The w*-algebra in the standard representation is used to define a vector sp
ace for representations of Lie algebras. The Poincare group is studied as i
n thermofield dynamics (TFD) with the result that the notion of phase space
is introduced from the structure of the Poincare-Lie algebra. The basis of
quantum-field kinetic theory is analyzed in association with TFD. As a par
ticular case, the Juttner distribution is derived.