Berry Curvature and Four-Dimensional Monopoles in the Relativistic Chiral Kinetic Equation

Abstract

We derive a relativistic chiral kinetic equation with manifest Lorentz covariance from Wigner functions of spin-$1/2$ massless fermions in a constant background electromagnetic field. It contains vorticity terms and a four-dimensional Euclidean Berry monopole which gives an axial anomaly. By integrating out the zeroth component of the 4-momentum $p$, we reproduce the previous three-dimensional results derived from the Hamiltonian approach, together with the newly derived vorticity terms. The phase space continuity equation has an anomalous source term proportional to the product of electric and magnetic fields (${F}_{\ensuremath{\sigma}\ensuremath{\rho}}{\stackrel{\texttildelow{}}{F}}^{\ensuremath{\sigma}\ensuremath{\rho}}\ensuremath{\sim}{E}_{\ensuremath{\sigma}}{B}^{\ensuremath{\sigma}}$). This provides a unified interpretation of the chiral magnetic and vortical effects, chiral anomaly, Berry curvature, and the Berry monopole in the framework of Wigner functions.