Vorticity dependent quantum kinetic equation in three dimensions

Abstract

In this thesis, we formulate a 3-dimensional transport theory for massive Dirac particles in vortical systems by means of QKE that includes the vorticity tensor and the Wigner function. Furthermore, we investigate the results of such formalism. The Wigner function is the quantum mechanical analog of the classical distribution function that governs the collective behavior of non-equilibrium systems. 3-dimensional distribution functions and the equations that govern them can be extracted from the Wigner function after decomposing it in terms of the Clifford algebra basis elements. We employ a method called equal-time formalism which can be roughly summarized as the method that obtains the 3-dimensional Wigner function by integrating the 4-dimensional Wigner function over the zeroth component of the momentum variable. Another method called the covariant formalism prefers to stick with the 4-dimensional fields. Both of the formalism includes the semiclassical expansion which allows us to proceed order by order in terms of Planck constant as the zeroth order corresponding to the classical limit. The zeroth order kinetic equation for the Wigner function is the quadratic mass-shell equation. Thus, the solutions involves the delta function that enforces the mass-shell condition with negative- and positive-energy. The classical relations between the distribution functions follows directly from the zeroth order constraint equations that is obtained after performing the momentum's zeroth component integration. The first order equations can be found by expanding the on-shell delta function mentioned in the previous paragraph around the shell. It follows from this expansion that the energy shift terms needs to be determined in order to proceed. A natural and the way we will follow is to obtain this energy shifts from the connection between the covariant and equal-time formulations. In order to progress, we semiclassicaly expand our equations. After plugging the energy shift terms we are able to express the full set of the first order components of distribution functions in terms of the vector and axial vector distribution functions. Finally, we find the zeroth and first order kinetic equations for vector and axial vector distribution functions. By fixing the spatial component of the axial vector distribution function, we find the kinetic equation for the temporal component of this function. A special choice of the spatial axial vector distribution function allows us to establish kinetic equations for the left- and right-handed distribution functions. We can find the rate of changes of phase space variables. Hence insofar, we obtain the current density for the equilibrium chiral distribution function and therefore calculate it both in its zero temperature and massless limits. The results in these limits are compatible with the ones obtained in other studies.