Classical/Quantum=Commutative/Noncommutative?

Details Classical/Quantum=Commutative/Noncommutative? Classical/Quantum=Commutative/Noncommutative?

2012-04-09

arxiv.org/abs/1204.1858v2

Physics

Quantum Physics

8 pages, AMS-LaTeX, no figures. v2: small improvements and additional references

Scientific paper

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.

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