Erlangen Programme at Large 3.2: Ladder Operators in Hypercomplex Mechanics

We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator (the elliptic case) we similarly treat the repulsive oscillator (hyperbolic case) and the free particle (the parabolic case). The respective hypercomplex numbers turn to be handy on this occasion, this provides a further illustration to Similarity and Correspondence Principle. 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, Fock--Segal--Bargmann representation, Schr\"odinger representation, dynamics equation, harmonic and unharmonic oscillator, contextual probability, symplectic group, metaplectic representation, Shale--Weil representation