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Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics

In the spirit of geometric quantisation we consider representations of the Heisenberg(--Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit h->0. Keywords: Heisenberg group, Kirillov's method of orbits, geometric quantisation, quantum mechanics, classical mechanics, Planck constant, dual numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics, interference, Segal--Bargmann representation, Schroedinger representation, dynamics equation, harmonic and unharmonic oscillator, contextual probability