Symmetry, Geometry, and Quantization with Hypercomplex Numbers

These notes describe some links between the group ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$, 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.