Canonical Form of an Element of the Weyl Algebra

Q: Is the choice of putting the x's first significant?

Mathematically, no: the reversed monomials ∂^β x^α are equally a basis, and the two are related by the explicit change of basis in the Properties section. Practically it matters a great deal, because coefficients differ between conventions and software packages do not all agree.

Q: Where exactly does characteristic zero get used?

In one step of the independence proof: after evaluating at x^σ the surviving expression is σ! ∑ _α c_ασx^α , and concluding that the coefficients vanish requires σ! ≠ 0 in K . In characteristic p that fails for |σ| ≥ p , and the theorem genuinely fails for the operator algebra.

Q: How do I multiply two elements in canonical form?

Only the middle block needs attention. To compute x^α∂^β ⋅ x^γ∂^δ , expand ∂^β x^γ by (2.3), then attach x^α on the left and ∂^δ on the right of every term and merge. For n = 1 and small degrees this is quick by hand; the term count grows as ∏ _i( (β_i,γ_i)+1) .

Q: How is this related to the Poincare-Birkhoff-Witt theorem?

It is an instance of the same phenomenon. A_n is the enveloping algebra of the Heisenberg Lie algebra modulo the relation that the central generator acts as 1 , and PBW gives ordered monomial bases for enveloping algebras. The direct proof given here is shorter and makes the role of the characteristic explicit.

Q: Can I read the order of an operator off its canonical form?

Yes: the order is the largest |β| occurring with a non-zero coefficient, and the Bernstein degree is the largest |α|+|β| . Both are well defined precisely because the canonical form is unique. Read off the wrong expression — one that is not in canonical form — and either number can come out too large.

Q: What is the canonical form of (x∂)^m?

∑ _j=1^m S(m,j),x^j∂^j , where S(m,j) are the Stirling numbers of the second kind. For m = 3 this is x^3∂^3 + 3x^2∂^2 + x∂ , computed in the worked example. The inverse relation expresses x^j∂^j as a signed combination of powers of x∂ using Stirling numbers of the first kind.

Overview

An element of $A_n$ arrives as some expression in the generators — a product of operators, or a sum of such products, in whatever order the construction happened to produce. Because the generators do not commute, the same operator has infinitely many such expressions: $∂ x$ , $x∂ + 1$ and $12(∂ x + x∂) + 12$ are all the same element of $A_1$ . Without a rule for choosing among them, one cannot even decide whether two expressions denote the same operator.

The canonical form supplies the rule. Fix the convention that every multiplication operator is written to the left of every derivative. Then each element of $A_n$ has exactly one expression of the form

D = ∑ _α,β ∈ N^n c_αβ, x^α ∂^β,

with finitely many non-zero coefficients $c_αβ ∈ K$ . Equivalently, the monomials $x^α∂^β$ form a basis of $A_n$ as a $K$ -vector space — the canonical basis.

There are two things to prove, and they are of quite different character. That the monomials span is a rewriting procedure: the commutation relations let any expression be pushed into the required order, and the procedure terminates. That they are linearly independent uses the fact that elements of $A_n$ are operators: a non-trivial combination is shown to act non-trivially on a carefully chosen polynomial. This page gives the rewriting in full and the independence argument in outline; the complete proof is on the dedicated proof page.

Definition

The statement is cleanest in multi-index notation. A multi-index is an element $α = (α_1, … ,α_n) ∈ N^n$ ; it abbreviates

x^α = x_1^α_1 ⋅ s x_n^α_n, ∂^β = ∂_1^β_1 ⋅ s∂_n^β_n, |α| = α_1 + ⋅ s + α_n, α! = α_1! ⋅ sα_n!.

No ambiguity arises in either product because the $x_i$ commute among themselves and the $∂_i$ commute among themselves. A pair $(α,β)$ of multi-indices in $N^n$ is itself a multi-index in $N^2n$ , so it too has a length, namely $|α| + |β|$ .

The canonical basisCoutinho (2.1)

Let $K$ be a field of characteristic zero. The set

B = , x^α∂^β ;:; α,β ∈ N^n ,

is a basis of $A_n$ as a $K$ -vector space. Consequently every $D ∈ A_n$ is a finite sum $D = ∑ c_αβx^α∂^β$ with uniquely determined coefficients $c_αβ ∈ K$ .

Canonical form

That unique expression is the canonical form of $D$ . Two elements of $A_n$ are equal if and only if their canonical forms have the same coefficients, so canonical form solves the word problem for $A_n$ outright.

Remark

The choice to put the $x$ 's on the left is a convention and nothing more. The reversed monomials $∂^β x^α$ are equally a basis, as is the Weyl-symmetric family obtained by symmetrising each product. What matters is that some order is fixed, so that the expression is unique. Comparing a formula against a source or a software package requires knowing which convention that source uses.

Core Concepts

Why a basis is not obvious

Since $A_n$ is generated by the $2n$ operators, everything in it is certainly a linear combination of words in those generators. The claim is stronger: the sorted words already suffice, and there are no relations among them. Both halves need argument. In a general algebra given by generators and relations, sorted monomials need not span (the relations may not be orientable into a terminating rewriting system) and need not be independent (the relations may collapse them). Here both work, and the reason is that the commutator of two generators is a scalar, which is strictly simpler than either factor.

Rewriting terminates because commutators are cheap

The rewriting step is $∂_i x_j → x_j ∂_i + δ_ij$ . Applying it to a word replaces one occurrence of a \"bad\" adjacent pair (a derivative immediately left of a multiplication) by a word with one fewer inversion, plus possibly a strictly shorter word. Because the number of inversions of a word is a non-negative integer that strictly decreases, and the correction terms are shorter, the process cannot run forever. This is the same mechanism that makes the Poincare-Birkhoff-Witt theorem work for enveloping algebras.

Independence is a statement about operators

Nothing in the rewriting shows that the sorted monomials are independent — rewriting alone can never rule out an unexpected collapse. The independence is imported from the definition of P0 as operators: to prove a combination $D = ∑ c_αβx^α∂^β$ non-zero, one exhibits a polynomial $f$ with $D(f) ≠ 0$ . The right $f$ is a monomial $x^σ$ where $σ$ is chosen to be of minimal length among the derivative multi-indices actually occurring. Applying $D$ to $x^σ$ then annihilates every term except those with $β = σ$ , and what survives is $σ!$ times a non-zero polynomial in the $x$ 's alone.

What the canonical form buys

Three things, all of which are used constantly. Equality becomes coefficient comparison, so the algebra has a decidable word problem and a usable data structure. The two natural degree functions become readable off the exponents: the Bernstein degree is $(|α| + |β|)$ and the order is $|β|$ . And the symbol map $x^α∂^β ↦ x^αξ^β$ becomes well defined, which is what lets a non-commutative problem be attacked with commutative algebra.

Construction and Proof

The monomials span

Every element of $A_n$ is a $K$ -linear combination of words in the $2n$ generators, so it is enough to bring a single word into canonical form.

Termination of the rewriting

Let $w$ be a word in $x_1, … ,x_n,∂_1, … ,∂_n$ . Call a position in $w$ an inversion if a derivative stands immediately to the left of a multiplication operator. If $w$ has no inversions, all the $x$ 's precede all the $∂$ 's, and since the $x$ 's commute among themselves and the $∂$ 's do too, $w$ is already of the form $x^α∂^β$ .

If $w$ has an inversion, apply (2.2) at that position, replacing $… ∂_i x_j …$ by $… x_j∂_i … + δ_ij( … )$ . The first word has the same length as $w$ and strictly fewer inversions; the second, when it is present, is two letters shorter. Order words by the pair (length, number of inversions), lexicographically. Each rewriting step strictly decreases this pair, and the pair takes values in a well-ordered set, so no infinite chain of rewritings is possible. The procedure therefore terminates, and it terminates with a linear combination of elements of $ℬ$ .

Remark

A slicker phrasing of the same argument: by (2.2) the subspace spanned by $ℬ$ is stable under left multiplication by each $x_i$ (obvious) and by each $∂_i$ (because $∂_i ⋅ x^α∂^β = x^α∂^β+e_i + α_i x^α - e_i∂^β$ , both of which lie in the span). A subspace containing $1$ and stable under left multiplication by all generators is a left ideal containing $1$ , hence all of $A_n$ .

The monomials are independent

Here is the argument in outline; it is carried out in full, with all the multi-index bookkeeping, on the proof page.

Evaluating a derivative monomialCoutinho (2.2)

Let $σ, β ∈ N^n$ with $|σ| ≤ |β|$ . Then $∂^β(x^σ) = σ!$ if $σ = β$ , and $∂^β(x^σ) = 0$ otherwise.

Proof

Directly, $∂^β(x^σ) = ∏ _i ∂_i^β_i(x_i^σ_i)$ , and the $i$ -th factor is $0$ unless $σ_i ≥ β_i$ . So a non-zero value requires $σ_i ≥ β_i$ for every $i$ , hence $|σ| ≥ |β|$ . Combined with the hypothesis $|σ| ≤ |β|$ this forces $|σ| = |β|$ and then $σ_i = β_i$ for every $i$ . In that case each factor is $σ_i!$ , and the product is $σ!$ .

Independence of the canonical monomials (outline)

Suppose $∑ c_αβx^α∂^β = 0$ as an operator, with some coefficient non-zero. Among all $β$ with $c_αβ ≠ 0$ for some $α$ , choose one of minimal length and call it $σ$ . Evaluate the operator at $x^σ$ .

Terms with $|β| < |σ|$ do not occur, by minimality. Terms with $|β| ≥ |σ|$ are covered by the lemma, which kills them unless $β = σ$ . What remains is

0 = ∑ _α c_ασ,x^α,∂^σ(x^σ) = σ! ∑ _α c_ασ, x^α.

The multi-index factorial $σ!$ is a non-zero element of $K$ because $K$ has characteristic zero, and the monomials $x^α$ are linearly independent in $K[X]$ . Hence $c_ασ = 0$ for all $α$ , contradicting the choice of $σ$ .

Where characteristic zero enters

The step that divides by $σ!$ is where characteristic zero is used, and it is not cosmetic. Over a field of characteristic $p$ the operator $∂_1^p$ is a non-zero element of the abstractly presented Weyl algebra but acts as zero on $K[X]$ , so the monomials in $ℬ$ are not independent as operators. The canonical basis theorem, stated for the operator algebra, is false in characteristic $p$ .

Key Equations

The canonical form itself:

D = ∑ _α,β ∈ N^n c_αβ, x^α∂^β, c_αβ ∈ K and almost all c_αβ = 0.

(2.1)

The single rewriting rule that produces it, from the commutation relations:

∂_i x_j = x_j ∂_i + δ_ij, more generally ∂_i f = f∂_i + (∂ f)/(∂ x_i) (f ∈ K[X]).

(2.2)

Iterating (2.2) gives the closed-form reordering rule, which is how an implementation multiplies two basis elements:

∂^β x^γ = ∑ _κ ≤ (β,γ) βκ γκ κ!; x^γ-κ,∂^β-κ,

(2.3)

the sum over multi-indices $κ$ with $κ_i ≤ (β_i,γ_i)$ , and $βκ = ∏ _i β_iκ_i$ .

The lemma that powers the independence proof evaluates a derivative monomial on a monomial of no larger degree:

∂^β (x^σ) = cases σ! & if σ = β, 0 & if σ ≠ β, cases whenever |σ| ≤ |β|.

(2.4)

Applied to (2.1) at the minimal $σ$ , this gives the evaluation that proves independence:

D(x^σ) = σ! ∑ _α c_ασ, x^α when c_αβ = 0 for all α and all |β| < |σ|.

(2.5)

Finally, the two degree functions read off the canonical form:

_B D = |α| + |β| : c_αβ ≠ 0, ord D = |β| : c_αβ ≠ 0.

(2.6)

Variable Definitions

$K$: the ground field, of characteristic zero
$K[X]$: the polynomial ring $K[x_1, … ,x_n]$ on which $A_n$ acts
$α, β, γ, κ, σ$: multi-indices in $N^n$
$x^α$ , $∂^β$: the monomials $x_1^α_1 ⋅ s x_n^α_n$ and $∂_1^β_1 ⋅ s∂_n^β_n$
$|α|$: the length $α_1 + ⋅ s + α_n$ of a multi-index, equal to the degree of $x^α$
$α!$: the multi-index factorial $α_1! ⋅ sα_n!$
$κ ≤ β$: the componentwise order: $κ_i ≤ β_i$ for every $i$
$c_αβ$: the coefficients in the canonical form of an operator, uniquely determined by it
$ℬ$: the canonical basis $x^α∂^β$
$_B D$ , $ord D$: the Bernstein degree and the order of $D$ , read off the canonical form by (2.6)

Properties and Behaviour

Dimension

$_K A_n$ is countably infinite, since $ℬ$ is indexed by $N^n × N^n$ . In particular $A_n$ is a proper subalgebra of $End_K(K[X])$ , which has uncountable dimension.

Freeness over the polynomial subalgebras

$A_n$ is free as a left module over the subalgebra $K[X]$ of multiplication operators, with basis $∂^β : β ∈ N^n$ ; and free as a right module over $K[∂_1, … ,∂_n]$ with basis $x^α$ . Both statements are just the canonical form regrouped, and the first is the one used when the order filtration is analysed, since each piece of that filtration is a finitely generated free $K[X]$ -module.

The reversed basis

The reversed monomials $∂^β x^α$ also form a basis. One way to see this is to apply the algebraic Fourier automorphism $x_i ↦ ∂_i$ , $∂_i ↦ -x_i$ , which is an automorphism of $A_n$ carrying one family to the other up to signs. The change-of-basis coefficients are given by (2.3); in $A_1$ ,

x^a∂^b = ∑ _k ≥ 0 (-1)^k ak bk k!; ∂^,b-k x^,a-k,

which for $a = b = 1$ reads $x∂ = ∂ x - 1$ , as it must.

The symbol map is linear, not multiplicative

The symbol map $σ : A_n → K[x_1, … ,x_n,ξ_1, … ,ξ_n]$ sending $x^α∂^β ↦ x^αξ^β$ is a well-defined $K$ -linear bijection — but it is not a ring homomorphism. Since $∂ x = x∂ + 1$ , its symbol is $σ(∂ x) = xξ + 1$ , whereas $σ(∂)σ(x) = xξ$ ; the two differ by exactly the commutator. It becomes multiplicative only after passing to the associated graded algebra, where the discrepancy is of lower degree and is discarded.

Reading off the filtrations

Canonical form makes the two filtrations transparent. If $D$ has canonical coefficients supported in $|α|+|β| ≤ m$ then $D ∈ B_m$ , and conversely; similarly $D ∈ F_m$ exactly when $c_αβ = 0$ whenever $|β| > m$ . It also makes visible that $_K B_m = 2n+m2n$ , since $B_m$ has as basis the monomials of $ℬ$ of total degree at most $m$ in $2n$ exponents.

Examples and Special Cases

Small reorderings

In $A_1$ : $∂ x = x∂ + 1$ ; $∂^2 x = x∂^2 + 2∂$ ; $∂ x^2 = x^2∂ + 2x$ ; $∂^2 x^2 = x^2∂^2 + 4x∂ + 2$ . Each is one application of (2.3) with $n = 1$ .

Only same-index pairs cost anything

In $A_2$ , the operator $∂_1 x_2 ∂_1 x_2$ has two inversions, but each pairs a derivative with a variable of a different index. Since $∂_1$ and $x_2$ commute, both inversions clear with no correction terms and the canonical form is simply $x_2^2∂_1^2$ . Compare the same-index case in $A_1$ : $(∂ x)^2 = x^2∂^2 + 3x∂ + 1$ , where every reordering costs a term. Cross pairs are free; only the diagonal pairs are expensive.

Classical operators are already canonical

The Bessel operator $x^2∂^2 + x∂ + (x^2-ν^2)$ and the Airy operator $∂^2 - x$ are already in canonical form. The Legendre operator, written as $(1-x^2)∂^2 - 2x∂$ , is too: no reordering is needed because every coefficient sits to the left. Classical differential operators are almost always presented in canonical form, which is why the issue only becomes visible once operators are composed.

Products leave the basis

Canonical form is not preserved by composition, which is the whole point. $(x∂)(x∂) = x^2∂^2 + x∂$ , so the square of a canonical monomial is not a canonical monomial. Any implementation must renormalise after every product.

Worked Example

The canonical form of $(x∂)^3$ in $A_1$

Step 1 - square the Euler operator
Write $θ = x∂$ . The only inversion in $θ^2 = x∂ x∂$ is the central $∂ x$ , so one application of (2.2) suffices:

$θ^2 = x(∂ x)∂ = x(x∂ + 1)∂ = x^2∂^2 + x∂.$

Check on $x^k$ : since $θ(x^k) = kx^k$ , the left side gives $k^2 x^k$ ; the right side gives $k(k-1)x^k + kx^k = k^2 x^k$ .
Step 2 - multiply by θ once more
Now $θ^3 = x∂(x^2∂^2 + x∂)$ . Two reorderings are needed, $∂ x^2 = x^2∂ + 2x$ and $∂ x = x∂ + 1$ :

$x∂,x^2∂^2 = x(x^2∂ + 2x)∂^2 = x^3∂^3 + 2x^2∂^2,$

$x∂,x∂ = x(x∂+1)∂ = x^2∂^2 + x∂.$

Adding the two lines and collecting like basis elements,

$θ^3 = x^3∂^3 + 3x^2∂^2 + x∂.$
(2.7)
Step 3 - verify on the general monomial
Apply both sides of (2.7) to $x^k$ . The left side is $k^3 x^k$ . The right side is

$k(k-1)(k-2),x^k + 3k(k-1),x^k + k,x^k = (k^3 - 3k^2 + 2k )x^k + (3k^2 - 3k )x^k + k,x^k = k^3 x^k.$

They agree identically in $k$ , so (2.7) is an identity of operators, not merely a coincidence at small values. The coefficients $1, 3, 1$ are the Stirling numbers of the second kind $S(3,3), S(3,2), S(3,1)$ , and in general $θ^m = ∑ _j=1^m S(m,j), x^j∂^j$ .
Step 4 - use uniqueness to detect non-zero
Apply the independence criterion (2.5) to $D = θ^3$ directly. The derivative multi-indices occurring are $β = 3, 2, 1$ , so the minimal one is $σ = 1$ . Evaluating at $x^σ = x$ :

$D(x) = x^3∂^3(x) + 3x^2∂^2(x) + x∂(x) = 0 + 0 + x = x,$

which is $σ!,c_1,1,x^1 = 1 ⋅ 1 ⋅ x$ , exactly as (2.5) predicts. The two higher terms are annihilated because $|β| > |σ|$ , which is the mechanism of the whole independence proof visible in one line.
Step 5 - read off the invariants
From (2.7) and (2.6): the Bernstein degree of $θ^3$ is $3+3,,2+2,,1+1 = 6$ , and the order is $3,2,1 = 3$ . Its symbol with respect to the order filtration is $x^3ξ^3$ , the top-order part only; its symbol with respect to the Bernstein filtration is also $x^3ξ^3$ , since that term is alone in top total degree. Neither symbol records the terms $3x^2∂^2$ or $x∂$ — a reminder that the symbol forgets a great deal that the canonical form retains.

Result

$(x∂)^3 = x^3∂^3 + 3x^2∂^2 + x∂$ , verified by acting on $x^k$ (both sides multiply it by $k^3$ ). The coefficients are Stirling numbers of the second kind; the operator has Bernstein degree $6$ , order $3$ and principal symbol $x^3ξ^3$ .

Applications and Industry Use

In a mathematics topic, this section covers downstream use inside mathematics, computing and engineering rather than a manufactured product.

Data structures for computer algebra. Every system that implements $A_n$ stores elements as sparse maps from exponent pairs to coefficients. Canonical form is what makes that representation faithful and equality testing exact.
Non-commutative Gröbner bases. Buchberger's algorithm generalises to $A_n$ precisely because the canonical monomials admit compatible term orders, so leading terms are well defined; this underlies every algorithm on characteristic varieties and holonomic systems.
Filtrations and dimension theory. The counts $_K B_m = 2n+m2n$ that drive Bernstein's inequality are direct consequences of the canonical basis, since $B_m$ has an explicit monomial basis.
Normal ordering in physics. The same reordering problem, with derivatives to the left, is the normal-ordering procedure for creation and annihilation operators; the coefficients that appear in (2.3) are the combinatorial content of Wick's theorem.
Combinatorial identities. Expanding $(x∂)^m$ produces Stirling numbers, and more elaborate reorderings produce whole families of identities; the same machinery drives automatic identity proving.

Design Considerations

For a mathematical object, design considerations are the modelling choices: which ring, which filtration, which category to work in.

Which side to normalise

Putting the $x$ 's on the left matches how differential operators are written classically, $∑ a_β(x)∂^β$ , and makes $A_n$ visibly a free left $K[X]$ -module. Putting the $∂$ 's on the left is more natural when the module of interest is a space of functionals or when Fourier duality is in play. The Weyl-symmetric ordering is the choice that behaves best under the symplectic symmetries of the relations, at the price of coefficients with denominators — which is why it is avoided in exact computation.

Which grading to attach

The canonical form supports both standard filtrations without modification, and the choice depends on the goal. Giving $x_i$ and $∂_i$ weight $1$ gives the Bernstein filtration, whose pieces are finite dimensional over $K$ — this is what makes Hilbert-polynomial arguments work. Giving $x_i$ weight $0$ and $∂_i$ weight $1$ gives the order filtration, whose associated graded ring is the coordinate ring of the cotangent space — this is what makes geometry work. Both are read off the same exponent vectors.

Term orders for computation

For Gröbner-basis work a total order on the exponent pairs must be chosen that is compatible with multiplication. Degree-reverse-lexicographic on $(α,β)$ is the usual default, and orders that refine the order filtration are used when the characteristic variety is wanted. The choice can change running times by orders of magnitude without changing the answer.

Standards and Codes

For mathematics, the relevant standards are notation, numeric and markup standards together with reference implementations.

ISO 80000-2 fixes the notation used here for $∂$ , for $ℕ$ , and for the factorial and binomial symbols appearing in (2.3).
ISO/IEC 40314 (MathML 3.0) is the encoding of the expressions on this page, so multi-index superscripts remain machine-readable.
There is no standard name: canonical form, normal form and normally ordered form all denote the same thing, and physics uses normal ordering for the opposite convention. Coutinho's term is canonical form, which is the one used throughout this collection.
Software normalisation sides differ; Macaulay2, Singular and SageMath each state theirs in the documentation of the constructor.
Result citation follows Coutinho's numbering: the canonical basis is (2.1) and the evaluation lemma is (2.2) in Ch. 1.

Computational Notes

Read this as the manufacturing section of the template: how the object is actually built by machine, at what cost, and where the computation stops being decidable.

Canonical form is the specification of the data structure and of the two primitive operations.

Representation. A sparse dictionary from exponent pairs $(α,β) ∈ N^n × N^n$ to non-zero coefficients in $K$ .
Addition. Merge the dictionaries, adding coefficients and deleting zeros. Linear in the number of terms.
Multiplication. For each pair of terms apply (2.3) to the middle block $∂^β x^γ$ , multiply through, and merge. One pair of terms produces $∏ _i( (β_i,γ_i)+1)$ new terms.
Equality. Compare dictionaries. This is exact, and it is the reason canonical form is worth computing at all.

The cost profile is worth knowing. Term counts grow multiplicatively with $n$ , so an operator in $A_4$ with derivative degree $5$ in each variable can generate up to $6^4 = 1296$ terms from a single product of basis elements. Coefficients acquire factorials from (2.3), so over $ℚ$ the integers grow quickly; modular techniques with reconstruction are standard practice for large computations.

Because canonical form is a rewriting system, its correctness has a general explanation: the rule (2.2) is confluent and terminating, which is Bergman's diamond lemma applied to this presentation. That is also the reason the analogous statement holds for enveloping algebras (the Poincare-Birkhoff-Witt theorem) and for Ore algebras generally.

Implementations: WeylAlgebra in Macaulay2, Weyl in Singular's nctools.lib, DifferentialWeylAlgebra and ore_algebra in SageMath, and HolonomicFunctions.m in Mathematica. Each documents its normalisation side; do not compare coefficients across systems without checking it.

Limits of Validity

Characteristic zero is required for the theorem as stated. The spanning half holds over any commutative ring. The independence half divides by $σ!$ and fails in characteristic $p$ : the operator algebra generated by $x_i$ and $∂_i$ on $K[X]$ then satisfies $∂_i^p = 0$ , so $ℬ$ is not independent. The abstractly presented Weyl algebra does still have $ℬ$ as a basis in characteristic $p$ — the two objects have parted company. See the positive characteristic page.
Uniqueness is relative to a convention. \"The\" canonical form of an operator is only defined once the ordering rule is fixed. Nothing distinguishes $x$ -left from $∂$ -left mathematically; the coefficients differ, though the operator does not.
Finiteness is essential. The statement is about finite linear combinations. Infinite formal sums $∑ c_αβx^α∂^β$ define elements of larger rings — rings of formal or of infinite-order differential operators — which are genuinely different objects with different ideal theory.
It does not make the algebra commutative. A basis of ordered monomials is a statement about the underlying vector space. The multiplication is still the one dictated by (2.3), and confusing the two is the most common error in this area.

Failure Modes and Common Mistakes

Reading the basis as a ring isomorphism

Because $x^α∂^β$ and $x^αξ^β$ are indexed the same way, it is tempting to conclude $A_n ≅ K[x_1, … ,x_n,ξ_1, … ,ξ_n]$ . The bijection is one of vector spaces only. As rings they differ at the first opportunity: $∂ x - x∂ = 1$ but $ξ x - xξ = 0$ . The commutative ring is the associated graded ring of $A_n$ , a coarser invariant.

Reordering more than is necessary

When multiplying $x^α∂^β$ by $x^γ∂^δ$ , only the middle block $∂^β x^γ$ needs reordering; the outer $x^α$ and $∂^δ$ are already on the correct sides. Reordering the whole product from scratch wastes work and multiplies the opportunities for a dropped binomial coefficient.

Dropping the fully contracted term

The term $κ = (β,γ)$ in (2.3) — the one where the derivatives are entirely consumed — is the one most often dropped, because it looks degenerate. In $∂^2x^2 = x^2∂^2 + 4x∂ + 2$ it contributes the constant $2$ , and losing it makes the operator act incorrectly on constants while still appearing to work on higher monomials. Always test a reordering on $f = 1$ as well as on a general monomial.

Testing equality on too few polynomials

An operator can be recognised from the way it acts, but two operators are equal only if they agree on all of $K[X]$ . Checking agreement on $1, x, x^2$ proves nothing in general. Either compare canonical forms coefficient by coefficient, or check the action on the general monomial $x^k$ and compare polynomials in $k$ , as done in the worked example.

Confusing the canonical form with the symbol

The principal symbol keeps only the top-degree part of the canonical form, and which part that is depends on the filtration. For $θ^3 = x^3∂^3 + 3x^2∂^2 + x∂$ the lower terms are invisible to the symbol, and any argument that requires them must be made before passing to the graded algebra.

Historical Notes

Ordered-monomial bases predate the Weyl algebra. Poincare in 1900, and then Birkhoff and Witt in 1937, established that ordered monomials in a basis of a Lie algebra form a basis of its universal enveloping algebra. Since $A_n$ is a quotient of the enveloping algebra of the Heisenberg Lie algebra, the canonical basis theorem can be deduced from that result — but the direct proof by evaluation on polynomials, which is Coutinho's, is shorter and shows exactly where characteristic zero is used.

The general principle behind such theorems was isolated by Bergman in 1978 as the diamond lemma: if a rewriting system on words is terminating and its ambiguities resolve, the irreducible words form a basis. Independently, Bokut and Shirshov developed the same idea in the Soviet literature under the name of composition lemmas. Buchberger's 1965 thesis on Gröbner bases is the commutative counterpart, and the non-commutative Gröbner theory used in D-module computation today descends from all three lines.

In physics, the reordering problem appeared as normal ordering in quantum field theory, where the combinatorics of (2.3) is the content of Wick's theorem of 1950. That the same binomial-and-factorial coefficients arise in both settings is not a coincidence: they are two readings of the identical algebra.

Comparison

Three ordering conventions for the same algebra, each giving a basis.
	$x$ -left (used here)	$∂$ -left (normal ordering)	Weyl-symmetric
Basis monomials	$x^α∂^β$	$∂^β x^α$	symmetrised products of $x$ 's and $∂$ 's
Reordering rule	$∂_i x_j → x_j∂_i + δ_ij$	$x_j ∂_i → ∂_i x_j - δ_ij$	average over all interleavings
$A_n$ is free over	$K[X]$ on the left	$K[∂]$ on the left	neither, in an obvious way
Coefficients	integers	integers, with signs	rationals with denominators
Natural for	classical differential operators	Fourier duality, quantum field theory	symplectic invariance
$x∂$ becomes	$x∂$	$∂ x - 1$	$12(x∂ + ∂ x) - 12$

Key Takeaways

Key points

The monomials $x^α∂^β$ , with all multiplications to the left of all derivatives, form a $K$ -basis of $A_n$ ; the resulting unique expression is the canonical form.
Spanning is a terminating rewriting procedure driven by $∂_i x_j = x_j∂_i + δ_ij$ ; each step removes an inversion and any correction term is strictly shorter.
Independence is proved by evaluation: apply the operator to $x^σ$ for $σ$ of minimal length among the derivative indices occurring, and use $∂^β(x^σ) = σ!,δ_σβ$ for $|σ| ≤ |β|$ .
Characteristic zero is needed exactly once, to know $σ! ≠ 0$ ; in characteristic $p$ the theorem is false for the operator algebra because $∂_i^p$ acts as zero.
Canonical form makes equality decidable, the Bernstein degree and the order readable off exponents, and the symbol map well defined.
The basis is a vector-space statement only: $A_n$ is not isomorphic as a ring to the commutative polynomial ring in $2n$ variables, which is instead its associated graded ring.

FAQs

Why is a canonical form needed at all?

Because without one there is no way to tell whether two expressions denote the same operator. $∂ x$ and $x∂ + 1$ look different and are equal; $∂ x$ and $x∂$ look similar and are not. Canonical form reduces the question to comparing finitely many coefficients.

Is the choice of putting the

x

's first significant?

Mathematically, no: the reversed monomials $∂^β x^α$ are equally a basis, and the two are related by the explicit change of basis in the Properties section. Practically it matters a great deal, because coefficients differ between conventions and software packages do not all agree.

Does the canonical form make

A_n

commutative?

No. It says the underlying vector space has a monomial basis indexed like a polynomial ring in $2n$ variables. The multiplication remains non-commutative, and reduction back to canonical form after every product is exactly the extra work that non-commutativity costs.

Where exactly does characteristic zero get used?

In one step of the independence proof: after evaluating at $x^σ$ the surviving expression is $σ! ∑ _α c_ασx^α$ , and concluding that the coefficients vanish requires $σ! ≠ 0$ in $K$ . In characteristic $p$ that fails for $|σ| ≥ p$ , and the theorem genuinely fails for the operator algebra.

How do I multiply two elements in canonical form?

Only the middle block needs attention. To compute $x^α∂^β ⋅ x^γ∂^δ$ , expand $∂^β x^γ$ by (2.3), then attach $x^α$ on the left and $∂^δ$ on the right of every term and merge. For $n = 1$ and small degrees this is quick by hand; the term count grows as $∏ _i( (β_i,γ_i)+1)$ .

How is this related to the Poincare-Birkhoff-Witt theorem?

It is an instance of the same phenomenon. $A_n$ is the enveloping algebra of the Heisenberg Lie algebra modulo the relation that the central generator acts as $1$ , and PBW gives ordered monomial bases for enveloping algebras. The direct proof given here is shorter and makes the role of the characteristic explicit.

Can I read the order of an operator off its canonical form?

Yes: the order is the largest $|β|$ occurring with a non-zero coefficient, and the Bernstein degree is the largest $|α|+|β|$ . Both are well defined precisely because the canonical form is unique. Read off the wrong expression — one that is not in canonical form — and either number can come out too large.

What is the canonical form of

(x∂)^m

$∑ _j=1^m S(m,j),x^j∂^j$ , where $S(m,j)$ are the Stirling numbers of the second kind. For $m = 3$ this is $x^3∂^3 + 3x^2∂^2 + x∂$ , computed in the worked example. The inverse relation expresses $x^j∂^j$ as a signed combination of powers of $x∂$ using Stirling numbers of the first kind.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 1 §2, Proposition (2.1) and Lemma (2.2).
G. M. Bergman, The diamond lemma for ring theory, Advances in Mathematics 29 (1978), 178-218 - the general rewriting principle behind canonical forms.
J. Dixmier, Sur les algèbres de Weyl, Bulletin de la Société Mathématique de France 96 (1968), 209-242.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics 30, revised edition, American Mathematical Society, 2001 - Ch. 1 and Ch. 8, for ordered bases and filtrations.
J.-E. Björk, Rings of Differential Operators, North-Holland Mathematical Library 21, North-Holland, 1979 - Ch. 1, for symbols and filtrations built on the canonical form.
M. Saito, B. Sturmfels and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6, Springer, 2000 - Ch. 1, for term orders and Gröbner bases in the Weyl algebra.
Macaulay2 Dmodules and WeylAlgebra documentation; SageMath ore_algebra documentation - for the normalisation conventions cited above.
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization.
ISO/IEC 40314:2016, Information technology - Mathematical Markup Language (MathML) Version 3.0, International Organization for Standardization.

AI Suggested Questions

Compute the canonical form of $∂^2 x^3 ∂$ in $A_1$ and verify it on the general monomial $x^k$ .
Prove that $θ^m = ∑ _j S(m,j)x^j∂^j$ by induction, using $∂ x^j = x^j∂ + jx^j-1$ .
Derive the reordering formula (2.3) from repeated application of $∂_i x_i = x_i∂_i + 1$ .
Express $x^2∂^2$ in the reversed basis $∂^b x^a$ and check the result on $x^k$ .
Give an explicit non-trivial linear relation among the monomials $x^a∂^b$ over a field of characteristic $3$ .
Show that $A_n$ is a free left $K[X]$ -module with basis $∂^β$ and describe the induced filtration by order.
Count the number of terms produced by multiplying two generic elements of $A_2$ of Bernstein degree $4$ .