  We have gone through a detailed calculation of the two-point correlation
function of vector currents at finite density and magnetic field by employing
the real time formalism of finite temperature field theory and Schwinger's
proper time formalism. With respect to the direction of external magnetic
field, the parallel and perpendicular components of electric conductivity for
the degenerate relativistic fermionic matter are obtained from the zero
momentum limit of the current-current correlator, owing to Kubo formula. Our
quantum field theoretical expressions and numerical estimations are compared
with the same, obtained from the relaxation time approximation methods of
kinetic theory and its Landau quantized extension, which may be called as
classical and quantum results respectively. All the results are merged in the
classical domain i.e. high density and low magnetic field region but in the
remaining (quantum) domain, quantum results carry a quantized information like
Shubnikov-de Haas oscillation along density and magnetic field axes. We have
obtained completely new quantum field theoretical expression for perpendicular
conductivity of degenerate relativistic fermionic matter. Interestingly, our
quantum field theoretical calculation provide a new mathematical form of
cyclotron frequency with respect to its classical definition, which might
require more future research to interpret the phenomena.
