We study the properties of $K$ and $bar K$ mesons in nuclear matter at finite temperature from a chiral unitary approach in coupled channels which incorporates the $s$- and p-waves of the kaon-nucleon interaction. The in-medium solution accounts for Pauli blocking effects, mean-field binding on all the baryons involved, and $pi$ and kaon self-energies. We calculate $K$ and $bar K$ (off-shell) spectral functions and single particle properties. The $bar K$ effective mass gets lowered by about -50 MeV in cold nuclear matter at saturation density and by half this reduction at T=100 MeV. The p-wave contribution to the ${bar K}$ optical potential, due to $Lambda$, $Sigma$ and $Sigma^*$ excitations, becomes significant for momenta larger than 200 MeV/c and reduces the attraction felt by the $bar K$ in the nuclear medium.The $bar K$ spectral function spreads over a wide range of energies, reflecting the melting of the $Lambda (1405)$ resonance and the contribution of $YN^{-1}$ components at finite temperature. In the $KN$ sector, we find that the low-density theorem is a good approximation for the $K$ self-energy close to saturation density due to the absence of resonance-hole excitations. The $K$ potential shows a moderate repulsive behavior, whereas the quasi-particle peak is considerably broadened with increasing density and temperature. We discuss the implications for the decay of the $phi$ meson at SIS/GSI energies as well as in the future FAIR/GSI project.
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