Noether's Theorem and the Stress Energy Tensor
A common derivation that you will find in QFT texts is the derivation of the Stress Energy Tensor using Noether's theorem. I am just starting to study QFT using David Tong's notes and Schwartz' book . However, I have found that in both the expositions are incomplete and the terminology and notation is sometimes confusing. As such, I didn't feel I grokked the derivations. So to fill in the gaps, I am writing out all the gory details for my own edification and the hope that this might help someone else facing similar frustrations. After starting this, I also found these notes by Hagen Kleinart to be particularly clear. Noether's theorem is a statement about continous symmetries of an action. With such a symmetry there is an associated conserved current. The symmetry we will be looking at is a space-time translation. To that end we consider an infinitesimal space time translation. $$x^{\nu} \rightarrow x^{\nu} - \epsilon^