Out of equilibrium thermal field theories: Finite time after switching on the interaction and Wigner transforms of projected functions

We study out of equilibrium thermal field theories with switching on the interaction occurring at finite time using the Wigner transforms of two-point functions. For two-point functions we define the concept of a projected function: it is zero if any of the times refers to the time before switching on the interaction; otherwise it depends only on the relative coordinates. This definition includes bare propagators, one-loop self-energies, etc. For the infinite-average-time limit of the Wigner transforms of projected functions we define the analyticity assumptions: (1) The function of energy is analytic above (below) the real axis. (2) The function goes to zero as the absolute value of energy approaches infinity in the upper (lower) semiplane. Without use of the gradient expansion, we obtain the convolution product of projected functions. We sum the Schwinger-Dyson series in closed form. In the calculation of the Keldysh component (both resummed and single self-energy insertion approximation) contributions appear which are not the Fourier transforms of projected functions, signaling the limitations of the method. In the Feynman diagrams there is no explicit energy conservation at vertices; there is an overall energy-smearing factor taking care of the uncertainty relations. The relation between the theories with the Keldysh time path and with the finite time path enables one to rederive the results, such as the cancellation of pinching, collinear, and infrared singularities, hard thermal loop resummation, etc.