Physically distinct quantizations

Question

In J. Phys. A: Math. Gen. 22 (1989) 811-822, Crehan considered the classical Hamiltonian,

\begin{align} H=\frac{p^2}{2}+\frac{q^2}{2}+\lambda(p^2+q^2)^3\,. \end{align}

Due to the presence of the third term, the process of quantizing $H$ is ambiguous as we need to worry about the order of the operators. For instance, we could write $\hat{H}$ using standard ordering, anti-standard ordering, or Weyl ordering, to name a few possibilities.

Crehan showed that the eigenfunction of $H$ for all possible quantizations is the eigenfunction of the SHO, but with an eigenvalue of

\begin{align} E_n=\frac{1}{2}\hbar(2n+1)+\lambda\hbar(2n+1)^3+\lambda(3\hbar^2\alpha-4)(2n+1)\,, \end{align}

where $\alpha$ is a parameter of the quantization.

How do we know which value of $\alpha$ gives the correct quantization/operator ordering for this problem? As different quantizations yield different operator orderings and hence different energies, we have physically distinct predictions.

Answer 1

But...did you specify the problem? Which problem? Of course you have physically distinct predictions. Which ones do you want to use and where? Crehan's paper finds all 2-parameter (ħ,α) deformations of this cubic oscillator, subject to his plausible constraints, but you seem to have extra conditions based on unstated physics principles? If you do not state them, your question cannot be answerable.

Either you look at your experimental system modeled by this deformed oscillator and see which α best fits its spectrum, a situation often occurring with small systems relying on such simple models, e.g. in nonlinear optics; or, else, with Robnik (cited), you search for convenient recipes and models easiest handled by some particular class of αs. (Also see the stochastic transition to quantum non-integrable systems in de Carvalho, R. E. (1993). "Classical and quantal aspects of resonant normal forms". Nonlinearity 6(6), 973.)

In a range of problems, like quantization on nonstandard manifolds (spheres, etc..) you pick the α that best preserves classical symmetry algebras through quantization-- often you want to preserve those.

Ivan Todorov's tasteful and edifying "Quantization is a mystery" covers the waterfront.

But quantization is the quintessential one-to-many map (it contains additional information over and above the classical limit---otherwise how could people have invented QM, and why?), and you never know you have the correct quantum hamiltonian, operator, etc... until it is checked to describe an experimental situation. QM is not a coordinate-change-like functor of classical mechanics, it is an extension with new information over and above what survives the classical limit.

Frankly, I'd shudder to think you observe two different quantum systems in the lab with different αs and spectra, etc, but the same classical limit, and somehow decide that only one of them is "correct" on unstated capricious metaphysical principles...