  Conventional wisdom teaches us that the electrical conductivity in a material
is the inverse of its resistivity. In this work, we show that when both of
these transport coefficients are defined in linear response through the Kubo
formulae as two-point correlators of quantum fields, this Ohm's reciprocity
relation is generically violated in theories with dynamical electromagnetism.
We also elucidate how in special limits (e.g., in the DC limit in the presence
of thermal effects, in certain 2+1$d$ conformal theories, and in holographic
supersymmetric theories) the reciprocity is again reinstated, and show that if
the response of a material is measured with respect to the total electric field
that includes quantum corrections, then the reciprocity relation is satisfied
by definition. However, in that case, the transport coefficients are given by
the photon vacuum polarisation and not the correlators of conserved currents
that dominate macroscopic late-time transport.
