# nLab Weyl algebra

Contents

To be distinguished from Weil algebra.

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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.

## References

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

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