  In local scalar quantum field theories (QFTs) at finite temperature
correlation functions are known to satisfy certain non-perturbative
constraints, which for two-point functions in particular implies the existence
of a generalisation of the standard K\"{a}ll\'{e}n-Lehmann representation. In
this work, we use these constraints in order to derive a spectral
representation for the shear viscosity arising from the thermal asymptotic
states, $\eta_{0}$. As an example, we calculate $\eta_{0}$ in $\phi^{4}$
theory, establishing its leading behaviour in the small and large coupling
regimes.
