Dirac parçacıklarının yarı klasik kinetik kuramı

The semiclassical kinetic theory of massive spin-1/2 particles interacting with the external electromagnetic fields is formulated in terms of differential forms which are matrix valued in spin space. Semiclassical approximation is performed by employing the wave packet constructed as superposition of positive energy plane wave solutions of the free Dirac equation. A symplectic two-form is derived using the wave packet. It is a matrix in "spin indices" and possesses a term related to the Berry curvature obtained from a non-Abelian Berry gauge field. Time evolution of phase space variables in terms of phase space themselves are attained by making use of the volume form which is also a matrix. Continuity equation for particle number density and the particle current density are obtained by introducing a change of basis in order to define distribution functions in the helicity basis. The massless limit is derived by constructing the helicity states explicitly. When one deals with a non-relativistic formulation of massive particles the equations of motion should be corrected with a relativistic kinematic factor known as Thomas precession. Its origin lies in the fact that when one would like to write two successive Lorentz boost as one Lorentz boost it should be accompanied with a rotation whose angle depends on the related velocities. It is shown that Thomas precession can be included straightforwardly into the semiclassical formulation adopted in the thesis. It alters the equations of motion and cancels the anomalous velocity terms appearing due to the Berry curvature. Initially, I will derive the semiclassical block diagonal Hamiltonian for a Dirac particle in the electromagnetic field including all terms at the first order in Planck constant using the Gosselin-Berard-Mohrbach method. In this method the unitary transformation which block diagonalizes the Hamiltonian possesses terms related to the Berry gauge fields. In general curvature of the Berry gauge fields appear as the phase factor of a quantum state transported adiabatically. When the block diagonalization is carried out by unitary transformation, the dynamical operators should also be transformed and they become non-commutative. I will use these non-commutative phase space operators to derive the time evolution of spin matrices which will be introduced in the course of finding the semiclassical formulation. The one-form corresponding to first order Lagrangian is defined by making use of the wave packet built with the positive energy solutions of the Dirac equation. This one-form can be written as a matrix whose indices correspond to the positive energy solutions which are called spin indices. It has a term depending on the non-Abelian Berry gauge fields given by the degenerate positive energy solutions. Then the symplectic two-form derived from this one-form includes a term which depends on the Berry curvature. I use the differential form formalism to obtain the equations of motion of phase space variables. A straightforward method is applied to find solutions of the equations of motion for the phase space velocities in terms of the phase space variables employing Liouville equation and the differential form formalism. To get the kinetic theory of Dirac particles I need the distribution function which can be used to define the particle number density. However, it is a matrix whose elements should be interpreted appropriately. The mostly adopted procedure is to choose a specific configuration where the third component of spin is a conserved quantity. Then one can set the off-diagonal terms to zero. In general spin is not a conserved quantity but helicity operator gives a vanishing commutator with the free Dirac Hamiltonian. Moreover when I discuss the massless limit it would be essential to split the right-handed and the left-handed contributions. Therefore, the appropriate basis is the one where the helicity operator is diagonal. I define this new basis and obtain the continuity equation for the Dirac particle using the distribution function which is diagonal. Then I derive the continuity equation for the particle number density and the particle number current density. Obviously, because of possessing the solutions of the equations of motion for the velocities in terms of phase space variables one can directly obtain the particle current. Obtaining themassless limit in the helicity basis is straightforward. It yields the continuity equation which has an anomaly term. The particle current possesses an anomalous velocity term and a term leading to the chiral magnetic effect. Thomas precession which shows up as the relativistic correction in the equations of motion are obtained. I briefly discuss what is the source of the Thomas precession. Then I present how one should introduce this correction into the wave packet formalism. It gives a contribution to the initial one-form on the same footing with the Berry gauge field. In fact up to higher order terms in momentum it gives the opposite contribution of the Berry gauge field and cancels the anomalous velocity terms given by the Berry curvature. This result coincides with the ones obtained within the relativistic formulations of the Dirac particles. Originally the Thomas precession is used to obtain the corrections to the non-relativistic formulation of the time evolution of spin matrices. However, the formalism which I adopted is not aware of the time evolution of spin. For completeness I show that it can be integrated into the formalism by making use of the non-commutative charter of the dynamical variables obtained in the Gosselin-Berard-Mohrbach method. Time evolution of spin matrices are shown to be the same with the Bargmann-Michel-Telegdi equation. Lastly, the results obtained in the thesis and the possible extensions are discussed.