  We study shear viscosity in weakly coupled hot $\phi^4$ theory using the CTP
formalism . We show that the viscosity can be obtained as the integral of a
three-point function. Non-perturbative corrections to the bare one-loop result
can be obtained by solving a decoupled Schwinger-Dyson type integral equation
for this vertex. This integral equation represents the resummation of an
infinite series of ladder diagrams which contribute to the leading order
result. It can be shown that this integral equation has exactly the same form
as the Boltzmann equation. We show that the integral equation for the viscosity
can be reexpressed by writing the vertex as a combination of polarization
tensors. An expression for this polarization tensor can be obtained by solving
another Schwinger-Dyson type integral equation. This procedure results in an
expression for the viscosity that represents a non-perturbative resummation of
contributions to the viscosity which includes certain non-ladder graphs, as
well as the usual ladders. We discuss the motivation for this resummation. We
show that these resummations can also be obtained by writing the viscosity as
an integral equation involving a single four-point function. Finally, we show
that when the viscosity is expressed in terms of a four-point function, it is
possible to further extend the set of graphs included in the resummation by
treating vertex and propagator corrections self-consistently. We discuss the
significance of such a self-consistent resummation and show that the integral
equation contains cancellations between vertex and propagator corrections.
