  This paper is the first in a series revisiting the Faraday effect, or more
generally, the theory of electronic quantum transport/optical response in bulk
media in the presence of a constant magnetic field. The independent electron
approximation is assumed. At zero temperature and zero frequency, if the Fermi
energy lies in a spectral gap, we rigorously prove the Widom-Streda formula.
For free electrons, the transverse conductivity can be explicitly computed and
coincides with the classical result. In the general case, using magnetic
perturbation theory, the conductivity tensor is expanded in powers of the
strength of the magnetic field $B$. Then the linear term in $B$ of this
expansion is written down in terms of the zero magnetic field Green function
and the zero field current operator. In the periodic case, the linear term in
$B$ of the conductivity tensor is expressed in terms of zero magnetic field
Bloch functions and energies. No derivatives with respect to the quasi-momentum
appear and thereby all ambiguities are removed, in contrast to earlier work.
