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Gauge symmetry and W-algebra in higher derivative systems. (English) Zbl 1298.81153
Summary: The problem of gauge symmetry in higher derivative Lagrangian systems is discussed from a Hamiltonian point of view. The number of independent gauge parameters is shown to be in general \(less\) than the number of independent primary first classc onstraints, thereby distinguishing it from conventional first order systems. Different models have been considered as illustrative examples. In particular we show a direct connection between the gauge symmetry and the W-algebra for the rigid relativistic particle.

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70H45 Constrained dynamics, Dirac’s theory of constraints
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References:
[1] Dirac, PAM, Generalized Hamiltonian dynamics, Can. J. Math., 2, 129, (1950)
[2] P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva University, U.S.A. (1964).
[3] E.C.G. Sudarshan and N. Mukunda, Classical dynamics-a modern perspective, Wiely-Interscience, U.K. (1974).
[4] A. Hanson, T. Regge and C. Tietelboim, Constrained hamiltonian system, Accademia Nazionale dei Lincei, Roma, Italy (1976).
[5] K. Sundermeyer, Constrained dynamics, Springer, U.S.A. (1982).
[6] H.J. Rothe and K.D. Rothe, Classical and quantum dynamics of constrained hamiltonian systems, World Scientific, Singapore (2010).
[7] Bergmann, PG; Komar, A., The coordinate group symmetries of general relativity, Int. J. Theor. Phys., 5, 15, (1972)
[8] Teitelboim, C., How commutators of constraints reflect the space-time structure, Ann. Phys., 79, 542, (1973)
[9] Castellani, L., Symmetries in constrained Hamiltonian system, Ann. Phys., 143, 357, (1982)
[10] Costa, MEV; Girotti, HO; Simoes, TJM, Dynamics of gauge systems and dirac’s conjecture, Phys. Rev., D 32, 405, (1985)
[11] Henneaux, M.; Teitelboim, C.; Zanelli, J., Gauge invariance and degree of freedom count, Nucl. Phys., B 332, 169, (1990)
[12] J.M. Pons, D.C. Salisbury and L.C. Shepley, Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories, gr-qc/9612037 [SPIRES].
[13] Banerjee, R.; Rothe, HJ; Rothe, KD, Hamiltonian approach to Lagrangian gauge symmetries, Phys. Lett., B 463, 248, (1999)
[14] Banerjee, R.; Rothe, HJ; Rothe, KD, Master equation for Lagrangian gauge symmetries, Phys. Lett., B 479, 429, (2000)
[15] Banerjee, R.; Rothe, HJ; Rothe, KD, Recursive construction of generator for Lagrangian gauge symmetries, J. Phys., A 33, 2059, (2000)
[16] Pons, JM, Generally covariant theories: the Noether obstruction for realizing certain space-time diffeomorphisms in phase space, Class. Quant. Grav., 20, 3279, (2003)
[17] Mukherjee, P.; Saha, A., Gauge invariances vis-á-vis diffeomorphisms in second order metric gravity, Int. J. Mod. Phys., A 24, 4305, (2009)
[18] Samanta, S., Diffeomorphism symmetry in the Lagrangian formulation of gravity, Int. J. Theor. Phys., 48, 1436, (2009)
[19] Banerjee, R.; Gangopadhyay, S.; Mukherjee, P.; Roy, D., Symmetries of the general topologically massive gravity in the Hamiltonian and Lagrangian formalisms, JHEP, 02, 075, (2010)
[20] Ostrogradsky, M., Mèmories sur LES èquations differentielles relatives au probléme des isopérimètres, Mem. Ac. St. Petersbourg, V14, 385, (1850)
[21] Battle, C.; Gomis, J.; Pons, JM; Roman-Roy, N., Lagrangian and Hamiltonian constraints for second order singular Lagrangians, J. Phys., A21, 2693, (1988)
[22] Plyushchay, MS, Canonical quantisation and mass spectrum of relativistic particle analogue of relativistic string with rigidity, Mod. Phys. Lett., A 3, 1299, (1988)
[23] Plyushchay, MS, Massless point particle with rigidity, Mod. Phys. Lett., A 4, 837, (1989)
[24] Plyushchay, MS, Massive relativistic point particle with rigidity, Int. J. Mod. Phys., A 4, 3851, (1989)
[25] Nesterenko, VV, The singular Lagrangians with higher derivatives, J. Phys., A 22, 1673, (1989)
[26] Buchbinder, IL; Lyahovich, SL; Krychtin, VA, Canonical quantization of topologically massive gravity, Class. Quant. Grav., 10, 2083, (1993)
[27] Morozov, A., Hamiltonian formalism in the presence of higher derivatives, Theor. Math. Phys., 157, 1542, (2008)
[28] Dunin-Barkowski, P.; Sleptsov, A., Geometric Hamiltonian formalism for reparametrization invariant theories with higher derivatives, Theor. Math. Phys., 158, 61, (2009)
[29] D.M. Gitman and I.V. Tyutin, Quantization of fields with constraints, Springer, U.S.A. (1990).
[30] Andrzejewski, K.; Gonera, J.; Machalski, P.; Maślanka, P., Modified Hamiltonian formalism for higher-derivative theories, Phys. Rev., D 82, 045008, (2010)
[31] Lee, TD; Wick, GC, Negative metric and the unitarity of the S matrix, Nucl. Phys., B 9, 209, (1969)
[32] Lee, TD; Wick, GC, Finite theory of quantum electrodynamics, Phys. Rev., D2, 1033, (1970)
[33] Gitman, DM; Lyakovich, SL; Tyutin, IV, Hamiltonian formulation of a theory with high derivatives, Sov. Phys. Journ., 26, 730, (1983)
[34] Pisarski, RD, Field theory of paths with a curvature-dependent term, Phys. Rev., D 34, 670, (1986)
[35] Souza Dutra, A.; Natividade, CP, Consistent higher derivative quantum field theory: A model without tachyons and ghosts, Mod. Phys. Lett., A 11, 775, (1996)
[36] Hawking, SW; Hertog, T., Living with ghosts, Phys. Rev., D 65, 103515, (2002)
[37] Rivelles, VO, Triviality of higher derivative theories, Phys. Lett., B 577, 137, (2003)
[38] Smilga, AV, Benign vs. malicious ghosts in higher-derivative theories, Nucl. Phys., B 706, 598, (2005)
[39] Kruglov, SI, Higher derivative scalar field theory in the first order formalism, Annales Fond. Broglie, 31, 343, (2006)
[40] Carone, CD; Lebed, RF, A higher-derivative Lee-Wick standard model, JHEP, 01, 043, (2009)
[41] Mukherjee, P., Poincaré gauge theory from higher derivative matter Lagrangean, Class. Quant. Grav., 27, 215008, (2010)
[42] Thiring, W., Regularization as a consequence of higher order equations, Phys. Rev., 77, 570, (1950)
[43] Pais, A.; Uhlenbeck, GE, On field theories with nonlocalized action, Phys. Rev., 79, 145, (1950)
[44] Stelle, KS, Renormalization of higher derivative quantum gravity, Phys. Rev., D 16, 953, (1977)
[45] Fradkin, ES; Tseytlin, AA, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys., B 201, 469, (1982)
[46] I.L. Buchbinder, S.D. Odnitsov and I.L. Shapiro, Effective action in quantum gravity, IOP, Bristol U.K. (1992).
[47] Sotiriou, TP; Faraoni, V., \(f\)(\(R\)) theories of gravity, Rev. Mod. Phys., 82, 451, (2010)
[48] Banerjee, R.; Mukherjee, P.; Saha, A., Interpolating action for strings and membranes: a study of symmetries in the constrained Hamiltonian approach, Phys. Rev., D 70, 026006, (2004)
[49] Banerjee, R.; Mukherjee, P.; Saha, A., Genesis of ADM decomposition: a brane-gravity correspondence, Phys. Rev., D 72, 066015, (2005)
[50] Gangopadhyay, S.; Hazra, AG; Saha, A., Noncommutativity in interpolating string: a study of gauge symmetries in noncommutative framework, Phys. Rev., D 74, 125023, (2006)
[51] Gracia, X.; Pons, JM, Gauge transformations for higher-order Lagrangians, J. Phys., A 28, 7181, (1995)
[52] Ramos, E.; Roca, J., On \(W\)(3) morphisms and the geometry of plane curves, Phys. Lett., B 366, 113, (1996)
[53] Ramos, E.; Roca, J., W symmetry and the rigid particle, Nucl. Phys. B, 436, 529, (1995)
[54] Ramos, E.; Roca, J., Extended gauge invariance in geometrical particle models and the geometry of W symmetry, Nucl. Phys., B 452, 705, (1995)
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