Abstract
We investigate the light-cone SU(n) Yang-Mills mechanics formulated as the leading order of the long-wavelength approximation to the light-front SU(n) Yang-Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups SU(2) and SU(3), we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance: in addition to the local SU(n) gauge rotations, there is a new local two-parameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone SU(2) Yang-Mills mechanics, in contrast to the well-known instant-time SU(2) Yang-Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Gröbner bases in the theory of polynomial ideals.
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References
P. A. M. Dirac, Rev. Modern Phys., 21, 392–399 (1949).
P. A. M. Dirac, On the Development of Quantum Field Theory [in Russian] (Library of Theoretical Physics, Vol. 7), Nauka, Moscow (1990).
K. Sundermeyer, Constrained Dynamics: With Applications to Yang-Mills Theory, General Relativity, Classical Spin, Dual String Model, Springer, Berlin (1982).
G. Baseian, S. Matinyan, and G. Savvidy, Sov. Phys. JETP, 29, 587 (1979).
M. Lüscher, Nucl. Phys. B, 219, 233–261 (1983).
B. Simon, Ann. Phys., 146, 209–220 (1983).
B. V. Chirikov and D. L. Shepelyansky, Sov. Phys. JETP, 34, 163 (1981); E. S. Nikolaevsky and L. N. Shchur, Sov. Phys. JETP, 36, 218 (1982).
B. V. Medvedev, Theor. Math. Phys., 60, 782–797 (1984).
I. Ya. Aref’eva, A. S. Koshelev, and P. B. Medvedev, Modern Phys. Lett. A, 13, 2481–2493 (1998).
B. Dahmen and B. Raabe, Nucl. Phys. B, 384, 352–380 (1992).
S. A. Gogilidze, A. M. Khvedelidze, D. M. Mladenov, and H. P. Pavel, Phys. Rev. D, 57, 7488–7500 (1998).
A. M. Khvedelidze, H. P. Pavel, and G. Röpke, Phys. Rev. D, 61, 025017 (2000).
A. M. Khvedelidze and H. P. Pavel, Phys. Lett. A, 267, 96–100 (2000).
A. M. Khvedelidze and D. M. Mladenov, Phys. Rev. D, 62, 125016 (2000).
H. P. Pavel, Phys. Lett. B, 648, 97–106 (2007); arXiv:hep-th/9701283v1 (2007).
T. Becker and V. Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer, New York (1993).
D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (2nd ed.), Springer, New York (1996).
B. Buchberger and F. Winkler, eds., Gröbner Bases and Applications, Cambridge Univ. Press, Cambridge (1998).
V. P Gerdt and S. A. Gogilidze, “Constrained Hamiltonian systems and Gröbner bases,” in: Computer Algebra in Scientific Computing (Proc. 2nd Workshop CASC’99, Munich, Germany, May 31–June 4, 1999, V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, eds.), Springer, Berlin (1999), pp. 138–146; arXiv:math/9909113v2 [math.NA] (1999).
V. Gerdt, A. Khvedelidze, and D. Mladenov, “Analysis of constraints in light-cone version of SU(2) Yang-Mills mechanics,” in: Computer Algebra and Its Applications to Physics CAAP-2001 (Proc. Intl. Workshop, Dubna, Russia, June 28–30, 2001, V. P. Gerdt, ed.), Joint Inst. Nucl. Res., Dubna (2002), pp. 83–92; arXiv:hep-th/9209107v1 (2002).
V. Gerdt, A. Khvedelidze, and D. Mladenov, “Light-cone SU(2) Yang-Mills theory and conformal mechanics,” arXiv:hep-th/9210022v4 (2002).
V. Gerdt, A. Khvedelidze, and D. Mladenov, “On application of involutivity analysis of differential equations to constrained dynamical systems,” in: Symmetries and Integrable Systems (Selected Papers of the Seminar, 2000–2005, Vol. 1, A. N. Sissakian, ed.), Joint Inst. Nucl. Res., Dubna (2006), pp. 132–150; arXiv:hep-th/9311174v1 (2003).
V. Gerdt, A. Khvedelidze, and Yu. Palii, “Towards an algorithmisation of the Dirac constraint formalism,” in: Global Integrabilty of Field Theories (Proc. GIFT 2006, J. Calmet, W. M. Seiler, and R. W. Tucker, eds.), Cockroft Institute, Daresbury, UK (2006), p. 135–154; arXiv:math-ph/9611021v1 (2006).
T. Heinzl, Light-Cone Quantization: Foundations and Applications (Lect. Notes Phys., Vol. 572), Springer, Berlin (2001).
M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, Princeton, N. J. (1992).
S. A. Gogilidze, V. V. Sanadze, Yu. S. Surovtsev, and F. G. Tkebuchava, J. Phys. A, 27, 6509–6523 (1994).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 62–73, April, 2008.
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Gerdt, V.P., Palii, Y.G. & Khvedelidze, A.M. Light-cone Yang-Mills mechanics: SU(2) vs. SU(3). Theor Math Phys 155, 557–566 (2008). https://doi.org/10.1007/s11232-008-0046-3
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DOI: https://doi.org/10.1007/s11232-008-0046-3