Geodesic
An integral formula of hyperbolic type for solutions of the Dirac equation on Minkowski space with initial conditions on a hyperboloid
Archivum mathematicum, Tome 46 (2010) no. 5, pp. 363-376 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The Dirac equation for spinor-valued fields $f$ on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet $H^+$ of the hyperboloid. In particular, we derive an integral formula expressing the value of $f$ in a chosen point $p$ as an integral over a compact cycle given by the intersection of the null cone with $H^+$ in the Minkowski space ${\mathbb{M}}$.
The Dirac equation for spinor-valued fields $f$ on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet $H^+$ of the hyperboloid. In particular, we derive an integral formula expressing the value of $f$ in a chosen point $p$ as an integral over a compact cycle given by the intersection of the null cone with $H^+$ in the Minkowski space ${\mathbb{M}}$.
Classification : 30E20, 30G35, 35Q41
Keywords: Clifford analysis; integral formula of hyperbolic type; hyperboloid; Minkowski space 
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     journal = {Archivum mathematicum},
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Sikora, Martin. An integral formula of hyperbolic type for solutions of the Dirac equation on Minkowski space with initial conditions on a hyperboloid. Archivum mathematicum, Tome 46 (2010) no. 5, pp. 363-376. http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a6/

[1] Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. Pitman Advanced Pub. Program, 1982. | MR | Zbl

[2] Bureš, J., Souček, V.: The Penrose transform on isotropic Grassmannians, in 75 Years of Radon transform. Conf. Proc. Lecture Notes Math. Phys., 1994. | MR

[3] Delanghe, R., Lávička, R., Souček, V.: On polynomial solutions of generalized Moisil-Theodoresco systems and Hodge-de Rham systems. arXiv:0908.0842, pp.11, 08 2009.

[4] Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and spinor-valued functions. A function theory for the Dirac operator. vol. 53, Math. Appl., 1992. | MR

[5] Dodson, M., Souček, V.: Leray residues applied to the solution of the Laplace and wave equations. Geometry seminars 1984 (Italian) (Bologna, 1984), Univ. Stud. Bologna, 1985, pp. 93–107. | MR

[6] Eelbode, D.: Clifford analysis on the hyperbolic unit ball. Ph.D. thesis, Ghent University, 2005. | MR

[7] Frenkel, I., Libine, M.: Quaternionic analysis, representation theory and physics. Adv. Math. 218 (2008), 1806–1877. | DOI | MR | Zbl

[8] Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley $\&$ Sons, Inc., 1997.

[9] Gürsey, F., Tze, C.-H.: On the role of division, Jordan and related algebras in particle physics. World Scientific Publishing Co., 1996. | MR

[10] Leutwiler, H.: Modified Clifford analysis. Complex Variables and Elliptic Equations 17 (3,4) (1992), 153–171, | DOI | MR | Zbl

[11] Libine, M.: Hyperbolic Cauchy integral formula for the split complex number. arXiv:0712.0375v1, pp.6, 12 2007.

[12] Souček, V.: Complex-quaternionic analysis applied to spin–1/2 massless fields. Complex Variables and Elliptic Equations 1 (4) (1983), 327–346, | DOI | MR

[13] Sudbery, A.: Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–225. | DOI | MR | Zbl