Noether formalism with gauge-invariant variations
The energy–momentum tensor is usually obtained by symmetrizing the canonical tensor, which is sometimes difficult. The fact that the energy–momentum tensor must always be gauge invariant, and insight gained by investigating the Maxwell kinetic equations, lead one to conclude that this tensor should be readily obtainable for a certain class of Lagrange densities. The appropriate method to achieve this consists in performing gauge-invariant variations within the framework of Noether's theory. These variations are equivalent to shifts in space and time, and are considered here instead of the usual shift variations, which are not gauge invariant. These features indicate the existence of a certain underlying structure which is made evident by the proposed method. The Lagrange densities of the class in question are characterized by having a gauge-invariant contribution additional to the Maxwell part. For vanishing potentials, such Lagrange densities lead directly to a symmetric canonical tensor; this implies that there are only scalar fields. Maxwell's equations, being the basic part of such theories, are treated first. Then, the coupled system of electromagnetic fields and the usual scalar fields describing charged matter is considered. Finally, a different kind of scalar field which occurs in a structural non-standard form in collisionless phase-space theories (Vlasov or collisionless Boltzmann-like equations) is treated.(Received April 12 2003)
(Revised July 20 2003)
(Accepted August 13 2003)
c1 Corresponding author.