# A Stress-Energy Tensor - derivation

1. May 20, 2017

### spaghetti3451

In page 64 of David Tong's notes (http://www.damtp.cam.ac.uk/user/tong/string/four.pdf) on conformal field theory, Tong mentions that

1. the stress-energy tensor is defined as the matrix
of conserved currents which arise from translational invariance,
$$\delta\sigma^{a} = \epsilon^{a},$$
where $\sigma^{a}$ are the worldsheet coordinates, and that

2. in flat spacetime, a translation is a special case of a conformal transformation.

This I understand. But then he goes on to say that

we can derive conserved currents by promoting the constant parameter $\epsilon$ that appears in the symmetry to a function of the spacetime coordinates.

I do not follow this. If the translational invariance of the worldsheet coordinates under $\delta\sigma^{a} = \epsilon^{a}$ give us the stress-energy tensor of the theory, why do we need to gauge the symmetry in order to derive the stress-tensor of the theory?

How do you then go ahead to show that the change in the action must then be of the form,
$$\delta S = \int d^{2}\sigma J^{\alpha}\partial_{\alpha}\epsilon$$
for some function of the fields, $J^{\alpha}$?

2. May 26, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.