Symmetry, Geometry, and Quantization with Hypercomplex Numbers

TL;DR

This paper explores the connections between hypercomplex numbers, group theory, and mechanics, revealing how classical mechanics emerges from quantum mechanics through the use of dual numbers and induced representations.

Contribution

It introduces a novel framework linking hypercomplex numbers with group representations to unify quantum and classical mechanics.

Findings

Classical mechanics can be derived from quantum mechanics using dual numbers.

A Calderón–Vaillancourt-type estimate for relative convolutions is established.

Relations between group actions, hypercomplex numbers, and physical theories are clarified.

Abstract

These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual-/double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are discussed. Finally, we prove a Calder\'on--Vaillancourt-type norm estimation for relative convolutions.