Erlangen Programme at Large 3.2: Ladder Operators in Hypercomplex Mechanics

TL;DR

This paper extends the construction of creation and annihilation operators in quantum mechanics to hypercomplex settings, covering elliptic, hyperbolic, and parabolic cases using representation theory and hypercomplex numbers.

Contribution

It introduces a unified framework for ladder operators in hypercomplex quantum mechanics, including hyperbolic and parabolic cases, using representation theory and hypercomplex numbers.

Findings

Unified treatment of harmonic, hyperbolic, and free particle oscillators

Use of hypercomplex numbers for operator construction

Illustration of the Similarity and Correspondence Principle

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 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