1

I have a problem providing or finding a general proof for this statement i found in Mussardo's statistical field theory book, section $10.3.2$:

Due to the locality of the theory there exists a local field $T_{\mu\nu}(x)$, called the stress-energy tensor, defined by the variation of the local action $S[\varphi]$ under the infinitesimal transformation $x^{\,\mu} \rightarrow x^{' \mu} = x^{\,\mu} + \epsilon^{\,\mu}(x)$

$$ \delta S \, =\, \frac{1}{(2 \pi)^{D-1}} \int d^Dx \, T_{\mu\nu}(x)\, \partial^{\,\mu} \epsilon^{\,\nu} $$

where it says that the factor $(2 \pi)^{D-1}$ is there for later convenience

Now I've seen the proof for a real scalar field which is clearly a field without local interactions.

What is the proof in the general case? Is there a place (book or article) where I can find the proof?

RenatoRenatoRenato
  • 2,327
  • 1
    It is known Noether's theorem. – my2cts Dec 06 '18 at 17:20
  • @my2cts Sorry, my bad, both for the question and the lack of research. Can I mark it as a duplicate on my own since there are questions about Noether's theorem that answer this one? Or there is a different procedure for that? – RenatoRenatoRenato Dec 06 '18 at 19:08
  • 1
    Possible duplicate of Energy momentum tensor from Noether's theorem – RenatoRenatoRenato Dec 06 '18 at 20:38
  • @Qmechanic I'm waiting to close it cause I think you put it in your favourites to do something later. I'm thinking this cause in another question of mine someone put in favourites and then after a bit you provided an answer. After that it wasn't a favourite question for anyone anymore. So if it's not you or if you don't intend to act on the question I close it, if it's the opposite I wait. – RenatoRenatoRenato Dec 06 '18 at 20:49
  • Does Mussardo explicitly assume that the mentioned infinitesimal transformation is a symmetry of the action? – Qmechanic Dec 06 '18 at 21:46
  • I 'm not sure he does, for sure before the statement I've written in the question he says "Let us now present the theorem due to Polyakov on the conformal symmetry of physical systems with local interactions that are invariant under translations, rotations, and a global dilatation." So the $\epsilon$ may only be related to those symmetries. – RenatoRenatoRenato Dec 06 '18 at 23:27

0 Answers0