Generally in a non-interacting QFT one can solve the Klein-Gordon equation to get a (complete) set of states $\frac{e^{i\omega_k t-ikx}}{\sqrt{2\omega_k}}$. It is not clear to me how to construct the density matrix $\rho_0$ for zero-temperature QFT, but presumably it can be done, and the resulting matrix corresponds to a pure state (zero entropy).
On the other hand, at finite temperature one has to solve the ``imaginary'' time K-G equation and it yeilds a set of eigenmodes $e^{i(2\pi/\beta)n\tau-k_nx}$ for integer $n$. This state has dispersion, and as well, the entropy corresponding to the free-particle states is that of a Bose gas at temperature $\beta$.
My question is: what do the eigenmodes correspond to in this case? Are they complete, and if so, do they correspond to a ``mixed state''?