
SIGMA 4 (2008), 004, 17 pages arXiv:0801.1892
https://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 973314605, 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 higherorder
extensions obtained by Lie differentiation
with respect to conformal Killing vectors.
In particular, the results yield a complete classification
of generalized symmetries of the DiracWeyl 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
 Anco S.C., Pohjanpelto J.,
Classification of local conservation laws of Maxwell's equations,
Acta Appl. Math. 69 (2001), 285327, mathph/0108017.
 Anco S.C., Pohjanpelto J.,
Conserved currents of massless fields of spin s ≥ 1/2,
Proc. R. Soc. Lond. A. 459 (2003), 12151239, mathph/0202019.
 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, 112, mathph/0306072.
 Anderson I.M., Torre C.G.,
Classification of local generalized symmetries
for the vacuum Einstein equations,
Comm. Math. Phys. 176 (1996), 479539, grqc/9404030.
 Benn I.M., Kress J.M., First order Dirac symmetry operators,
Classical Quantum Gravity 21 (2004), 427431.
 Bluman G., Anco S.C.,
Symmetry and integration methods for differential equations,
Springer, New York, 2002.
 Durand S., Lina J.M., Vinet L., Symmetries of the massless Dirac
equations in Minkowski space,
Phys. Rev. D 38 (1988), 38373839.
 Fushchich W.I., Nikitin, A.G.,
On the new invariance algebras and superalgebras
of relativistic wave equations,
J. Phys. A: Math. Gen. 20 (1987), 537549.
 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), 18931900.
 Kalnins E.G., McLenaghan R.G., Williams G.C.,
Symmetry operators for Maxwell's
equations on curved spacetime,
Proc. R. Soc. Lond. A 439 (1992), 103113.
 Kumei S.,
Invariance transformations, invariance group transformations
and invariance groups of the sineGordon equations,
J. Math. Phys. 16 (1975), 24612468.
 Martina L., Sheftel M.B., Winternitz P.,
Group foliation and noninvariant solutions of the heavenly equation,
J. Phys. A: Math. Gen. 34 (2001), 92439263, mathph/0108004.
 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, 115184.
 Miller W. Jr.,
Symmetry and separation of variables,
AddisonWesley, Reading, Mass., 1977.
 Nikitin A.G., A complete set of symmetry operators
for the Dirac equation, Ukrainian Math. J. 43 (1991), 12871296.
 Olver P.J.,
Applications of Lie groups to differential equations,
2nd ed.,
Springer, New York, 1993.
 Penrose R., Rindler W.,
Spinors and spacetime. Vol. 1: Twospinor calculus
and relativistic fields, Vol. 2: Spinor and twistor methods
in spacetime geometry,
Cambridge University Press, Cambridge, 1984.
 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, 560589.
 Pohjanpelto J., Classification of generalized symmetries
of the YangMills fields with a semisimple structure group,
Differential Geom. Appl. 21 (2004), 147171, mathph/0109021.
 Ward R.S., Wells R.O. Jr.,
Twistor geometry and field theory,
Cambridge University Press, Cambridge, 1990.

