Let us insist on linear response theory and the definition of the Green’s function. Take a system and apply an external source to it, i.e \(\text {source}(\omega , k)\), which is frequency and momentum dependent. Read the response of the system: \(\text {response}(\omega , k)\). At leading order, the two are related by a linear map. Here, we introduce another very simple example: the electric conductivity. We apply an external electric field E, which will be our source, and we measure the produced electric current J, which is going to be our response.
A wise man can learn more from a foolish question than a fool can learn from a wise answer.
In contrast with the normal frequencies with \(\omega _n\) real.
3.
This has been verified by explicit computations in several examples. The most famous one is that of holographic superconductors studied in [205].
4.
If you do not use gauge-invariant variables, there are additional complications in the procedure. See [205, 207] for details. My advise is: use GI variables!
5.
We thank Amadeo Jimenez for pointing this out.
6.
In absence of momentum relaxation, the DC conductivity would be infinite. This can be understood by simply thinking at a collection of electrons accelerated by an electric field. If there is no way by which these electrons can lose their momenta, once accelerated, they will go forever and they will produce an infinite conductivity. Probably the best way to understand this is to look back at the famous Drude model [211].
7.
The specific breaking pattern presents several subtleties. For those of you interested, I suggest to read [186, 187, 208].
8.
You can prove by yourself that the others are redundant.
9.
We thank Karl Landsteiner for suggesting this simple derivation.
10.
See, for example, [152].
11.
For simplicity, we have taken a system with a purely imaginary quasinormal mode. The same method can be applied for a complex mode. In this case, every equation and every function have to be split in a real and an imaginary part. For example, in the case of a single field considered in the main text, the matching conditions would be four now, instead of two.
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Authors and Affiliations
Physics Department, IFT Instituto de Fisica Teorica Madrid, Madrid, Spain
Baggioli, M. (2019). Holographic Transport via Analytic and Numerical Techniques.
In: Applied Holography. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-35184-7_4
