I will recall the basics of geometric quantization and what is usually meant by quantizing a mechanical (a hamiltonian) system.
Will then consider two one-parameter families of Kahler polarizations on the plane intersecting at one point and degenerating to two important real polarizations: the vertical or Schrodinger one and the (singular) harmonic-oscilator-energy one.
Eventhough the quantizations associated with the above real degenerations are equivalent, on the coherent-space-transform driven quantum path from one to the other [more specifically all along the half way from the crossing point to the energy representation] I will argue that the Kahler quantizations are inequivalent to the two real ones above.
Will then comment briefly on the relevance of this analysis to loop quantum gravity.
Based on joint work with William Kirwin and João P. Nunes.