Contrasting Real-time Dynamics with Screening Phenomena at Finite Temperature

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.