Example of a quantum-mechanical theory with nontrivial classical limit I am looking for a toy model example of a well defined quantum-mechanical theory with the following properties:


*

*It can be constructed via canonical quantization starting from some classical theory with a phase space $(q^a, p_a)$ and a Hamiltonian $H(q^a, p_a)$.

*Its classical limit (dynamical evolution of coherent states in the $\hbar \rightarrow 0$ limit) is a classical theory different (i.e. experimentally distinguishable) from the original theory. (Note that choosing a different ordering for $\hat{H}$ is not a good example: theories with different orderings reduce to the same thing in the $\hbar \rightarrow 0$ limit.)


If you are aware of multiple examples, please provide the least complicated one.
 A: In quantization of a symplectic manifold (phase space) $M$, there are classical limit theorems of convergence of the operator algebra of the quantum system to the Poisson algebra of functions on the symplectic manifold, that we started from, in the limit $\hbar \rightarrow 0$:
$$ \lim_{\hbar \rightarrow 0}||T_f^{\frac{1}{\hbar}}|| = |f|_{\infty}$$ 
$$ \lim_{\hbar \rightarrow 0}||[T_f^{ \frac{1}{\hbar}}, T_g^{ \frac{1}{\hbar}}]- i T_{\left \{f, g \right\}}^{ \frac{1}{\hbar}} ||= 0$$ 
Where $ T_f^{\frac{1}{\hbar}} $ is the Toeplitz operator representing the observable $\hat{f}_{\frac{1}{\hbar} }$ in the quantum Hilbert space (acting as a convolution kernel on the wave functions):
$$(\hat{f}_{\frac{1}{\hbar} }\circ \psi)(x) = \int_M d\mu_{L}(M) h^{\frac{1}{\hbar}} (x,y) T_f^{\frac{1}{\hbar}}(x,y) \psi(y) $$
Where: $ d\mu_{L}$ is the Liouville measure on $M$ and $h$ is a metric on the fibers of the quantization line bundle.
The Toeplitz operators may be expressed, given a coherent state basis, as: $|x\rangle_{\frac{1}{\hbar}}$,
$$ T_f^{\frac{1}{\hbar}}(x,y)  = _{\frac{1}{\hbar}}\langle y| \hat{f}_{\frac{1}{\hbar} } | |x\rangle_{\frac{1}{\hbar}}$$
The value of $\hbar$ is controlled by means of the choice of the metric on the fibers or equivalently, the coherent state basis.
Bordemann, Meinrenken and Schlichenmaier  proved the above theorem in the case of compact Kähler manifolds. Their proof is valid for Berezin-Toeplitz quantization as well as for geometric quantization, whose Toeplitz operators are related to the Berezin Toeplitz operators by the Tuynman formula:
$$Q_f ^{\frac{1}{\hbar}} = T_{f-\frac{\hbar}{2}\Delta}^{\frac{1}{\hbar}} $$ 
This theorem was generalized by Ma and Marinescu for Berezin-Toeplitz quantization of non-compact Kähler manifolds and orbifolds and general symplectic manifolds.
Charles and Polterovich obtained sharper estimates for the semiclassical limits in the case of compact manifolds.
The above story is valid, when we quantize a given manifold at the start. But sometimes, we know only the operator algebra and the Hamiltonian, such as in the case of spin models. In this case, (please see Gnutzmann, Haake and Kuś), there are certain singular cases when the operator algebra can be represented isomorphically by means of Toeplitz operators on two (or more) distinct phase spaces, which emerge as classical limits. In this case, the classical theories are completely different, when the Poisson structure is nondegenerate, the classical limit is integrable, when it is degenerate, the classical limit is chaotic.
