Symmetric energy–momentum tensor in Maxwell, Yang–Mills, and Proca theories obtained using only Noether's theorem
Merced Montesinos*, Ernesto Flores**
* Departamento de Física, Cinvestav, Av. Instituto Politecnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de Mexico, México,e–mail: merced@fis.cinvestav.mx
Associate Member of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
** Facultad de Física e Inteligencia Artificial, Universidad Veracruzana, 91000, Xalapa, Veracruz, México
]]> Recibido el 6 de junio de 2005
Abstract
The symmetric and gauge–invariant energy–momentum tensors for source–free Maxwell and Yang–Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source–free neutral Proca field, the same procedure yields also the symmetric energy–momentum tensor. In all cases, the key point to get the right expressions for the energy–momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end.
Keywords: Energy–momentum tensor; Noether's theorem; gauge field theory.
Resumen
Los tensores de energía–momento invariantes de norma y simetricos para las teorías de Maxwell y Yang–Mills sin fuentes son obtenidos mediante traslaciones en el espacio–tiempo mediante una aplicacion sistemática del teorema de Noether. Para el campo de Proca neutral y sin fuentes, el mismo procedimiento proporciona tambien el tensor de energía–momento simetrico. En todos los casos, el punto clave para obtener las expresiones correctas de los tensores de energía–momento es el manejo adecuado de las ecuaciones de movimiento y de las identidades de Bianchi. Debe ser enfatizado que estos resultados son obtenidos sin usar las tecnicas de simetrización de Belinfante las cuales son usualmente empleadas para este fin.
Descriptores: Tensor de energía–momento; teorema de Noether; teoría de campo de norma.
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Acknowledgments
Warm thanks to G.F. Torres del Castillo, Jose David Vergara, and Abdel Perez–Lorenzana for their detailed reading and criticisms to the first version of this paper. We also thank the referee for pointing out Refs. 10 and 11. This work was supported in part by the CONACyT grant SEP–2003–C02–43939.
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