Abstract
We present a new class of covariantly constant solutions of the Yang–Mills equations. These solutions correspond to the solution of the field equation for the spin connection of the general form.
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References
Actor, A.: Classical solutions of SU(2) Yang–Mills theories. Rev. Mod. Phys. 51, 461–525 (1979)
Belavin, A., Polyakov, A., Schwartz, A., Tyupkin, Y.: Pseudoparticle solutions of the Yang–Mills equations. Phys. Lett. B 59, 85 (1975)
Benn, I., Tucker, R.: An Introduction to Spinors and Geometry with Applications in Physics. Adam Hilger, Bristol (1987)
de Alfaro, V., Fubini, S., Furlan, G.: A new classical solution of the Yang–Mills field equations. Phys. Lett. B 65, 163 (1976)
Graf, W.: Differential forms as spinors. Ann. Inst. Henri Poincare 29(1), 85–109 (1978)
Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (2001)
Marchuk, N.G.: On a field equation generating a new class of particular solutions to the Yang–Mills equations. Proc. Steklov Inst Math. 285, 197–210 (2014)
Marchuk, N.: Field theory equations. Amazon, ISBN 9781479328079 (2012)
Marchuk, N.G., Shirokov, D.S.: General solutions of one class of field equations. Rep. Math. Phys. 78(3), 305–326 (2016)
Marchuk, N.G., Shirokov, D.S.: Unitary spaces on Clifford algebras. Adv. Appl. Clifford Algebr. 18(2), 237–254 (2008)
Marchuk, N.G., Shirokov, D.S.: Constant solutions of Yang–Mills equations and generalized proca equations. J. Geom. Symmetry Phys. 42, 53–72 (2016)
Salingaros, N.A., Wene, G.P.: The Clifford algebra of differential forms. Acta Appl. Math. 4, 271–292 (1985)
Schimming, R.: On constant solutions of the Yang–Mills equations. Arch. Math. 24(2), 65–73 (1988)
Schimming, R., Mundt, E.: Constant potential solutions of the Yang–Mills equation. J. Math. Phys. 33, 4250 (1992)
Shirokov, D.S.: Extension of Pauli’s theorem to Clifford algebras. Dokl. Math. 84(2), 699–701 (2011)
Shirokov, D.S.: Method of averaging in Clifford algebras. Adv. Appl. Clifford Algebras 27(1), 149–163 (2017). arXiv:1412.0246
Shirokov, D.S.: Symplectic, orthogonal and linear Lie groups in Clifford algebra. Adv. Appl. Clifford Algebras 25(3), 707–718 (2015)
Shirokov, D.S.: On some Lie groups containing spin group in Clifford algebra. J. Geom. Symmetry Phys. 42, 73–94 (2016)
Shirokov, D.S.: Classification of Lie algebras of specific type in complexified Clifford algebras. In: Linear and Multilinear Algebra (2017). arXiv:1704.03713 (Published online)
Wu, T., Yang, C.: Some solutions of the classical isotopic gauge field equations. In: Mark, H., Fernbach, S. (eds.) Properties of Matter Under Unusual Conditions. Wiley Interscience, New York (1968)
Zhdanov, R., Lahno, V.: Symmetry and exact solutions of the Maxwell and SU(2) Yang–Mills equations. Adv. Chem. Phys. Mod. Nonlinear Opt. 119(part II), 269–352 (2001)
Acknowledgements
The author is grateful to N. G. Marchuk for fruitful discussions and participants of ICCA 11 Conference for useful comments. The author is grateful to anonymous reviewer for careful reading of the manuscript and helpful comments on how to improve the presentation.
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The reported study was funded by RFBR according to the research project No. 16-31-00347 mol_a.
This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.
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Shirokov, D. Covariantly Constant Solutions of the Yang–Mills Equations. Adv. Appl. Clifford Algebras 28, 53 (2018). https://doi.org/10.1007/s00006-018-0868-5
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DOI: https://doi.org/10.1007/s00006-018-0868-5