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

  1. It can be constructed via canonical quantization starting from some classical theory with a phase space $(q^a, p_a)$ and a Hamiltonian $H(q^a, p_a)$.
  2. 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.

Qmechanic
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Prof. Legolasov
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  • This post (v2) seems like a list question. – Qmechanic Oct 19 '17 at 08:16
  • More on classical limit. – Qmechanic Oct 19 '17 at 08:26
  • @Qmechanic in which way is this a list question? I only used the list in formatting to make the question clearer. – Prof. Legolasov Oct 19 '17 at 08:30
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    I think @Qmechanic means that your question seems to ask for an open-ended list of examples rather than a unique identifiable correct answer. However, I think that your question is less seeking for a list of examples and more asks if there exists a single example at all, which is not too broad. – ACuriousMind Oct 19 '17 at 14:32
  • By construction, the classical limit of a quantum theory corresponds to the limit $h\rightarrow 0$, and process of quantisation consists in defining a non-commutative algebra proportional to $h$. Clearly there are many many quantum model giving back to the same classical one. Certainly your question can be answered rigorously in the context of deformation quantisation, see e.g. https://physics.stackexchange.com/a/56196/16689 and references therein. The example you are looking for must violate the deformation quantisation procedure. – FraSchelle Oct 22 '17 at 13:56
  • Hi @FraSchelle, I couldn’t agree more. Anyway, I’m still waiting for somebody to give this example.. – Prof. Legolasov Oct 22 '17 at 16:37
  • I were you, I'd look for a demonstration of its non-existence... :-) It should be in the seminal papers about quantisation theory. In deformation theory, you start from a classical theory and you deform it by the generation of a non-commutative star-product. As I said before, by construction, the star-product goes back to the classical one in the classical limit. So for the deformation quantisation procedure to be consistent I would say that an exemple as you are looking for must be impossible to construct. – FraSchelle Oct 23 '17 at 10:15
  • It should exist a rigorous demonstration in the context of deformation quantisation (or other way like Berezin or Sourriaux / Kostant methods) ruling out your hypothetical example. But I'm not aware of such a demonstration, sorry. – FraSchelle Oct 23 '17 at 10:16

1 Answers1

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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.

David Bar Moshe
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