  We have explored the multi-component structure of electrical conductivity of
relativistic Fermionic and Bosonic fluid in presence of magnetic field by using
Kubo approach. This is done by explicitly evaluating the thermo-magnetic vector
current spectral functions using the real time formalism of finite temperature
field theory and the Schwinger proper time formalism. In absence of magnetic
field, the one-loop diagramatic representation of Kubo expression of any
transport coefficients is exactly same with relaxation time approximation (RTA)
based expression, but this equality does not hold for finite magnetic field
picture due to lacking of proper implementation of quantum effect in latter
approach. We have shown this discrepancy for particular transport coefficient -
electrical conductivity, whose starting point in Kubo approach will be
electromagnetic current-current correlator and its one-loop skeleton diagram
carrying two scalar/Dirac propagators for scalar/Dirac fluid. Through a
numerical comparison between RTA and Kubo expressions of conductivity
components (parallel and perpendicular), we have attempted to interpret detail
quantum field theoretical effect, contained by Kubo expression but not by RTA
expression. In classical RTA expression we get magnetic field independent
parallel conductivity due to zero Lorentz force but in field theoretical Kubo
expression, it decreases and increases with the magnetic field for scalar and
Dirac medium respectively due to Landau quantization effect. This parallel
component of conductivity can be interpreted as zero momentum limit of quantum
fluctuation with same Landau level internal lines. While for perpendicular
component of conductivity, fluctuation with Landau level differences $\pm 1$
are noticed, which might be a new realization of transportation in field
theoretical sector.
