How to calculate the properties of Photon-Quasiparticles in recent questions like "How are classical optics phenomena explained in QED (Snell's law)?" and "Do photons gain mass when they travel through glass?" we could learn something about effective properties of matter interacting with a force field in terms of the path integral and quasiparticles.
Surely, both approaches must be equivalent but come from a different philosophy. Widely used is the quasiparticle approach in solid state physics e.g. calculating dispersion relations of phonons.
I would really like to know if there are simple examples for explicit calculations of the properties of photon-quasiparticles coming from a rigorous approach like a matter description via QED and finding an effective action e.g. using the Wetterich equation (see e.g. Introduction to the functional RG and applications to gauge theories).
Any calculations and/or references would be very nice.
Thank you in advance, sincerely,
Robert
 A: The quasiparticle description of photons is well known in condensed matter physics as the frequency dependent complex dielectric constant/magnetic-permeability of a material. You are asking if this quantity can be calculated from QED by a first principles method.
This seems less difficult compared to other quasiparticles, because the photon usually stays noninteracting in the medium at ordinary temperatures, except for the absorption and dispersion. There is a mismatch of scale between the photon frequency/wavelength and other quasiparticles, which travel much slower than light. Even so, I didn't find many papers adressing this question, because we have good measurements of the dielectric constant as a function of frequency for any material. The material is doing the calculation for you.
For dilute gasses, Feynman adresses the question of calculating the dielectric constant from a first principles calculation in his 1963 Acta Physica Polonica article which introduced ghost fields. The frequency dependent dielectric constant is determined from the scattering phase shift in the forward direction, which reproduces the qualitative behavior of even crazy cases like when you pass an atomic resonance. The reference is below.


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*RP Feynman, Acta Physica Polonica, 24 (1963) 697


Some classic reference for QED in media, which include calculations of the effects of dielectric properties at finite frequencies, are these original papers:


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*E. M. Lifshitz, “"The Theory of Molecular Attractive Forces”, Sov. Phys. JETP 2, 73 (1956).

*I.E. Dzyaloshinskii, E. M. Lifshitz, and L.P. Pitaevskii, ”General theory of van der Waals' forces”, Sov. Phys. JETP 10, 161 (1960)


These papers use the given photon quasiparticle description to calculate the Van-der-Waals forces for an arbitrary configuration of macroscopic matter.
