Erlangen Programme at Large 3.2 Ladder Operators in Hypercomplex Mechanics

Abstract

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 out to be handy on this occasion. This provides a further illustration to the 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 representatio