The God has created a man in order that he creates that the God fails to do

Sunday, 4 December 2011

Why a classical system admits different non-equivalent quantization

A classical mechanical system admits equivalent description in different variables whose transformation law need not be linear.

In particular, a Hamiltonian classical system is equivalently described by variables related by arbitrary canonical transformations. 

If we have a completely integrable Hamiltonian system, its descriptions in original variables and the action-angle ones also are equivalent, though the transformation law between these variables is neither linear nor canonical in general.

In contrast with classical variables, quantum operators are linear operators in Hilbert spaces of quantum states and, therefore, they admit only linear transformations.  For instance, let a classical system be described in an equivalent way with respect to different variables (q,p) and (q’,p’) which possess some non-linear transformation law q’=Q(q,p), p’=P(q,p). Let (q, p) and (q’,p’) be quantization of these variables by operators in Hilbert spaces E and E’, respectively. Then the quantum systems characterized by quantum operators (q, p) and (q’,p’)  fail to be equivalent because there is no Hilbert space morphism E->E’ which transform (q, p)->(q’,p’).

In particular, there is no quantum partner of classical canonical transformations ubless they are linear.

Quantization of a completely integrable Hamiltonian system with respect to original variables and the action-angle ones is not equivalent and leads to different energy spectrums. For instance, this is the case of a Kepler system, whose familiar Schrodinger quantization provides the well-known energy spectrum of a hydrogen atom, but its quantization with respect to action-angle variables leads to a different energy spectrum.

Thus, a classical system can admit non-equivalent quantization. A problem is that nobody generally knows what its quantization is true.


G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005)

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables, Phys. Lett. A 301 (2002) 53-57; arXiv: quant-ph/0112083

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