文档介绍：Currents and the Energy-Momentum Tensor
in Classical Field Theory:
A Fresh Look at an Old Problem
Michael Forger 1 ∗ and Hartmann R¨omer 2 †
1 Departamento de Matem´atica Aplicada,
Instituto de Matem´atica e Estat´ıstica,
Universidade de S˜aoPaulo,
Caixa Postal 66281,
BR–05311-970 S˜aoPaulo, ., Brazil
2 Fakult¨at f¨urPhysik
Albert-Ludwigs-Universit¨atFreiburg im Breisgau
Hermann-Herder-Straße 3
D–79104 Freiburg ., Germany
Universit¨at Freiburg
THEP 03/10
Universidade de S˜aoPaulo
RT-MAP-0301
July 2003
1E-mail address: ******@
2E-mail address: hartmann.******@-
∗Partially supported Pq and FAPESP, Brazil, and by DFG, Germany
†Partially supported by FAPESP, Brazil
Abstract
We give prehensive review of various methods to define currents and the
energy-momentum tensor in classical field theory, with emphasis on a geometric
point of view. The necessity of “improving” the expressions provided by the
canonical Noether procedure is addressed and given an adequate geometric frame-
work. The main new ingredient is the explicit formulation of a principle of
“ultralocality” with respect to the symmetry generators, which is shown to fix the
ambiguity inherent in the procedure of improvement and guide it towards a unique
answer: bined with the appropriate splitting of the fields into sectors,
it leads to the well-known expressions for the current as the variational derivative
of the matter field Lagrangian with respect to the gauge field and for the energy-
momentum tensor as the variational derivative of the matter field Lagrangian with
respect to the metric tensor. In the second case, the procedure is shown to work
even when the matter field Lagrangian depends explicitly on the curvature, thus
establishing the correct relation between scale invariance, in the form of local
Weyl invariance “on shell”, and tracelessness of the energy-momentum tensor,
required for a consistent definition of the conc