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

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