What is the Quantum Boltzmann equation and what does it describe? I think it describes the propagation of electrons and photons but I am not sure.
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Quantum Boltzmann/kinetic equation usually refers to the formalism pioneered by Kadanoff and Baym, which is a quasi-classical theory describing kinetic phenomena in metals - easier than the full Green's function approach, but more rigorous than the popular kinetic/Boltzmann equation treatment that is ubiquitous in many books on solid state physics.
Quantum field-theoretical methods in transport theory of metals by Rammer and Smith is probably the best place to start - the Bolttzmann equation is derived here coherently from the Green's function formalism, different levels of quantum corrections are calculated, and collision integrals for various types of scattering processes are treated in details.
For a more intuitive treatment see Quantum Kinetics in Transport and Optics of Semiconductors by Haug and Jauho.
