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VOL. 18 | 2017 Symmetry, Geometry and Quantization with Hypercomplex Numbers


These notes describe some links between the group $SL_2(\mathbb{R})$, 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 also discussed. Finally, we prove a Calderón-Vaillancourt-type norm estimation for relative convolutions.


Published: 1 January 2017
First available in Project Euclid: 14 January 2017

zbMATH: 1394.81143
MathSciNet: MR3616912

Digital Object Identifier: 10.7546/giq-18-2017-11-76

Rights: Copyright © 2017 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences


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