Groenewold's theorem

Builds on Canonical quantization Poisson bracket Hermite functions

Statement

Let $\hbar > 0$ and let $\mathcal{P}_{\le k}$ be the space of real polynomials of degree at most $k$ in two variables $q, p$ , with the Poisson bracket $\{f, g\} = \frac{\partial f}{\partial q}\frac{\partial g}{\partial p} - \frac{\partial f}{\partial p}\frac{\partial g}{\partial q}$ . Let $V$ be the space of functions $x \mapsto h(x) e^{-x^2/2}$ on $\R$ with $h$ a complex polynomial, which is the span of the Hermite functions. Let $X$ and $P$ be the operators on $V$ given by $(X\psi)(x) = x\psi(x)$ and $P\psi = -i\hbar\, \psi'$ . Then there is no linear map $Q \colon \mathcal{P}_{\le 4} \to \End(V)$ such that:

$Q(1) = I$ , $Q(q) = X$ and $Q(p) = P$ ;
$[Q(f), Q(g)] = i\hbar\, Q(\{f, g\})$ for all $f, g \in \mathcal{P}_{\le 3}$ .

Remarks

The idea of the proof: since $V$ is an irreducible module over the algebra generated by $X$ and $P$ , an operator on $V$ that commutes with both is a scalar. Using this and clause 2 with $f$ or $g$ of degree $1$ , the map $Q$ is forced, step by step. It must agree with Weyl quantization on all polynomials of degree at most $3$ . The polynomial $q^2 p^2$ then can be reached in two ways, since $\{q^3, p^3\} = 9 q^2 p^2$ and $\{q^2 p, q p^2\} = 3 q^2 p^2$ . The two commutators $\frac{1}{9}[Q(q^3), Q(p^3)]$ and $\frac13[Q(q^2 p), Q(q p^2)]$ , computed with the Weyl operators, differ by a nonzero multiple of $\hbar^3 I$ , which contradicts the linearity of $Q$ .

Example of what does work: on $\mathcal{P}_{\le 2}$ alone, Weyl quantization satisfies clause 2 exactly, since the bracket of two quadratic polynomials is again quadratic. So the obstruction appears only at degree $3$ and $4$ .

Going deeper

For a student

The theorem explains why canonical quantization needs extra choices beyond the canonical commutation relations. Weyl quantization satisfies the bracket rule only up to terms of order $\hbar^2$ : the Moyal bracket replaces the Poisson bracket. Deformation quantization accepts this and builds a star product whose commutator agrees with the Poisson bracket at first order in $\hbar$ . Van Hove later extended the result to more general phase spaces and quantization settings.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`EDPPWXMXA4`
URI	`m2://EDPPWXMXA4`
Kind	statement, theorem
Writing	W0 stub
MSC	81S08, 81S30
Status	This copy of the site has no build data. Run `m2 status EDPPWXMXA4`.
About	Canonical quantization
Uses	Canonical quantization, Poisson bracket, Hermite functions
Changed	3 November 2026