Symmetry, Geometry and Quantization with Hypercomplex Numbers

Abstract

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.