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