Weyl algebra

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In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable),

f_n(X) \partial_X^n + \cdots + f_1(X) \partial_X + f_0(X).

More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.

The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.

You can also construct the Weyl algebra as a quotient of the free algebra on two generators, X and Y, by the ideal generated by the single relation

YX − XY − 1.

The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by Xi and \part_{X_i}.

Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Lie algebra of the Heisenberg group, by setting the element 1 of the Lie algebra equal to the unit 1 of the universal enveloping algebra.

One may give an abstract construction of the algebras An in terms of generators and relations. We do so in a more sophisticated way: Start with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be

W(V) := T(V) / (\!( v \otimes w - w \otimes v - \omega(v,w), \text{ for } v,w \in V )\!),

where the notation (\!( )\!) means "the ideal generated by". In other words, W(V) is the algebra generated by V subject only to the relation vw − wv = ω(v,w). Then, W(V) is isomorphic to A_{n}\, (it does not depend on the choice of ω). In this form, one sees that W(V) is a quantization of the symmetric algebra Sym(V). If V is over a field of characteristic zero, then W(V) is naturally isomorphic to the symmetric algebra Sym(V) equipped with the deformed Moyal product (considering the symmetric algebra to be polynomial functions on V * , where the variables span the vector space V, and replacing i \hbar in the Moyal product formula with 1). The isomorphism is given by the symmetrization map from Sym(V) to W(V): a_1 \cdots a_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)}. If one prefers to have the i \hbar and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by Xi and i \hbar \part_{X_i} (as is frequently done in quantum mechanics).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication. In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra.

For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to integrable ("most") functions, not just polynomial functions), see Weyl quantization.

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