Wigner Function For Spin-1/2 Fermions In Electromagnetic Fields

We study the Wigner function for massive spin-1/2 fermions in electromagnetic fields.The covariant Wigner function is a four by four matrix function in 8-dimensional phase space {x?,p?},whose components give various physical quantities such as the particle distribution,the current density,and the spin distribution,etc..The kinetic equations for the Wigner function are obtained from the Dirac equation.We derive the Dirac form equations with first order differential operators,as well as the Klein-Gordon form equations with second order differential operators,both are matrix equations in Dirac space.We prove that some component equations are automatically satisfied if the rest are fulfilled,which means both the Dirac form and the Klein-Gordon form equations have redundancy.In this thesis two methods are proposed for calculating the Wigner function,which are proved to be equivalent.In addition to the covariant Wigner function,the equal-time Wigner function will also be introduced.The equal-time one is a function of time and 6-dimensional phase space variables {x,p},which can be derived from the covariant one by taking an integration over energy p0.The equal-time Wigner function is not Lorentz-covariant but it is a powerful tool to deal with dynamical problems.In this thesis,it is used to study the Schwinger pair-production in the presence of an electric field.The Wigner function can be analytically calculated following the standard second-quantization procedure.We consider three cases:free fermions with or without chiral imbalance,and fermions in constant magnetic field with chiral imbalance.The com-putations are achieved via firstly deriving a set of orthonormal single-particle wave-functions from the Dirac equation,then constructing the quantized field operator,and finally inserting the field operator into the Wigner function and determine the expecta-tion values of operators under the wave-packet description.The Wigner functions are computed to leading order in spatial gradients.In Strong electric field the vacuum can decay into a pair of particle and anti-particle.The pair-production process is studied using the equal-time Wigner function.General solutions are obtained for pure constant electric fields and for constant parallel electromagnetic fields.We also solve the case of a Sauter-type electric field numerically.For an arbitrary space-time dependent electromagnetic field,the Dirac equation does not have an analytical solution and neither has the Wigner function.A semi-classical expansion with respect to the reduced Planck's constant h are performed for the Wigner function as well as the kinetic equations.We calculate the Wigner func-tion(and all of its components equivalently)to leading order in h,in which order the spin component start playing a role.Up to this order,the Wigner function contains four independent degrees of freedoms,three of which describe the polarization density and the remaining one describes the net particle number density.A generalized Bargmann-Michel-Telegdi(BMT)equation and a generalized Boltzmann equation are obtained for these undetermined parts,which can be used to construct spin-hydrodynamics in the future.Using analytical results and semi-classical solutions,we compute physical quanti-ties in thermal equilibrium.In semi-classical expansion,we introduce the chiral chem-ical potential ?5 in the thermal distribution.This naive treatment is straightforward extension of the massless case but provides a good estimate of physical quantities when?5 is comparable or smaller than the typical energy scale,i.e.,the temperature in a thermal system.Meanwhile,by making comparison of the results of the semi-classical expansion and the ones in a constant magnetic field,we find that the semi-classical method works well for the chiral effects,including the Chiral Magnetic Effect,the Chi-ral Separation Effect,as well as the energy flux along the direction of the magnetic field.But when the mass and chemical potentials are much larger than the temperature,the semi-classical results over estimate these chiral effects.The magnetic field strength de-pendence of physical quantities is discussed.If we fix the thermodynamical variables,the net fermion number density,energy density,and the longitudinal pressure are pro-portional to the field strength,while the axial-charge density and the transverse pressure are inversely proportional to it.Schwinger pair-production rates in a thermal background are computed for a Sauter-type electric field and a constant parallel electromagnetic field,respectively.For the Sauter-type field,the total number of newly generated pairs is proportional to the field strength and the life time of the field.On the other hand,a parallel magnetic field will enhance the pair-production rate.Due to Pauli's exclusion principle,the creation of pairs is forbidden for particles already exsiting in the same quantum state.Thus in both cases,the pair-production rate is proved to be inversely proportional to the chemical potential and temperature.