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Quaternionic analysis, representation theory and physics. (English) Zbl 1167.30030
After my first look upon the title of this article I thought it to be yet another re-invention of quaternionic analysis. But I soon changed my mind after starting to read the paper. It is a rich and well written presentation of quaternionic analysis – real as well as complex. The viewpoint is unusual as the paper is written using mainly (complex) representations of the quaternions. The theory is really used for applications in physics, what is also not usual.
The content may be best described by citing parts of the abstract: “Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures.”

MSC:
30G35 Functions of hypercomplex variables and generalized variables
22E30 Analysis on real and complex Lie groups
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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