nLab Weyl algebra


To be distinguished from Weil algebra.



In general, the term Weyl algebra refers to noncommutative associative algebras controlled by canonical commutation relations (CCR) which are the hallmark of quantum mechanics.

More specifically, by the Weyl algebra with Weyl relations one refers to the exponentiated form of these CCR where the algebra generators are (represented by) unitary operators, introduced by Weyl 1927, pp. 27 and further highlighted in von Neumann 1931. It is this form of “Weyl algebra” that the Stone-von Neumann theorem directly applies to.


Given a field kk, the nn-th Weyl algebra A n,kA_{n,k} is an associative unital algebra over kk generated by the symbols x 1,,x n, 1,, nx^1,\ldots,x^n,\partial_1,\ldots,\partial_n modulo the “canonical commutation relationsx ix j=x jx ix^i x^j = x^j x^i, i j= j i\partial_i\partial_j = \partial_j\partial_i and ix jx j i=δ i j\partial_i x^j - x^j \partial_i = \delta_i^j (the Kronecker delta).

In characteristic zero, this agrees with the algebra of regular differential operators on the nn-dimensional affine space.

Sometimes one considers the Weyl algebras over an arbitrary kk-algebra RR, including noncommutative RR, when the definition is simply A n,k kRA_{n,k}\otimes_k R. Another generalization are the symplectic Weyl algebras.

In quantum physics, one often studies Weyl algebras over the complex numbers (see below); the usual notation there is p jp_j for i j- \mathrm{i} \partial_j (where i\mathrm{i} is the imaginary unit).


Relation to Heisenberg Lie algebra

Consider the standard symplectic form on the Cartesian space 2n\mathbb{R}^{2n}, making a symplectic vector space. This gives rise to the corresponding Heisenberg Lie algebra.

Depending on conventions, the universal enveloping algebra of the Heisenberg Lie algebra either already is the Weyl algebra on 2n2n generators or else it becomes so after after forming the quotient algebra in which the central generator is identified with the unit element of the ground field – whereas in the former case (considered eg. in Kravchenko 2000, Def. 2.1; Bekaert 2005, p. 9) the central generator plays the role of the formal Planck constant \hbar with the Weyl algebra regarded as a formal deformation quantization of the symplectic manifold 2m\mathbb{R}^{2m}.

Accordingly, given a Heisenberg Lie n n -algebra it makes sense to call its universal enveloping E n E_n -algebra a Weyl n n -algebra.


The term “Weyl algebra” for algebras freely generated subject to canonical commutation relations is due to

(there attributed to a suggestion by Irving Segal) and referring to the original discussion in

However, beware that the invention of Weyl 1927 was not the canonical commutation relations but their exponential reformulation via the Weyl relations, whose relevance was then picked up by

More on the history:

Further discussion (of either notion):

On continuous fields of Weyl algebras as continuous deformation quantizations of symplectic topological vector spaces:

On Weyl algebras as groupoid algebras being strict deformation quantizations of Lie-Poisson structures given by tangent Lie algebroids:

On group algebras of (underlying discrete) Heisenberg groups as strict deformation quantizations of pre-symplectic topological vector spaces by continuous fields of Weyl algebras:

See also:

A categorification of the Weyl algebra is introduced in operadic language in

Last revised on December 7, 2023 at 16:15:32. See the history of this page for a list of all contributions to it.