Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 021, 18 pages      arXiv:1112.1860      https://doi.org/10.3842/SIGMA.2012.021
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Lagrange Anchor and Characteristic Symmetries of Free Massless Fields

Dmitry S. Kaparulin, Simon L. Lyakhovich and Alexey A. Sharapov
Department of Quantum Field Theory, Tomsk State University, 36 Lenin Ave., Tomsk 634050, Russia

Received December 28, 2011, in final form April 09, 2012; Published online April 12, 2012

Abstract
A Poincaré covariant Lagrange anchor is found for the non-Lagrangian relativistic wave equations of Bargmann and Wigner describing free massless fields of spin s>1/2 in four-dimensional Minkowski space. By making use of this Lagrange anchor, we assign a symmetry to each conservation law and perform the path-integral quantization of the theory.

Key words: symmetries; conservation laws; Bargmann-Wigner equations; Lagrange anchor.

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References

  1. Anco S.C., Bluman G., Direct construction of conservation laws from field equations, Phys. Rev. Lett. 78 (1997), 2869-2873.
  2. Anco S.C., Pohjanpelto J., Classification of local conservation laws of Maxwell's equations, Acta Appl. Math. 69 (2001), 285-327, math-ph/0108017.
  3. Anco S.C., Pohjanpelto J., Conserved currents of massless fields of spin s≥1/2, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), 1215-1239, math-ph/0202019.
  4. Barnich G., Brandt F., Henneaux M., Local BRST cohomology in the antifield formalism. I. General theorems, Comm. Math. Phys. 174 (1995), 57-91, hep-th/9405109.
  5. Barnich G., Henneaux M., Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket, J. Math. Phys. 37 (1996), 5273-5296, hep-th/9601124.
  6. Bluman G.W., Cheviakov A.F., Anco S.C., Applications of symmetry methods to partial differential equations, Applied Mathematical Sciences, Vol. 168, Springer, New York, 2010.
  7. DeWitt B.S., Dynamical theory of groups and fields, Gordon and Breach Science Publishers, New York, 1965.
  8. Dickey L.A., Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, Vol. 12, World Scientific Publishing Co. Inc., River Edge, NJ, 1991.
  9. Dubois-Violette M., Henneaux M., Talon M., Viallet C.M., Some results on local cohomologies in field theory, Phys. Lett. B 267 (1991), 81-87.
  10. Fairlie D.B., Conservation laws and invariance principles, Nuovo Cimento 37 (1965), 897-904.
  11. Fang J., Fronsdal C., Massless fields with half-integral spin, Phys. Rev. D 18 (1978), 3630-3633.
  12. Fronsdal C., Massless fields with integer spin, Phys. Rev. D 18 (1978), 3624-3629.
  13. Fushchich W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, Allerton Press Inc., New York, 1994.
  14. Kaparulin D.S., Lyakhovich S.L., Sharapov A.A., Rigid symmetries and conservation laws in non-Lagrangian field theory, J. Math. Phys. 51 (2010), 082902, 22 pages, arXiv:1001.0091.
  15. Kazinski P.O., Lyakhovich S.L., Sharapov A.A., Lagrange structure and quantization, J. High Energy Phys. 2005 (2005), 076, 42 pages, hep-th/0506093.
  16. Kazinski P.O., Lyakhovich S.L., Sharapov A.A., Local BRST cohomology in (non-)Lagrangian field theory, J. High Energy Phys. 2011 (2011), 006, 34 pages, arXiv:1106.4252.
  17. Kibble T.W.B., Conservation laws for free fields, J. Math. Phys. 6 (1965), 1022-1026.
  18. Konstein S.E., Vasiliev M.A., Zaikin V.N., Conformal higher spin currents in any dimension and AdS/CFT correspondence, J. High Energy Phys. 2000 (2000), 018, 12 pages, hep-th/0010239.
  19. Kosmann-Schwarzbach Y., The Noether theorems, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011.
  20. Lipkin D.M., Existence of a new conservation law in electromagnetic theory, J. Math. Phys. 5 (1964), 696-700.
  21. Lyakhovich S.L., Sharapov A.A., Quantizing non-Lagrangian gauge theories: an augmentation method, J. High Energy Phys. 2007 (2007), 047, 40 pages, hep-th/0612086.
  22. Lyakhovich S.L., Sharapov A.A., Schwinger-Dyson equation for non-Lagrangian field theory, J. High Energy Phys. 2006 (2006), 007, 27 pages, hep-th/0512119.
  23. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
  24. Morgan T.A., Two classes of new conservation laws for the electromagnetic field and for other massless fields, J. Math. Phys. 5 (1964), 1659-1660.
  25. Nastasescu C., Van Oystaeyen F., Graded and filtered rings and modules, Lecture Notes in Mathematics, Vol. 758, Springer, Berlin, 1979.
  26. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  27. Penrose R., Rindler W., Spinors and space-time, Vols. I and II, Cambridge University Press, Cambridge, 1987.
  28. Pohjanpelto J., Anco S.C., Generalized symmetries of massless free fields on Minkowski space, SIGMA 4 (2008), 004, 17 pages, arXiv:0801.1892.
  29. Streater R.F., Wightman A.S., PCT, spin and statistics, and all that, W.A. Benjamin, Inc., New York - Amsterdam, 1964.
  30. Vasiliev M.A., Consistent equations for interacting gauge fields of all spins in 3+1 dimensions, Phys. Lett. B 243 (1990), 378-382.
  31. Vasiliev M.A., Higher spin gauge theories in various dimensions, Fortschr. Phys. 52 (2004), 702-717, hep-th/0401177.
  32. Vasiliev M.A., Gelfond O.A., Skvortsov E.D., Higher-spin conformal currents in Minkowski space, Theoret. Math. Phys. 154 (2008), 294-302, hep-th/0601106.

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