  We study out of equilibrium thermal field theories with switching on the
interaction occurring at finite time using the Wigner transforms (in relative
space-time) of two-point functions.
  For two-point functions we define the concept of projected function: it is
zero if any of 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 Wigner
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.
