On Groenewold's Theorem and Classical and Quantum Hamiltonians

I recently encountered Groenewold's Theorem or the Groenewold-Van Hove Theorem which shows that there is no function which can satisfy the following mapping

$$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$

Does this show that there exist Quantum Hamiltonians which cannot be derived by the process of Quantization?

If so in what way would the derivation of such a Hamiltonian be approach and what are some examples of such cases?

This is related to a previous question I asked.

Answer

You probably need to internalize Ivan Todorov's accessible Quantization is a mystery. Your best bet for addressing your questions is Geometric quantization, not phase space quantization that you appear to be hung up on. Since I did not follow the logic of your conclusion/question after the Dirac 1925 thesis heuristic vision map you wrote, and I have no broad interest in quantization trickery, I will probably not be as helpful in your quest. The "related" link I stuck in the comments might be more helpful to you. For a thorough discussion of quantization, gebnerations have relied on Abraham & Marsden pp 425-452.

Since, however, Groenewold's breathtaking 1946 theorem closed the book on deformation quantization in phase space, I might as well review it to send you packing elsewhere--probably Geometric quantization.

• (Today, "deformation quantization" merely means "quantum theory in phase space, regarded as a deformation of classical mechanics there"; it definitely does not mean quantization: systematic derivation of a consistent quantum theory through a unique functor from classical mechanics. That is because Groenewold proved that Weyl's and von Neumann's misplaced expectations of such were off. So, more so than in the past, people now quantize heuristically, pulling quantum theories out of well-informed hats, and are at peace with it.)

Groenewold's correspondence principle theorem enunciates that, in general, there is no invertible linear map from all functions of phase space $f(x,p), g(x,p),...,$ to hermitean operators in Hilbert space ${\mathfrak Q}(f)$, ${\mathfrak Q}(g),...,$ such that the PB structure is preserved, $$ {\mathfrak Q} (\{ f,g\})= \frac{1}{i \hbar} ~ \Bigl [ ~ {\mathfrak Q}(f) , {\mathfrak Q} (g)~\Bigr ] ~, $$ as envisioned in Dirac's functor heuristics.

Instead, the Weyl 1927 correspondence map from phase-space functions to ordered operators in Hilbert space, $$ {\mathfrak W}(f) \equiv \frac{1}{(2\pi)^2}\int d\tau d\sigma dx dp ~ f(x,p) \exp (i\tau ({ {\mathfrak p}}-p)+i\sigma ({ {\mathfrak x}}-x)), $$ determines the $\star$-product in
${\mathfrak W} (f\star g) ={\mathfrak W} (f) ~ {\mathfrak W} (g)$, and thus the Moyal Bracket Lie algebra,
$$ {\mathfrak W} (\{\!\!\{ f,g \}\!\!\} )= \frac{1}{i \hbar} ~ \Bigl [ {\mathfrak W}(f) , {\mathfrak W} (g) \Bigr ] . $$

It is the MB, then, instead of the PB, which maps invertibly to the quantum commutator.

The two infinite dimensional Lie algebras, MB and PB are essentially different. (Actually the PB is a Wigner-İnönü contraction of the MB, analogous to $SU(\infty)$ being a contraction of SU(N).)

That is to say, the "deformation" involved in phase-space quantization is nontrivial: the quantum (observable) functions, in general, need not coincide with the classical ones, and contain more information than those (Groenewold).

For example, the Wigner transform (inverse Weyl transform) of the square of the angular momentum ${\mathfrak L}\cdot {\mathfrak L}$ turns out to be $L^2 - 3 \hbar^2/2$, significantly for the ground-state Bohr orbit.

Groenewold's celebrated counterexample noted that the classically vanishing PB expression $$ \{x^3 ,p^3\}+ \frac{1}{12} \{\{p^2,x^3\} , \{x^2,p^3\}\} = 0 $$ is anomalous in implementing Dirac's heuristic proposal to substitute commutators of ${\mathfrak Q}(x), {\mathfrak Q}(p),...$, for PBs upon quantization: Indeed, this substitution, or the equivalent substitution of MBs for PBs, yields a Groenewold anomaly , $-3 \hbar^2$, for this specific expression.

People tried in vain for decades to instead (!?) find a "better" ordering prescription than Weyl's that would magically (philosopher's stone in quantization chrysopoeia?) produce quantum observables "correctly" out of classical ones. Until they appreciated how all ordering prescriptions are technically equivalent, so, then, related to Weyl's by an equivalence (basis) transformation. All prescriptions produce brackets isomorphic to commutators and hence MBs, the Lie algebra of QM:

• QM observables are an irrep of the MB, not the PB infinite-dimensional Lie algebra.

So, what do people do in practice? What you saw above. It is trivial to quantize the hamiltonian (unless you are dealing with velocity-dependent potentials mixing coordinates and momenta), by dialing the desired spectra and symmetries of the problem. They then cobble up the suitable observables such as angular momentum, as Pauli did in his solution of the Hydrogen atom using SO(4) symmetry. They can then verify many observables violate "consistent quantization desiderata" as here (further observables "acting up" like this are the Boltzmann exponential), and smile, shrug, and move on. A fact of life. What possessed early practitioners to expect classical mechanics would suffice to completely specify quantum mechanics with its ferocious Planck constant dependence through a functor? As though $\hbar$ lacked extra information beyond classical physics? Were they thinking of magical analyticity constraints? You tell me.