They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.
The Weyl algebra arises naturally in the context of quantum mechanics and the process of canonical quantization.
Consider a classical phase space with canonical coordinates
Erwin Schrödinger proposed in 1926 the following:[1] With this identification, the canonical commutation relation holds.
The Weyl algebras have different constructions, with different levels of abstraction.
In the differential operator representation, similar to Schrödinger's canonical quantization, let
can be constructed as a quotient of a free algebra in terms of generators and relations.
One construction starts with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω.
In other words, W(V) is the algebra generated by V subject only to the relation vu − uv = ω(v, u).
Then, W(V) is isomorphic to An via the choice of a Darboux basis for ω.
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∗, 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) 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 qi and iħ∂qi (as per quantum mechanics usage).
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.
[4][5][6] Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.
[7] Specifically, the Weyl algebra corresponding to the polynomial ring
[8] Because "étale" means "(flat and) possessing null cotangent sheaf",[9] this means that every D-module over such a scheme can be thought of locally as a module over the
is clear from context) is inductively defined as a graded subalgebra of
, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit
representation, this equation is obtained by the general Leibniz rule.
Since the general Leibniz rule is provable by algebraic manipulation, it holds for
[12] By repeating the commutator relations, any monomial can be equated to a linear sum of these.
The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.
For the induction step, similarly to the above calculation, there exists some element
In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain.
In fact, there are stronger statements than the absence of finite-dimensional representations.
To any finitely generated An-module M, there is a corresponding subvariety Char(M) of V × V∗ called the 'characteristic variety'[clarification needed] whose size roughly corresponds to the size[clarification needed] of M (a finite-dimensional module would have zero-dimensional characteristic variety).
Then Bernstein's inequality states that for M non-zero, An even stronger statement is Gabber's theorem, which states that Char(M) is a co-isotropic subvariety of V × V∗ for the natural symplectic form.
The situation is considerably different in the case of a Weyl algebra over a field of characteristic p > 0.
[21] For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to a class of integrable functions larger than the polynomial functions), see Wigner–Weyl transform.
Consider a polynomial ring Then a differential operator is defined as a composition of