nLab Weyl algebra

To be distinguished from Weil algebra.

Idea

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.

Definition

Given a field $k$, the $n$-th Weyl algebra $A_{n,k}$ is an associative unital algebra over $k$ generated by the symbols $x^1,\ldots,x^n,\partial_1,\ldots,\partial_n$ modulo the “canonical commutation relations” $x^i x^j = x^j x^i$, $\partial_i\partial_j = \partial_j\partial_i$ and $\partial_i x^j - x^j \partial_i = \delta_i^j$ (the Kronecker delta).

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

Sometimes one considers the Weyl algebras over an arbitrary $k$-algebra $R$, including noncommutative $R$, when the definition is simply $A_{n,k}\otimes_k R$. 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 $p_j$ for $- \mathrm{i} \partial_j$ (where $\mathrm{i}$ is the imaginary unit).

Properties

Relation to Heisenberg Lie algebra

Consider the standard symplectic form on the Cartesian space $\mathbb{R}^{2n}$, 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 $2n$ 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 $\mathbb{R}^{2m}$.

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

Related concepts

• Weyl relations

• canonical commutation relations

• Heisenberg algebra

References

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

• Jacques Dixmier, Sur les algèbres de Weyl, Bulletin de la Société Mathématique de France, 96 (1968) 209-242 [numdam:BSMF_1968__96__209_0]

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

• Hermann Weyl, (36) in: Quantenmechanik und Gruppentheorie, Zeitschrift für Physik 46 (1927) 1–46 [doi:10.1007/BF02055756]

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

• John von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Mathematische Annalen 104 (1931) 570–578 [doi:10.1007/BF01457956]

  (proving the Stone-von Neumann theorem)

More on the history:

• Severino C. Coutinho, Introduction to: A primer of algebraic D-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press (1995) [doi:10.1017/CBO9780511623653]

• Severino C. Coutinho, The Many Avatars of a Simple Algebra, The American Mathematical Monthly 104 7 (1997) 593-604 [doi:10.2307/2975052, jstor:2975052]

Further discussion (of either notion):

• Alan Weinstein, p. 392 of: Deformation quantization, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [numdam:SB_1993-1994__36__389_0]

• Olga Kravchenko, Deformation Quantization of Symplectic Fibrations, Compositio Mathematica 123 (2000) 131–165 [arXiv:math/9802070, doi:10.1023/A:1002452002677]

• Xavier Bekaert, Universal enveloping algebras and some applications in physics (2005) [cds:904799, pdf]

• Markus Pflaum, From Weyl quantization to modern algebraic index theory, in Groups and Analysis – The Legacy of Hermann Weyl, Cambridge University Press (2008) 84-99 [doi:10.1017/CBO9780511721410.005]

• Jason Gaddis, The Weyl algebra and its friends: a survey [arXiv:2305.01609]

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

• Ernst Binz, Reinhard Honegger, Alfred Rieckers, Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization, Annales Henri Poincaré 5 (2004) 327–346 [doi:10.1007/s00023-004-0171-y]

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

• Nicolaas P. Landsman, B. Ramazan, Ex. 11.3 in: Quantization of Poisson algebras associated to Lie algebroids, in: Groupoids in Analysis, Geometry, and Physics, Contemporary Mathematics 282 (2001) [arXiv:math-ph/0001005, ams:conm/282]

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

• Ernst Binz, Reinhard Honegger, Alfred Rieckers, Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization, International Journal of Pure and Applied Mathematics 38 1 (2007) [ijpam:2007-38-1/6, pdf]

• Reinhard Honegger, Alfred Rieckers, Heisenberg Group Algebra and Strict Weyl Quantization, Chapter 23 in: Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems, World Scientific (2015) [chapter:doi;10.1142/9789814696586_0023, book:doi:10.1142/9251-vol1]

See also:

• eom: J.-E. Björk, Weyl algebra

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

• Nikita Markarian, Weyl $n$-algebras, arxiv/1504.01931