  The $f$-sum rule and the Kohn formula are well-established general
constraints on the electric conductivity in quantum many-body systems. We
present their generalization to non-linear conductivities at all orders of the
response in a unified manner, by considering two limiting quantum
time-evolution processes: a quench process and an adiabatic process. Our
generalized formulas are valid in any stationary state, including the ground
state and finite temperature Gibbs states, regardless of the details of the
system such as the specific form of the kinetic term, the strength of the
many-body interactions, or the presence of disorders.
