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arXiv:cond-mat/0503684 · Reports on Mathematical Physics

Canonical quantization of classical systems with generalized entropies

Antonio Maria Scarfone

Published Apr 1, 2005• 2 citations
arXivHTMLPDF
DOI
10.1016/s0034-4877(05)00012-1

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Dirac's postulate of canonical quantization, [(The Symbol is abbreviated)]=-iħδ(subscript ij) for conjugate canonical variables, has been the most concise and general prescription on how to quantize a classical system. Since classical systems described by variables connected with canonical transformations are equivalent, [(The Symbol is abbreviated)]=-iħδ(subscript ij) must remain invariant under classical canonical transformations This invariance has not been proved except for the limited class of cascaded infinitesimal transformations. In this paper it is shown that if ((The Symbol is abbreviated)) are related to ((The Symbol is abbreviated)) by a classical canonical transformation, then [(The Symbol is abbreviated)]=-iħδ(subscript ij) In other words, the canonical quantization prescription is invariant for variables connected with classical canonical transformations.
It is known that the equations of motion of a classical system do not in general determine its canonical description uniquely. Thus the corresponding quantum system obtained by canonical quantization is also ambiguous. We use this freedom to quantize a free nonrelativistic particle so that the corresponding quantum particle has half-integral or integral helicity. Methods are developed for finding inequivalent canonical descriptions of a given classical system. It is emphasized that classical symmetries can be broken at the quantum level by a suitable choice of the canonical formalism prior to quantization. It is suggested that this may provide a new mechanism for breaking internal symmetries at the quantum level.