Spectral representation of the shear viscosity for local scalar QFTs at finite temperature

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.