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In quantization, we frequently run into ordering ambiguities. In general, this means that there can be inequivalent quantum theories corresponding to the same classical theory.

Has there ever been an experiment that favored one choice of ordering to another?

Below is a more detailed description of the same question.

For example, one of the terms in the Hamiltonian of the harmonic oscillator $$ v = q^2 $$ can be quantized either as $$ V = Q^2 = \frac{1}{2} (A + A^{\dagger})^2 = \frac{1}{2} \left( A^2 + A A^{\dagger} + A^{\dagger} A + A^{\dagger 2} \right), $$ or as $$ V_0 = :V: = \frac{1}{2} :(A + A^{\dagger})^2: = \frac{1}{2} \left( A^2 + 2 A^{\dagger} A + A^{\dagger 2} \right) = Q^2 - \frac{1}{2}. $$ The first choice of ordering is rather arbitrary and comes from naively plugging in the expression for $Q$ into the classical formula $v = q^2$ without noticing the ordering ambiguity at all. The second term is obtained through normal ordering, which is arguably the correct physical choice of ordering.

This ordering ambiguity is benign because it only contributes to a constant shift in energy levels, which is unobservable, because experiments only measure differences in energy of two different levels.

However, consider the quartic term $$ u = q^4. $$

This can be quantized as $$ U = Q^4, $$ or as $$ U_0 = :U: = Q^4 - \frac{3}{2} Q^2 + \frac{1}{2}. $$

Let's assume that the latter is the correct physical choice. The constant shift of $1/2$ is again unphysical, but the $-3Q^2/2$ term modifies the Hamiltonian and its energy levels. It is in principle observable, provided that the constant in front of the $Q^2$ term is fixed by some fundamental principle (otherwise, this term will get re-absorbed into the empirical value for that constant).

My question is – is there any known experiment where the ordering ambiguity is experimentally shown to be resolved one way or another?

Prof. Legolasov
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  • You are basically asking for realistic systems described by different quantum hamiltonians, which, nevertheless, share a common classical limit. A related question is here. But there is no such thing as "resolution" of the ambiguity : nature does not choose orderings. I assume you have boned up on the classic Quantization is a mystery. – Cosmas Zachos Mar 03 '20 at 14:03
  • @CosmasZachos I understand that nature is quantum and quantization is a heuristic procedure. But if we have two theories with distinct predictions, both obtained by heuristic procedures, in hopes to describe the same quantum phenomenon, experiment should decide which one is right and which is wrong, don't you agree? I'm looking for examples of this situation. – Prof. Legolasov Mar 03 '20 at 14:09
  • Examples of the different spectra of different hamiltonians? You could cook one up with number operators of an oscillator. – Cosmas Zachos Mar 03 '20 at 14:14
  • @CosmasZachos the form of the Hamiltonian is not God-given, it is usually determined by physical principles. E.g. the Coulomb potential gives the Hamiltonian for the hydrogen atom. I'm interested in examples of Hamiltonians that arise from physical principles, where there is an ordering ambiguity that is impossible to resolve (both resolutions are mathematically consistent) but the experiment agrees with one and not the other. This question is not about the mathematical formulation of quantum mechanics, it is about physics. – Prof. Legolasov Mar 03 '20 at 14:17
  • I'm not sure what you understand by "physical principles". There are complicated condensed matter systems that can be mapped to recondite inequivalent hamiltonians after a series of maps. Even the hydrogen atom may be mapped to freaky oscillator systems. So expecting "nature" to dictate a mathematical "choice" of orderings and therefore exclude systems you might discover as "unnatural" is a tall order. – Cosmas Zachos Mar 03 '20 at 14:21
  • Theoretically, computer simulators like "Kerr" hamiltonians, $N+\lambda N^2$ versus its ordering inequivalent brothers, with visibly different spectra, to monitor their long-term dynamical differences. But I am not knowledgeable which of them "nature" prefers more often, as there is an element of subjectivity in which systems physicists choose to describe that way more often. – Cosmas Zachos Mar 03 '20 at 14:32
  • @CosmasZachos I gave an example of what I mean by "physical principles" in my question. Say we somehow know that a certain system classically is described by $H = p^2 / (2m) + \lambda q^4$. There is no $q^2$ term. It makes sense to examine the energy levels of that system and compare against predictions given by the two different quantizations with $Q^4$ and $:Q^4:$. I realize that it is possible that for different systems different quantizations are empirially "correct" (read, more successful than others). I'm simply looking for a single such example. – Prof. Legolasov Mar 03 '20 at 14:39
  • Yes, my first comment. It is the "physical principles" theology which grates. There are all kinds of recondite systems in the lab that aren't conceivably describable by classical situations. – Cosmas Zachos Mar 03 '20 at 15:28
  • Anyway, an entry point to the literature on Kerr anharmonic oscillators is this one. – Cosmas Zachos Mar 03 '20 at 20:29
  • Your best bet: Savchenkov, Matsko, Liang, Kerr combs with selectable central frequency. Nature Photon 5, 293–296 (2011). But, of course, all ordering-difference features are tunable to each other. – Cosmas Zachos Mar 04 '20 at 17:03
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    +1 for the question. As far as I am concerned there is ordering ambiguity in the hydrogen atom Hamiltonian: The quantization of the kinetic energy term ought to be rotational invariant around the origin (i.e. the nucleus), but translation invariance is no longer well motivated: this leaves possibilities for quantizations of the form $\mathbf{p}^2 \rightarrow \frac{1}{f(|\mathbf{x}|)}\nabla\cdot (f(|\mathbf{x}|) \nabla ...)$ with $f$ arbitrary. I'm wondering whether the Laplace-Runge-Lenz symmetry (SO(4) symmetry instead of SO(3)) might fix $f=1$. How does Pauli '24 deal with the issue? – 5th decile Jul 07 '21 at 00:01
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    Sorry, it's Pauli '26: https://arxiv.org/pdf/quant-ph/0212010.pdf – 5th decile Jul 07 '21 at 00:08

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