Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem

Abstract

We give a comprehensive 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 framework. 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: when combined 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 energymomentum 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 concept of a conformal field theory.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

“…5 These quantities can be found in the literature (see for example [79]), however we briefly sketch here the computations in order to make this section self contained.…”

Canonical and Gravitational Stress-Energy Tensors

“…Unfortunately, in literature, the only cases which are explicitely treated are the scalar and the free electromagnetic field. Only recently, attention has been paid to the general relationship between different concepts of stress-energy [1,2,3,4], most authors concluding on equivalence. However, this is obviously not always the case (see Maxwell field with sources), and therefore, we see the need of a complete investigation of the subject.…”

Canonical variational completion of differential equations

“…As a result, met T ij obeys on-shell the covariant conservation law met T ij ;j = 0 and also, has gauge invariance properties, [2]. Moreover, met T ij is, by construction, symmetric.…”

“…This is usually solved by adding a divergence-free term, thus obtaining a tensor can T ij which is symmetric and still conserved, i.e., can T ij ,j = 0. There are multiple possibilities of choosing the symmetrization term, [2].…”

“…In special relativity, energy-momentum tensors are obtained by adding to the Noether current corresponding to the invariance of the matter Lagrangian to space-time translations a symmetrization term. The way of obtaining the symmetrization term is subject to an old debate, [2], [3]. The canonical variational completion method offers the possibility of recovering the expression of a full, symmetric energy-momentum tensor from just one of its terms -e.g., from a non-symmetrized Noether current.…”