  The widely used linear-response (LR) theory of thermal conduction in the
quantum regime rests on the yet unproven assumption, that the thermal
conductivity is invariant with respect to the gauge of the energy density of
the system. This assumption manifests itself clearly in, e.g., Hardy's
formulation of the heat-flux operator [Phys. Rev. 132, 168 (1963)]. In this
work, we rigorously prove this assumption. Our proof, being valid for the
nuclear and electronic subsystems, not only puts the state-of-the-art theory on
solid grounds, but also enables going beyond the scope of the widely-used
Boltzmann transport equation (BTE) within the LR framework for many-body
systems.
