Generalized Beth--Uhlenbeck entropy formula from the $\Phi-$derivable approach

We derive a generalized Beth-Uhlenbeck formula for the entropy of a dense fermion system with strong two-particle correlations, including scattering states and bound states. We work within the $\Phi-$derivable approach to the thermodynamic potential. The formula takes the form of an energy-momentum integral over a statistical distribution function times a unique spectral density. In the near mass-shell limit, the spectral density reduces, contrary to na\"{i}ve expectations, not to a Lorentzian but rather to a "squared Lorentzian" shape. The relation of the Beth-Uhlenbeck formula to the $\Phi$-derivable approach is exact at the two-loop level for $\Phi$. The formalism we develop, which extends the Beth-Uhlenbeck approach beyond the low-density limit, includes Mott dissociation of bound states, in accordance with Levinson's theorem, and the self-consistent back reaction of correlations in the fermion propagation. We discuss applications to further systems, such as quark matter and nuclear matter.