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?
