  Within the closed time path formalism a general nonperturbative expression is
derived which resums through the Bethe-Salpter equation all leading order
contributions to the shear viscosity in hot scalar field theory. Using a
previously derived generalized fluctuation-dissipation theorem for nonlinear
response functions in the real-time formalism, it is shown that the
Bethe-Salpeter equation decouples in the so-called (r,a) basis. The general
result is applied to scalar field theory with pure lambda*phi**4 and mixed
g*phi**3+lambda*phi**4 interactions. In both cases our calculation confirms the
leading order expression for the shear viscosity previously obtained in the
imaginary time formalism.
