  We reexamine physical causal propagators for scalar and pseudoscalar bound
states at finite temperature in a chiral $U_L(1)\times U_R(1)$ NJL model,
defined by four-point amputated functions subtracted through the gap equation,
and prove that they are completely equivalent in the imaginary-time and
real-time formalism by separating carefully the imaginary part of the
zero-temperature loop integral. It is shown that the thermal transformation
matrix of the matrix propagators for these bound states in the real-time
formalism is precisely the one of the matrix propagator for an elementary
scalar particle and this fact shows similarity of thermodynamic property
between a composite and an elementary scalar particle. The retarded and
advanced propagators for these bound states are also given explicitly from the
imaginary-time formalism.
