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


SIGMA 4 (2008), 004, 17 pages      arXiv:0801.1892      http://dx.doi.org/10.3842/SIGMA.2008.004
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Generalized Symmetries of Massless Free Fields on Minkowski Space

Juha Pohjanpelto a and Stephen C. Anco b
a) Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605, USA
b) Department of Mathematics, Brock University, St. Catharines ON L2S 3A1 Canada

Received November 01, 2007; Published online January 12, 2008

Abstract
A complete and explicit classification of generalized, or local, symmetries of massless free fields of spin s ≥ 1/2 is carried out. Up to equivalence, these are found to consists of the conformal symmetries and their duals, new chiral symmetries of order 2s, and their higher-order extensions obtained by Lie differentiation with respect to conformal Killing vectors. In particular, the results yield a complete classification of generalized symmetries of the Dirac-Weyl neutrino equation, Maxwell's equations, and the linearized gravity equations.

Key words: generalized symmetries; massless free field; spinor field.

pdf (283 kb)   ps (197 kb)   tex (20 kb)

References

  1. Anco S.C., Pohjanpelto J., Classification of local conservation laws of Maxwell's equations, Acta Appl. Math. 69 (2001), 285-327, math-ph/0108017.
  2. Anco S.C., Pohjanpelto J., Conserved currents of massless fields of spin s ≥ 1/2, Proc. R. Soc. Lond. A. 459 (2003), 1215-1239, math-ph/0202019.
  3. Anco S.C., Pohjanpelto J., Symmetries and currents of massless neutrino fields, electromagnetic and graviton fields, in Symmetry in Physics, Editors P. Winternitz, J. Harnad, C.S. Lam and J. Patera, CRM Proceedings and Lecture Notes, Vol. 34, AMS, Providence, RI, 2004, 1-12, math-ph/0306072.
  4. Anderson I.M., Torre C.G., Classification of local generalized symmetries for the vacuum Einstein equations, Comm. Math. Phys. 176 (1996), 479-539, gr-qc/9404030.
  5. Benn I.M., Kress J.M., First order Dirac symmetry operators, Classical Quantum Gravity 21 (2004), 427-431.
  6. Bluman G., Anco S.C., Symmetry and integration methods for differential equations, Springer, New York, 2002.
  7. Durand S., Lina J.-M., Vinet L., Symmetries of the massless Dirac equations in Minkowski space, Phys. Rev. D 38 (1988), 3837-3839.
  8. Fushchich W.I., Nikitin, A.G., On the new invariance algebras and superalgebras of relativistic wave equations, J. Phys. A: Math. Gen. 20 (1987), 537-549.
  9. Kalnins E.G., Miller W. Jr., Williams G.C., Matrix operator symmetries of the Dirac equation and separation of variables, J. Math. Phys. 27 (1986), 1893-1900.
  10. Kalnins E.G., McLenaghan R.G., Williams G.C., Symmetry operators for Maxwell's equations on curved space-time, Proc. R. Soc. Lond. A 439 (1992), 103-113.
  11. Kumei S., Invariance transformations, invariance group transformations and invariance groups of the sine-Gordon equations, J. Math. Phys. 16 (1975), 2461-2468.
  12. Martina L., Sheftel M.B., Winternitz P., Group foliation and non-invariant solutions of the heavenly equation, J. Phys. A: Math. Gen. 34 (2001), 9243-9263, math-ph/0108004.
  13. Mikhailov A.V., Shabat A.B., Sokolov V.V., The symmetry approach to classification of integrable equations, in What Is Integrability?, Editor V.E. Zakharov, Springer, Berlin, 1991, 115-184.
  14. Miller W. Jr., Symmetry and separation of variables, Addison-Wesley, Reading, Mass., 1977.
  15. Nikitin A.G., A complete set of symmetry operators for the Dirac equation, Ukrainian Math. J. 43 (1991), 1287-1296.
  16. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Springer, New York, 1993.
  17. Penrose R., Rindler W., Spinors and space-time. Vol. 1: Two-spinor calculus and relativistic fields, Vol. 2: Spinor and twistor methods in space-time geometry, Cambridge University Press, Cambridge, 1984.
  18. Pohjanpelto J., Symmetries, conservation laws, and Maxwell's equations, in Advanced Electromagnetism: Foundations, Theory and Applications, Editors T.W. Barrett and D.M. Grimes, World Scientific, Singapore, 1995, 560-589.
  19. Pohjanpelto J., Classification of generalized symmetries of the Yang-Mills fields with a semi-simple structure group, Differential Geom. Appl. 21 (2004), 147-171, math-ph/0109021.
  20. Ward R.S., Wells R.O. Jr., Twistor geometry and field theory, Cambridge University Press, Cambridge, 1990.

Previous article   Next article   Contents of Volume 4 (2008)