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arXiv · 1611.05650

Symmetry, Geometry, and Quantization with Hypercomplex Numbers

Vladimir V. Kisil
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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 Calderón--Vaillancourt-type norm estimation for relative convolutions.

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Vladimir V. Kisil. 2016-11-17. Symmetry, Geometry, and Quantization with Hypercomplex Numbers. https://doi.org/10.7546/giq-18-2017-11-76

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