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Abstract: In our joint papers FL1-FL2 we revive quaternionic analysis and show deeprelations between quaternionic analysis, representation theory andfour-dimensional physics. As a guiding principle we use representation theoryof various real forms of the conformal group. We demonstrate that therequirement of unitarity of representations naturally leads us to theextensions of the Cauchy-Fueter and Poisson formulas to the Minkowski space,which can be viewed as another real form of quaternions. However, the Minkowskispace formulation also brings some technical difficulties related to the factthat the singularities of the kernels in these integral formulas are nowconcentrated on the light cone instead of just a single point in the initialquaternionic picture. But the same phenomenon occurs when one passes from thecomplex numbers to the split complex numbers or hyperbolic algebra. So, as awarm-up example we proved an analogue of the Cauchy integral formula for thesplit complex numbers. On the other hand, there seems to be sufficient interestin such formula among physicists. For example, see KS and the referencestherein.



Autor: Matvei Libine

Fuente: https://arxiv.org/







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