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.”


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
Full Text: DOI arXiv


[1] Arazy, J.; Upmeier, H., Boundary measures for symmetric domains and integral formulas for the discrete Wallach points, Integral Equations Operator Theory, 47, 4, 375-434 (2003) · Zbl 1050.32016
[2] Bander, M.; Itzykson, C., Group theory and the hydrogen atom (I), (II), Rev. Modern Phys., 38, 330-345 (1966), 346-358
[3] Bogoliubov, N. N.; Shirkov, D. V., Introduction to the Theory of Quantized Fields (1959), Interscience Publishers: Interscience Publishers New York, translated from the Russian by G.M. Volkoff · Zbl 0088.21701
[4] Colombo, F.; Sabadini, I.; Sommen, F.; Struppa, D. C., Analysis of Dirac Systems and Computational Algebra, Progr. Math. Phys., vol. 39 (2004), Birkhäuser: Birkhäuser Boston · Zbl 1064.30049
[5] Davydychev, A. I.; Delbourgo, R., A geometrical angle on Feynman integrals, J. Math. Phys., 39, 9, 4299-4334 (1998) · Zbl 0986.81082
[6] Delanghe, R.; Sommen, F.; Souček, V., Clifford Algebra and Spinor-valued Functions (1992), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0747.53001
[7] Drummond, J. M.; Henn, J.; Smirnov, V. A.; Sokatchev, E., Magic identities for conformal four-point integrals, J. High Energy Phys., 0701, 064 (2007), 15 pp
[8] Faraut, J.; Korányi, A., Analysis on Symmetric Cones, Oxford Math. Monogr. (1994), Oxford University Press: Oxford University Press New York · Zbl 0841.43002
[9] I. Frenkel, M. Libine, Split quaternionic analysis and the separation of the series for \(\mathit{SL}(2, \mathbb{R})\mathit{SL}(2, \mathbb{C}) / \mathit{SL}(2, \mathbb{R})\); I. Frenkel, M. Libine, Split quaternionic analysis and the separation of the series for \(\mathit{SL}(2, \mathbb{R})\mathit{SL}(2, \mathbb{C}) / \mathit{SL}(2, \mathbb{R})\) · Zbl 1264.30037
[10] Fueter, R., Die Funktionentheorie der Differentialgleichungen \(\Delta u = 0\) und \(\Delta \Delta u = 0\) mit vier reellen Variablen, Comment. Math. Helv., 7, 1, 307-330 (1934)
[11] Fueter, R., Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv., 8, 1, 371-378 (1935)
[12] Gelfand, I. M.; Graev, M. I.; Vilenkin, N. Ya., Generalized Functions. Vol. 5: Integral Geometry and Representation Theory (1966), Academic Press: Academic Press New York-London, translated from the Russian by E. Saletan · Zbl 0144.17202
[13] Gürsey, F.; Tze, C.-H., On the role of division, (Jordan and Related Algebras in Particle Physics (1996), World Scientific Publishing Co.) · Zbl 0956.81502
[14] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces (2001), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0993.53002
[15] Jakobsen, H. P.; Vergne, M., Wave and Dirac operators, and representations of the conformal group, J. Funct. Anal., 24, 1, 52-106 (1977) · Zbl 0361.22012
[16] Jakobsen, H. P.; Vergne, M., Restrictions and expansions of holomorphic representations, J. Funct. Anal., 34, 1, 29-53 (1979) · Zbl 0433.22011
[17] Knapp, A. W.; Speh, B., Irreducible unitary representations of \(SU(2, 2)\), J. Funct. Anal., 45, 1, 41-73 (1982) · Zbl 0543.22011
[18] Kobayashi, T.; Ørsted, B., Analysis on the minimal representation of \(O(p, q)\) I, II, III, Adv. Math., 180, 2, 486-512 (2003), 513-550, 551-595 · Zbl 1046.22004
[19] Segal, I. E., Positive-energy particle models with mass splitting, Proc. Nat. Acad. Sci. USA, 57, 194-197 (1967)
[20] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser., vol. 32 (1971), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0232.42007
[21] Strichartz, R. S., Harmonic analysis on hyperboloids, J. Funct. Anal., 12, 341-383 (1973) · Zbl 0253.43013
[22] Sudbery, A., Quaternionic analysis, Math. Proc. Cambridge Philos. Soc., 85, 199-225 (1979) · Zbl 0399.30038
[23] Ussyukina, N. I.; Davydychev, A. I., Exact results for three- and four-point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B, 305, 1-2, 136-143 (1993)
[24] Vergne, M.; Rossi, H., Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math., 136, 1-2, 1-59 (1976) · Zbl 0356.32020
[25] Vilenkin, N. Ja., Special Functions and the Theory of Group Representations, Transl. Math. Monogr., vol. 22 (1968), American Mathematical Society: American Mathematical Society Providence, RI, translated from the Russian by V.N. Singh · Zbl 0172.18404
[26] Wagner, P., A volume formula for asymptotic hyperbolic tetrahedra with an application to quantum field theory, Indag. Math. (N.S.), 7, 4, 527-547 (1996) · Zbl 0876.51003
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