  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 novel spectral
representation for the shear viscosity $\eta$. As an example, we calculate
$\eta$ for the thermal particle states in $\phi^{4}$ theory, and establish the
leading asymptotic behaviour in the small and large coupling regimes.
