Classical/Quantum=Commutative/Noncommutative?
Abstract
In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Dirac's proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit, which squares to zero. Thus, the crucial r\^ole in quantum theory shall be attributed to the usage of complex numbers. The paper provides English and Russian versions.
Cite
@article{arxiv.1204.1858,
title = {Classical/Quantum=Commutative/Noncommutative?},
author = {Vladimir V. Kisil},
journal= {arXiv preprint arXiv:1204.1858},
year = {2012}
}
Comments
8 pages, AMS-LaTeX, no figures. v2: small improvements and additional references; v3: small improvements, additional references, Russian translation is added