  We discuss the interpretation of Euclidean correlation functions at finite
temperature ($T$) and their relationship with the corresponding real-time
Green's functions. The soluble 2+1 dimensional Gross-Neveu model in the
large-$N$ limit is used throughout as a working example. First, the real-time
bound state, identified as an elementary excitation at finite $T$, is solved.
The bound state mass, the dispersion relation at low momenta, the coupling
constant and decay constant are calculated. To characterize the structure of
the bound state the on-shell form factor is carefully introduced and
calculated. Then we examine the corresponding screening state and contrast the
screening mass, coupling constant, decay constant and the screening
Bethe-Salpeter amplitude with the real-time quantities. We find that, although
they can be used as qualitative indicators in the low-$T$ regime, the screening
states at finite $T$ in general do not reflect the properties of the
corresponding real-time bound states. Besides, other relevant issues, such as
the subtlety of the real-time manifestation of conservation laws due to some
internal symmetries at $T\ne 0$, the temperature dependence of the pseudoscalar
spectral function and its sum rule, and the high-$T$ limit of the screening
state and its implications to the dimensional reduction, are also discussed in
detail.
