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

Abstract

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

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References in corpus (8)

• Erlangen Programme at Large: An Overview
• A Quantum-Classical Brackets from p-Mechanics
• Considerations on the hyperbolic complex Klein-Gordon equation
• p-Mechanics as a Physical Theory. An Introduction
• PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
• Two-Dimensional Conformal Models of Space-Time and Their Compactification
• Comment on `Do we have a consistent non-adiabatic quantum-classical mechanics?'
• Computation and Dynamics: Classical and Quantum

Cited by in corpus (8)

• Erlangen Programme at Large: An Overview
• Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group
• Higher spin quaternion waves in the Klein-Gordon theory
• Uncertainty and Analyticity
• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform
• Metamorphism as a covariant transform for the SSR group
• Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle