I am trying to understand under exactly what conditions it is (or is not) meaningful to use a self energy to describe and understand experiments on a strongly correlated metal which may be (at least in some sense) a non-Fermi liquid. This is particularly motivated by a recent paper on the overdoped cuprates.
Below are some statements which I am trying to ascertain the truth of and relationship between them. I believe
1. and 2. are always true.
3. and 4. are equivalent but are not always true.
5. is true.
I am not sure about 6.
1. The one-electron Greens function G(k,E) is an analytic function of energy E.
2. One can always define a self energy by Dyson's equation
3. If E is treated as a complex variable G(k,E) has isolated simple poles in the complex plane. These poles correspond to quasiparticles. One can then write down a Boltzmann transport equation for these quasiparticles.
4. The self energy can be written as a convergent perturbation expansion. This ensures adiabatic continuity and the existence of quasi-particles.
5. If G(k,E) has a non-integer power law dependence on E there will be a branch cut in the complex plane. This means a description in terms of quasi-particle poles is inadequate.
[This is what happens in one dimension with Luttinger liquids].
6. Branch cuts in the plane may mess up the Kramers-Kronig relation which relates the real and imaginary parts of the self energy.
So, I welcome thoughts about the above claims.
Is there somewhere that this is all written down and discussed clearly?
I have gleaned the above from my subconscious memory of a diverse range of sources.