# An example of Hamiltonian which fails with canonical quantization

The Groenewold's theorem states that canonical quantization, regarded as a rule to replace $$\{A,B\}$$ by $$\frac{1}{i\hbar}[A,B]$$ is inconsistent for some 3rd order polynomials of canonical variables $$p$$ and $$x$$. However, all single-particle Hamiltonian systems that I've ever seen had the form

$$H=\frac{p^2}{2m}+V(x)$$

and don't suffer from all this ordering ambiguities that Groenewold's theorem is about (because $$p$$ and $$x$$ are not multiplied). So, my question is: what are some concrete examples of classical mechanical systems that cannot be unambiguously quantized via canonical quantization due to the restrictions of Groenewold's theorem?

• – ZeroTheHero Jan 7 at 0:48
• "Fails"... "difficult" are emotionally charged, subjective rubrics, no? Kerr-oscillator type systems like the near duplicate @ZeroTheHero links above are popular in quantum optics and favored in studies of quantum approaches to chaos, but the options of many paths to quantization are normally prized as extra tools in the toolbox, when solving a tricky problem, not liabilities. Groenewold's theorem had a substantially liberating effect. – Cosmas Zachos Jan 7 at 1:47
• @CosmasZachos I apologize for using potentially confusing language. My question seems pretty specific to me: an actual real-world (i.e. used somewhere) Hamiltonian system which cannot be unambiguously quantized via canonical quantization. I've read about Kerr oscilators, and they do indeed seem to provide a concrete example. Thank you. – lisyarus Jan 7 at 2:25
• The magnetic, semirelativistic version of the above hamiltonian is another (provided that the vector potential is not linear in $x$), i. e. $H = \sqrt{(p-A(x))^2 + m^2} + V(x)$ has no canonical quantization. – Max Lein Jan 7 at 4:43