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Using Painleve Test on the Kinetic System for Yang-Mills Theory

Ahmed Kamil
Department of Theoretical Physics, Faculty of Physics and Mathematics, Moscow, 117198 ,Russian
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We study the Yang-mills theory as a kintic system, and used C. Becchi, A. Rouet and R. Stora; transformation to Renormalization of gauge theories, test the random of the system by use the painleve test


Yang-Mills theory, Painleve Test.


Yang-Mills theory is a non-Abelian gauge theory, and we used a lot in QCD calculation, the yang- mills theory is the most important in physics in the last fifty years. Yang-Mills theory was first discovered in the 1950, at this time, quantum electrodynamics was known to describe electromagnetism [1]. Yang–Mills theory seeks to describe the behavior of elementary particles. In 1954, Yang and Mills published a paper on the isotopic SU(2) invariance of the proton-neutron system [2]. Yang-Mills theory plays a central role in explaining fundamental interactions ,because both the strong and weak interactions are described by Yang-Mills theories [3].Yang-Mills theory allow one to describe both the Maxwell electromagnetic interactions and the Fermi weak interactions and to obtain the known value of the (Z0) boson (weak boson) mass. Yang-Mills gauge theory with gauge group SU(3)×SU(2)×U(1). Here the first factor is the gauge group of QCD while (SU(2)×U(1)) gauge field is that transmitting what is called the electroweak force [4].




Using the lagrangian of Yang-Mills theory [5]. introduceing C. Becchi , A. Rouet , R. Stora and I.V. Tyutin for transformation to Renormalization of gauge theories [6]. test the random of the system by use the painleve test [7]. Finally We found that yang-mills theory is non-integrable according to Painleve test.


[1] Davies, C. 2002, Lattice QCD, London: Institute of Physics.

[2] C. N. Yang and R. Mills, Phys. Rev. 96, 191 (1954).

[3] Addison-Wesley, the role of gauge theory in describing fundamental interactions is described in C.Quigg, Gauge theories of the strong, weak and Electro-magnetic interactions ( New York, 1993).

[4] Andreas Aste, Michael Dütsch, Günter Scharf, Perturbative gauge invariance: electroweak theory II, Annalen Phys.8:389- 404,1999 (arXiv:hep-th/9702053).

[5] S.G. Matinyan, G.K. Savvidy & N.G. Ter-Arutunian Savvidy ; Stochasticity Of Classical Yang-mills Mechanics And Its Elimination By Higgs Mechanism, Published in JETP Lett. 34 (1981) 590-593.

[6] C. Becchi, A. Rouet & R. Stora ; Renormalization of gauge theories, Ann. Phys. 98, 2 (1976) pp. 287–321.

[7] F.Cariello and M.Tabor, painleve expansions for nonintegrable evoulution equations. Physica D, 39:77-94,1989.

[8] L.D. Faddeev, V.N. Popov ; Feynman Diagrams for the Yang-Mills Field., Published in Phys.Lett.B25:29-30, (1967) doi:10.1016/0370- 2693(67) 90067-6. English Version in: 50 years of Yang-Mills Theory by G. ‘t Hooft; World Scientific Books (2005).