Weyl algebra
Template:Short description In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.
In the simplest case, these are differential operators. Let <math>F</math> be a field, and let <math>F[x]</math> be the ring of polynomials in one variable with coefficients in <math>F</math>. Then the corresponding Weyl algebra consists of differential operators of form
- <math> f_m(x) \partial_x^m + f_{m-1}(x) \partial_x^{m-1} + \cdots + f_1(x) \partial_x + f_0(x) </math>
This is the first Weyl algebra <math>A_1</math>. The n-th Weyl algebra <math>A_n</math> are constructed similarly.
Alternatively, <math>A_1</math> can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by <math>([p,q] - 1)</math>. Similarly, <math>A_n</math> is obtained by quotienting the free algebra on 2n generators by the ideal generated by<math display="block"> ([p_i,q_j] - \delta_{i,j}), \quad \forall i, j = 1, \dots, n</math>where <math> \delta_{i,j}</math> is the Kronecker delta.
More generally, let <math> (R,\Delta) </math> be a partial differential ring with commuting derivatives <math> \Delta = \lbrace \partial_1,\ldots,\partial_m \rbrace </math>. The Weyl algebra associated to <math>(R,\Delta)</math> is the noncommutative ring <math> R[\partial_1,\ldots,\partial_m] </math> satisfying the relations <math> \partial_i r = r\partial_i + \partial_i(r) </math> for all <math> r \in R </math>. The previous case is the special case where <math> R=F[x_1,\ldots,x_n] </math> and <math> \Delta = \lbrace \partial_{x_1},\ldots,\partial_{x_n} \rbrace </math> where <math> F </math> is a field.
This article discusses only the case of <math>A_n</math> with underlying field <math>F</math> characteristic zero, unless otherwise stated.
The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.
MotivationEdit
Template:See also 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 <math>(q_1, p_1, \dots, q_n, p_n) </math>. These coordinates satisfy the Poisson bracket relations:<math display="block"> \{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}. </math>In canonical quantization, one seeks to construct a Hilbert space of states and represent the classical observables (functions on phase space) as self-adjoint operators on this space. The canonical commutation relations are imposed:<math display="block"> [\hat{q}_i, \hat{q}_j] = 0, \quad [\hat{p}_i, \hat{p}_j] = 0, \quad [\hat{q}_i, \hat{p}_j] = i\hbar \delta_{ij}, </math>where <math>[\cdot, \cdot]</math> denotes the commutator. Here, <math>\hat{q}_i</math> and <math>\hat{p}_i</math> are the operators corresponding to <math>q_i</math> and <math>p_i</math> respectively. Erwin Schrödinger proposed in 1926 the following:Template:Sfn
- <math>\hat{q_j}</math> with multiplication by <math>x_j</math>.
- <math>\hat{p}_j</math> with <math>-i\hbar \partial_{x_j}</math>.
With this identification, the canonical commutation relation holds.
ConstructionsEdit
The Weyl algebras have different constructions, with different levels of abstraction.
RepresentationEdit
The Weyl algebra <math>A_n</math> can be concretely constructed as a representation.
In the differential operator representation, similar to Schrödinger's canonical quantization, let <math>q_j</math> be represented by multiplication on the left by <math>x_j</math>, and let <math>p_j</math> be represented by differentiation on the left by <math>\partial_{x_j}</math>.
In the matrix representation, similar to the matrix mechanics, <math> A_1 </math> is represented byTemplate:Sfn<math display="block"> P=\begin{bmatrix} 0 & 1 & 0 & 0 & \cdots \\ 0 & 0 & 2 & 0 & \cdots \\ 0 & 0 & 0 & 3 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}, \quad Q=\begin{bmatrix} 0 & 0 & 0 & 0 & \ldots \\ 1 & 0 & 0 & 0 & \cdots \\ 0 & 1 & 0 & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} </math>
GeneratorEdit
<math> A_n</math> 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. Define the Weyl algebra W(V) to be
- <math>W(V) := T(V) / (\!( v \otimes u - u \otimes v - \omega(v,u), \text{ for } v,u \in V )\!),</math>
where T(V) is the tensor algebra on V, and the notation <math>(\!( )\!)</math> means "the ideal generated by".
In other words, W(V) is the algebra generated by V subject only to the relation Template:Math. Then, W(V) is isomorphic to An via the choice of a Darboux basis for Template:Mvar.
<math> A_n</math> is also a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely [q, p]) equal to the unit of the universal enveloping algebra (called 1 above).
QuantizationEdit
The algebra W(V) is a 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∗, 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 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.
Stated in another way, let the 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>.Template:Sfn
In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra.Template:SfnTemplate:Sfn
The Weyl algebra is also referred to as the symplectic Clifford algebra.Template:SfnTemplate:SfnTemplate:Sfn Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.Template:Sfn
D-moduleEdit
The Weyl algebra can be constructed as a D-module.Template:Sfn Specifically, the Weyl algebra corresponding to the polynomial ring <math>R[x_1, ..., x_n]</math> with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations <math>D_{\mathbb{A}^n_R / R}</math>.Template:Sfn
More generally, let <math>X</math> be a smooth scheme over a ring <math>R</math>. Locally, <math>X \to R</math> factors as an étale cover over some <math>\mathbb{A}^n_R</math> equipped with the standard projection.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Because "étale" means "(flat and) possessing null cotangent sheaf",<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> this means that every D-module over such a scheme can be thought of locally as a module over the <math>n^\text{th}</math> Weyl algebra.
Let <math>R</math> be a commutative algebra over a subring <math>S</math>. The ring of differential operators <math>D_{R/S}</math> (notated <math>D_R</math> when <math>S</math> is clear from context) is inductively defined as a graded subalgebra of <math>\operatorname{End}_{S}(R)</math>:
- <math>D^0_R=R</math>
- <math>
D^k_R=\left\{d \in \operatorname{End}_{S}(R):[d, a] \in D^{k-1}_R \text { for all } a \in R\right\} . </math>
Let <math>D_R</math> be the union of all <math>D^k_R</math> for <math>k \geq 0</math>. This is a subalgebra of <math>\operatorname{End}_{S}(R)</math>.
In the case <math>R = S[x_1, ..., x_n]</math>, the ring of differential operators of order <math>\leq n</math> presents similarly as in the special case <math>S = \mathbb{C}</math> but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize <math>\mathbb{Z}[x_1, ..., x_n]</math>, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit <math>D_{\mathbb{A}^n_\mathbb{Z} / \mathbb{Z}}</math>. One such example is the operator <math>\partial_{x_1}^{[p]} : x_1^N \mapsto {N \choose p} x_1^{N-p}</math>.
Explicitly, a presentation is given by
- <math>D_{S[x_1, \dots, x_\ell]/S}^n = S \langle x_1, \dots, x_\ell, \{\partial_{x_i}, \partial_{x_i}^{[2]}, \dots, \partial_{x_i}^{[n]}\}_{1 \leq i \leq \ell} \rangle</math>
with the relations
- <math>[x_i, x_j] = [\partial_{x_i}^{[k]}, \partial_{x_j}^{[m]}] = 0</math>
- <math>[\partial_{x_i}^{[k]}, x_j] = \left \{ \begin{matrix}\partial_{x_i}^{[k-1]} & \text{if }i=j \\ 0 & \text{if } i \neq j\end{matrix}\right.</math>
- <math>\partial_{x_i}^{[k]} \partial_{x_i}^{[m]} = {k+m \choose k} \partial_{x_i}^{[k+m]} ~~~~~\text{when }k+m \leq n</math>
where <math>\partial_{x_i}^{[0]} = 1</math> by convention. The Weyl algebra then consists of the limit of these algebras as <math>n \to \infty</math>.<ref>Template:Cite journal</ref>Template:Pg
When <math>S</math> is a field of characteristic 0, then <math>D^1_R</math> is generated, as an <math>R</math>-module, by 1 and the <math>S</math>-derivations of <math>R</math>. Moreover, <math>D_R</math> is generated as a ring by the <math>R</math>-subalgebra <math>D^1_R</math>. In particular, if <math>S = \mathbb{C}</math> and <math>R=\mathbb{C}[x_1, ..., x_n]</math>, then <math>D^1_R=R+ \sum_i R \partial_{x_i} </math>. As mentioned, <math>A_n = D_R</math>.Template:Sfn
Properties of AnEdit
Many properties of <math> A_1 </math> apply to <math> A_n </math> with essentially similar proofs, since the different dimensions commute.
General Leibniz ruleEdit
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Template:Math proofIn particular, <math display="inline">[q, q^m p^n] = -nq^mp^{n-1}</math> and <math display="inline">[p, q^mp^n] = mq^{m-1}p^n</math>. Template:Math theorem
DegreeEdit
This allows <math> A_1 </math> to be a graded algebra, where the degree of <math> \sum_{m, n} c_{m,n} q^m p^n </math> is <math> \max (m + n) </math> among its nonzero monomials. The degree is similarly defined for <math> A_n </math>.
Template:Math theorem That is, it has no two-sided nontrivial ideals and has no zero divisors. Template:Math proof
DerivationEdit
Template:SeeTemplate:Math theorem That is, any derivation <math display="inline">D</math> is equal to <math display="inline">[\cdot, f]</math> for some <math display="inline">f \in A_n</math>; any <math display="inline">f\in A_n</math> yields a derivation <math display="inline">[\cdot, f]</math>; if <math display="inline">f, f' \in A_n</math> satisfies <math display="inline">[\cdot, f] = [\cdot, f']</math>, then <math display="inline">f - f' \in F</math>.
The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.Template:Sfn
Template:Collapse top Since the commutator is a derivation in both of its entries, <math display="inline">[\cdot, f]</math> is a derivation for any <math display="inline">f\in A_n</math>. Uniqueness up to additive scalar is because the center of <math display="inline">A_n</math> is the ring of scalars.
It remains to prove that any derivation is an inner derivation by induction on <math display="inline">n</math>.
Base case: Let <math display="inline">D: A_1 \to A_1</math> be a linear map that is a derivation. We construct an element <math display="inline">r</math> such that <math display="inline">[p, r] = D(p), [q,r] = D(q)</math>. Since both <math display="inline">D</math> and <math display="inline">[\cdot, r]</math> are derivations, these two relations generate <math display="inline">[g, r] = D(g)</math> for all <math display="inline">g\in A_1</math>.
Since <math display="inline">[p, q^mp^n] = mq^{m-1}p^n</math>, there exists an element <math display="inline">f = \sum_{m,n} c_{m,n} q^m p^n</math> such that <math display="block">
[p, f] = \sum_{m,n} m c_{m,n} q^m p^n = D(p)
</math>
<math display="block">
\begin{aligned}
0 &\stackrel{[p, q] = 1}{=} D([p, q]) \\
&\stackrel{D \text{ is a derivation}}{=} [p, D(q)] + [D(p), q] \\
&\stackrel{[p,f] = D(p)}{=} [p, D(q)] + [[p,f], q] \\
&\stackrel{\text{Jacobi identity}}{=} [p, D(q) - [q, f]]
\end{aligned}
</math>
Thus, <math display="inline">D(q) = g(p) + [q, f]</math> for some polynomial <math display="inline">g</math>. Now, since <math display="inline">[q, q^m p^n] = -nq^mp^{n-1}</math>, there exists some polynomial <math display="inline">h(p)</math> such that <math display="inline">[q, h(p)] = g(p)</math>. Since <math display="inline">[p, h(p)] = 0</math>, <math display="inline">r = f + h(p)</math> is the desired element.
For the induction step, similarly to the above calculation, there exists some element <math display="inline">r \in A_n</math> such that <math display="inline">[q_1, r] = D(q_1), [p_1, r] = D(p_1)</math>.
Similar to the above calculation, <math display="block">
[x, D(y) - [y, r]] = 0
</math> for all <math display="inline">x \in \{p_1, q_1\}, y \in \{p_2, \dots, p_n, q_2, \dots, q_n\}</math>. Since <math display="inline">[x, D(y) - [y, r]]</math> is a derivation in both <math display="inline">x</math> and <math display="inline">y</math>, <math display="inline">[x, D(y) - [y, r]] = 0</math> for all <math display="inline">x\in \langle p_1, q_1\rangle</math> and all <math display="inline">y \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>. Here, <math display="inline">\langle \rangle</math> means the subalgebra generated by the elements.
Thus, <math display="inline">\forall y \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>, <math display="block">
D(y) - [y, r] \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle </math>
Since <math display="inline">D - [\cdot, r]</math> is also a derivation, by induction, there exists <math display="inline">r' \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math> such that <math display="inline">D(y) - [y, r] = [y, r']</math> for all <math display="inline">y \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>.
Since <math display="inline">p_1, q_1</math> commutes with <math display="inline">\langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>, we have <math display="inline">D(y) = [y, r + r']</math> for all <math>y \in \{p_1, \dots, p_n, q_1, \dots, q_n\}</math>, and so for all of <math>A_n</math>. Template:Collapse bottom
Representation theoryEdit
Zero characteristicEdit
In the case that the ground field Template:Mvar has characteristic zero, the nth Weyl algebra is a simple Noetherian domain.Template:Sfn It has global dimension n, in contrast to the ring it deforms, Sym(V), which has global dimension 2n.
It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of σ(q) and σ(Y) for some finite-dimensional representation σ (where Template:Nowrap).
- <math> \mathrm{tr}([\sigma(q),\sigma(Y)])=\mathrm{tr}(1)~.</math>
Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional.
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 Template:Nowrap called the 'characteristic variety'Template:What whose size roughly corresponds to the sizeTemplate:What of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,
- <math>\dim(\operatorname{char}(M))\geq n</math>
An even stronger statement is Gabber's theorem, which states that Char(M) is a co-isotropic subvariety of Template:Nowrap for the natural symplectic form.
Positive characteristicEdit
The situation is considerably different in the case of a Weyl algebra over a field of characteristic Template:Nowrap.
In this case, for any element D of the Weyl algebra, the element Dp is central, and so the Weyl algebra has a very large center. In fact, it is a finitely generated module over its center; even more so, it is an Azumaya algebra over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension p.
GeneralizationsEdit
The ideals and automorphisms of <math>A_1</math> have been well-studied.Template:SfnTemplate:Sfn The moduli space for its right ideal is known.Template:Sfn However, the case for <math>A_n</math> is considerably harder and is related to the Jacobian conjecture.Template:Sfn
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.
Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.
Affine varietiesEdit
Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring
- <math>R = \frac{\mathbb{C}[x_1,\ldots,x_n]}{I}.</math>
Then a differential operator is defined as a composition of <math>\mathbb{C}</math>-linear derivations of <math>R</math>. This can be described explicitly as the quotient ring
- <math> \text{Diff}(R) = \frac{\{ D \in A_n\colon D(I) \subseteq I \}}{ I\cdot A_n}.</math>