 (talk • contribs) 07:47, 21 May 2004

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. Hans Tilgner — Preceding unsigned comment added by 160.45.16.132 (talk • contribs) 06:51, 25 May 2004

I'm having trouble parsing this. What exactly is/are you issue[s] with the article? Myrkkyhammas (talk) 16:56, 17 February 2011 (UTC)

I do not want to make any change, but I want to ask Myrkkyhammas for the meaning of his comment/trouble with Tilgner's remarks. Please use plain english! Tilgner's first comment is a plain critic of the introduction to the article on Weyl algebras, this one being far too special (being restricted to two dimensions only and then to some derivatives). Please remain basisfree and general, which means - start the article from a symplectic vector space of dimension 2n, a concept which is dealt with in Wikipedia. Then use the commutative diagram, Weyl algebra - Clifford algebra, lying over symmetric algebra - exterior algebra, both lines being matched into Z(2)-graded generalizations also given by Tilgner in an old publication, long time ago mentioned by Roger Howe. A mathematician. — Preceding unsigned comment added by 194.94.3.130 (talk) 09:49, 7 June 2011 (UTC)

In this article Weyl algebra is not t h e Weyl algebra, it is only a special case. This should be corrected. Who writes an article on the real Weyl algebra. — Preceding unsigned comment added by 129.217.132.38 (talk) 09:21, 14 May 2012 (UTC)

It is almost impossible to understand the construction of A_n

In the introductory section, after A₁ is constructed, we find these two sentences:

"This is the first Weyl algebra A₁. The n-th Weyl algebra A_n are constructed similarly."

Note: In English a singular noun takes a singular verb, so "are" should be "is" (or else "algebra" should be "algebras").

This is extremely annoying writing, because it says nothing about how to "similarly" construct A_n. No reader has any idea what "similarly" means here.

Then we meet this passage:

"Alternatively, $A_{1}$ can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by $([p,q]-1)$. Similarly, $A_{n}$ is obtained by quotienting the free algebra on 2n generators by the ideal generated by$$([p_{i},q_{j}]-\delta _{i,j}),\quad \forall i,j=1,\dots ,n$$where $\delta _{i,j}$ is the Kronecker delta."

But the phrase "free algebra" has no meaning in the absense of any mention of what ring the free algebra is over.

Soon after, we meet this sentence:

"The Weyl algebra A_n can be concretely constructed as a representation."

But nothing is explained about how to construct A_n "as a representation".

Some matrices appear next, but their connection to A_n, if any, is left entirely unmentioned.

When one finally arrives at a concrete construction of A_n, it is in the section titled Generator — and this title is already extremely misleading, since it says nothing about A_n.

But then one reads the construction and encounters the symbol 𝜔, which is left totally unexplained (and which appears nowhere in the article prior to this use).

I hope that someone knowledgeable about this subject (and also capable of writing clearly) can fix this. — Preceding unsigned comment added by 98.36.148.11 (talk • contribs) 23:03, 17 November 2024 (UTC)