# Improved energy-momentum tensor

While still dealing with this issue, I've stumbled upon this answer to a question asking about the conserved quantity corresponding to a scaling transformation. It mentions that in accordance with Noether's theorem, an improved energy-momentum tensor and Noether current can be found for a large class of (scale invariant) theories, such that the conserved charge can be calculated.

Unfortunately, apart from these short remarks, the OP of the answer I cite left no reference with further information about this issue. For example I'd like to know how such an improved energy-momentum tensor can be derived generally, what form it and the corresponding conserved charge would take for some example theories, how this conserved charge can be physically interpreted, etc.

Finally I'm interested in applying these ideas to fluid dynamics,I'd like to know how to construct the conserved quantity corresponding to the scale invariance of the Navier Stokes equations for example. But references wherein this concept is explained dealing with QFTs I'd appreciate too :-).

• There are a set of lectures by Jaume Gomis on conformal field theory where he talks about the improved energy-momentum tensor which might be of interest to you. I'm sorry but I forget which lecture he talks about this particular issue though (there are ~14 lectures). pirsa.org/index.php?p=speaker&name=Jaume_Gomis Sep 5, 2012 at 14:02

## 1 Answer

This question was addressed by Forger and Romer. They formulate an "ultralocality principle" which characterizes the correction terms.

This principle can be described as follows:

The classical matter fields are sections of vector bundles $E$ over the configuration manifold. The Noether theorem induces a representation of the lie algebra of a Lie group acting on these bundles by bundle homomorphisms in terms of (projectable) vector fields on $E$. However, this representation does not extend to a local representation (valued in functions over the configuration manifold). The correction term is exactly, the one required to make this homomorphism local.

Forger and Romer explicitly work out, in the article, some well known examples of classical field theories and show that the correction term picked according to their principle is exactly the one which renders the energy-momentum tensor of locally Weyl invariant theories (on shell) traceless.