17.2 The Groenewold-van Hove no-go theorem

17.2 The Groenewold-van Hove no-go theorem

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation

If one wants to quantize polynomial functions on phase space of degree greater than two, it quickly becomes clear that the problem of “operator ordering ambiguities” is a significant one. Diferent prescriptions involving diferent ways of ordering the $P$ and $Q$ operators lead to diferent $O_f$ for the same function $f,$ with physically diferent observables (although the diferences involve the commutator of $P$ and $Q.$ , so higher-order terms in $\hslash )$

When physicists first tried to find a consistent prescription for producing an operator $O_f$ corresponding to a polynomial function on phase space of degree greater than two, they found that there was no possible way to do this consistent with the relation

$$O_{\{f,g\}}=-\frac{i}{\hslash }[O_f,O_g]$$

for polynomials of degree greater than two. Whatever method one devises for quantizing higher degree polynomials, it can only satisfy that relation to lowest order in $\hslash ,$ and there will be higher order corrections, which depend upon one’s choice of quantization scheme. Equivalently, it is only for the six dimensional Lie algebra of polynomials of degree up to two that the Schr¨odinger representation gives one a Lie algebra representation, and this cannot be consistently extended to a representation of a larger subalgebra of the functions on phase space. This problem is made precise by the following no-go theorem

Theorem (Groenewold-van Hove)

There is no map $f{O}_f$ from polynomials on ${\mathbf{R}}^2$ to self-adjoint operators on $L^2(\mathbf{R})$ satisfying

$$O_{\{f,g\}}=-\frac{i}{\hslash }[O_f,O_g]$$

and

$$O_p=P,O_q=Q$$

for any Lie subalgebra of the functions on ${\mathbf{R}}^2$ for which the subalgebra of polynomials of degree less than or equal to two is a proper subalgebra.

Proof. For a detailed proof, see section 5.4 of [8], section 4.4 of [26], or chapter 16 of [37]. In outline, the proof begins by showing that taking Poisson brackets of polynomials of degree three leads to higher order polynomials, and that furthermore for degree three and above there will be no finite dimensional subalgebras of polynomials of bounded degree. The assumptions of the theorem force certain specific operator ordering choices in degree three. These are then used to get a contradiction in degree four, using the fact that the same degree four polynomial has two diferent expressions as a Poisson bracket:

$$q^2{p}^2=\frac{1}{3}\{q^{2}p,p^{2}q\}=\frac{1}{9}\{q^3,p^3\}$$

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 197、198、199、200、201、202、203

来源版本：2025-10-20

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