# A Kubos formula for viscosity

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1. Mar 18, 2017

### Kurret

I am looking for a derivation of the following formula
$$\eta=\lim_{\omega\rightarrow0} \frac{1}{2\omega}\int dt dx\langle[T_{xy}(t,x),T_{xy}(0,0)]\rangle,$$
where $T_{xy}$ is a component of the stress-energy tensor. This is claimed in for instance https://arxiv.org/pdf/hep-th/0405231.pdf. There seems to be a derivation in https://arxiv.org/pdf/1207.7021.pdf, but it seems overly complicated and involved extra features. So before I dig into that paper to try to understand it, I would like to ask if someone knows a simple derivation of the above Kubo formula for the viscosity?

2. Mar 18, 2017

### vanhees71

The Kubo formulae are very important. I recommend to first look at the usual many-body approach before using special models like AdS/CFT ;-)). It's linear-response theory. For thermal relativistic QFT you find a nice treatment in

J. Kapusta, C. Gale, Finite-temperature field theory, Cambridge University Press

3. Mar 18, 2017

### Kurret

Actually the paper I referred to just applies the Kubo formula in an AdS/CFT context, but they don't derive it (it just happened to be the place where I saw it). I am interested in a derivation of this formula indeed using standard linear response theory (nothing to do with AdS/CFT). Thanks for the reference, I will have a look.