# Classical limit of quantum systems

I probably will run into a risk of this question being considered inappropriate for this forum. Still keeping fingers crossed!

One knows that given a classical system, if it is quantized in two different methods by say differing them in the ordering (normal ordered and anti-normal ordered) then the quantum spectrums one will get will differ only by a "constant".

I don't know how to make this notion of "quantizing in two different ways" precise except when one can distinguish them by just ordering.

But shouldn't it be possible that having quantized a classical system in two different methods in general it should be possible that their "classical limits" are actually different classical systems?

If the above is true then can one give some simple examples where this can be seen?

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BTW the standard is usually called Weyl ordering: books.google.com/… – Steve Huntsman Feb 17 '10 at 14:00
Also classical observables commute, so there should be a unique classical system unless I'm being obtuse (also a possibility). The Poisson brackets don't affect this. – Steve Huntsman Feb 17 '10 at 14:03
<i>But shouldn't it be possible that having quantized a classical system in two different methods in general it should be possible that their "classical limits" are actually different classical systems?</i>- I'm extremely confused about what this could mean. You seem to be suggesting that you could quantize a classical system and get something else as classical limit, which makes me wonder what your definition of "quantization" is. – Ben Webster Feb 17 '10 at 14:42
I second Ben's comment regarding quantization. I'm not sure what you mean. For example, are you talking about canonical versus loop versus geometric (etc.) quantization or are you talking about quantizing something differently using the same method (e.g. two canonical systems)? – Ian Durham Feb 17 '10 at 22:13
I am probably meaning "quantizing something differently using the same method" . Let us keep to canonical quantization. We know that in the simplest example of an harmonic oscillator if the classical hamiltonian is written as product of two complex conjugate numbers and then quantized then the energy spectrum doesn't have a zero-point energy unlike the usual case. Here the two different ways of quantizing only ended up in giving two spectrums shifted from each other by a constant. But the classical limit did not change. – Anirbit Feb 18 '10 at 10:29

This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two-dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.

Any filtered associative unital algebra $A$ whose associated graded algebra $\mathrm{Gr}A$ is commutative may be thought of as a quantisation of $\mathrm{Gr}A$ with the Poisson structure induced by the commutator in $A$. So an example of the situation you are after could be an associative unital algebra admitting two different filtrations with commutative associated graded algebras. I'd be surprised if one could not cook up something like this. It's another question altogether whether there are any "natural" examples.