Contractions on ranks and quaternion types in Clifford algebras

D. S. Shirokov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 1, pp. 117-135.

Résumé

In this paper we consider expressions in real and complex Clifford algebras, which we call contractions or averaging. We consider contractions of arbitrary Clifford algebra element. Each contraction is a sum of several summands with different basis elements of Clifford algebra. We consider even and odd contractions, contractions on ranks and contractions on quaternion types. We present relation between these contractions and projection operations onto fixed subspaces of Clifford algebras - even and odd subspaces, subspaces of fixed ranks and subspaces of fixed quaternion types. Using method of contractions we present solutions of system of commutator equations in Clifford algebras. The cases of commutator and anticommutator are the most important. These results can be used in the study of different field theory equations, for example, Yang-Mills equations, primitive field equation and others.

Détail

Keywords: Clifford algebras, projection operations
Mots-clés : contractions, quaternion type.

Bibliographie

[1] Clifford W. K., “Application of Grassmann's Extensive Algebra”, American Journal of Mathematics, 1:4 (1878), 350–358 | DOI | MR | Zbl

[2] Hamilton W. R., “II. On quaternions, or on a new system of imaginaries in algebra”, Philosophical Magazine Series 3, 25:163 (1844), 489–495 | DOI

[3] Grassmann H., Die Lineale Ausdehnungslehre ein neuer Zweig der Mathematik, Verlag von Otto Wigand, Leipzig, 1844, xxxii+282 pp. Internet Archive Identifier: ; Grassmann H., Die Lineale Ausdehnungslehre ein neuer Zweig der Mathematik, Cambridge University Press, Cambridge, 2012, xxxii+282 pp. dielinealeausde00grasgoog | DOI

[4] Lipschitz R., Untersuchungen über die Summen von Quadraten, Max Cohen und Sohn, Bonn, 1886, 147 pp.

[5] Chevalley C., Collected works, v. 2, The algebraic theory of spinors and Clifford algebras, eds. Pierre Cartier and Catherine Chevalley, Springer, Berlin, 1997, xiv+ 214 pp. | MR | Zbl

[6] Dirac P. A. M., “The Quantum Theory of the Electron”, Proc. R. Soc. (A), 117:778 (1928), 610–624 ; Dirac P. A. M., “The Quantum Theory of the Electron”, Special Theory of Relativity, The Commonwealth and International Library: Selected Readings in Physics, 1970, 237–256 | DOI | Zbl | DOI

[7] Hestenes D., Sobczyk G., Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics, Reidel Publishing Company, 1984, 314 pp. | MR

[8] Marchuk N. G., Uravneniia teorii polia i algebry Klifforda [Equations of Field Theory and Clifford Algebras], Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2009, 304 pp. (In Russian)

[9] Dixon J. D., “Computing Irreducible Representations of Groups”, Math. Comp., 24:111 (1970), 707–712 | DOI | MR

[10] Babai L., Friedl K., “Approximate representation theory of finite groups”, Foundations of Computer Science, 1991, 733–742 | DOI | MR

[11] Shirokov D. S., Method of averaging in Clifford algebras, 2015, 15 pp., arXiv: [math-ph] 1412.0246 | MR

[12] Marchuk N. G., Shirokov D. S., New class of gauge invariant solutions of Yang–Mills equations, 2014, 35 pp., arXiv: [math-ph] 1406.6665

[13] Shirokov D. S., Method of generalized contractions and Pauli's theorem in Clifford algebras, 2014, 14 pp., arXiv: [math-ph] 1409.8163

[14] Pauli W., “Contributions mathématiques à la théorie des matrices de Dirac”, Annales de l'institut Henri Poincaré, 6:2 (1936), 109–136 | MR

[15] Shirokov D. S., “Extension of Pauli's theorem to Clifford algebras”, Dokl. Math., 84:2 (2011), 699–701 | DOI | Zbl

[16] Shirokov D. S., “Pauli theorem in the description of n -dimensional spinors in the Clifford algebra formalism”, Theoret. and Math. Phys., 175:1 (2013), 454–474 | DOI | DOI | MR | Zbl

[17] Shirokov D. S., “The use of the generalized Pauli's theorem for odd elements of Clifford algebra to analyze relations between spin and orthogonal groups of arbitrary dimensions”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2013, no. 1(30), 279–287 (In Russian) | DOI

[18] Marchuk N. G., Shirokov D. S., “Unitary spaces on Clifford algebras”, Adv. Appl. Clifford Algebras, 18:2 (2008), 237–254, arXiv: [math-ph] 0705.1641 | DOI | MR | Zbl

[19] Lounesto P., Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series, 239, Cambridge University Press, Cambridge, 1997, ix+306 pp. ; Lounesto P., Clifford Algebras and Spinors (second edition), London Mathematical Society Lecture Note Series, 286, Cambridge University Press, Cambridge, 2001, ix+338 pp. | MR | Zbl | DOI | MR | Zbl

[20] Shirokov D. S., “Classification of elements of clifford algebras according to quaternionic types”, Doklady Mathematics, 80:1 (2009), 610–612 | DOI | MR | Zbl

[21] Shirokov D. S., “Quaternion typification of Clifford algebra elements”, Adv. Appl. Clifford Algebras, 22:1 (2012), 243-256 | DOI | MR | Zbl

[22] Shirokov D. S., “Development of the method of quaternion typification of clifford algebra elements”, Adv. Appl. Clifford Algebras, 22:2 (2012), 483–497, arXiv: [math-ph] 0903.3494 | DOI | MR | Zbl