Abstract:
The main purpose of this study is to discuss the properties of the solutions of the Dirac equation and calculate the Dirac propagator in the presence of a uniform background magnetic field. In particular the focus of the work is on the nature of the solutions, their orthonormality properties and how these solutions depend on the choice of the vector potential giving rise to the magnetic field. The spin-sum of the solutions is explicitly calculated. After the Dirac equation is exactly solved then we proceed to calculate the quantization of the solutions. Using these results the Dirac propagator in the presence of a uniform background magnetic field is calculated. The result have shown that the dispersion of the electron is seen to change from its form in the vacuum, the solution of the Dirac equation depends on the landau level, the energy of electron is seen to degenerate except at the lowest landau level, the orthonormality of the spinors closely resembles the corresponding results in vacuum. Comparing our spin-sum result with the result of others researchers work the spin sum also depends on the particular gauge we work in and would be different if we choose another gauge to represent the magnetic field. The Dirac propagator in the presence of magnetic field is different from the free field propagator it also depends on the Landau levels. Most of the quantities, the solutions of the Dirac equation, the normalization condition the spin sum and the propagator in the presence of uniform magnetic field depend on the choice of vector potential giving rise to the magnetic field.
