Editing Weyl algebra (section)

=== Quantization ===
The algebra ''W''(''V'') is a [[quantization (physics)|quantization]] of the [[symmetric algebra]] Sym(''V'').  If ''V'' is over a field of characteristic zero, then ''W''(''V'') is naturally isomorphic to the underlying vector space of the [[symmetric algebra]] Sym(''V'') equipped with a deformed product – called the Groenewold–[[Moyal product]] (considering the symmetric algebra to be polynomial functions on ''V''<sup>∗</sup>, where the variables span the vector space ''V'', and replacing ''iħ''  in the Moyal product formula with 1).

The isomorphism is given by the symmetrization map from Sym(''V'') to ''W''(''V'')
: <math>a_1 \cdots a_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)}~.</math>

If one prefers to have the ''iħ'' and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by ''q''<sub>''i''</sub> and ''iħ∂<sub>q<sub>i</sub></sub>''  (as per [[quantum mechanics]] usage).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the [[Moyal product|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.

Stated in another way, let the [[Moyal product|Moyal star product]] be denoted <math>f \star g</math>, then the Weyl algebra is isomorphic to <math>(\mathbb C[x_1, \dots, x_n], \star)</math>.{{sfn|Coutinho|1997|pp=602–603}}

In the case of [[exterior algebra]]s, the analogous quantization to the Weyl one is the [[Clifford algebra]], which is also referred to as the ''orthogonal Clifford algebra''.{{sfn|Lounesto|Ablamowicz|2004|p=xvi}}{{sfn|Micali|Boudet|Helmstetter|1992|pp=83-96}}

The Weyl algebra is also referred to as the '''symplectic Clifford algebra'''.{{sfn | Lounesto | Ablamowicz | 2004|p=xvi}}{{sfn | Micali | Boudet | Helmstetter | 1992 | pp=83-96}}{{sfn | Helmstetter | Micali | 2008 | p=xii}} Weyl algebras represent for symplectic [[bilinear form]]s the same structure that [[Clifford algebra]]s represent for non-degenerate symmetric bilinear forms.{{sfn | Helmstetter | Micali | 2008 | p=xii}}