Magnetized Dirac Electron
In this chapter the Dirac equation is solved for an electron in the presence of a magnetostatic field. To solve the Dirac equation requires a specific choice of gauge for the magnetostatic field, and a specific choice of the spin operator. It is then possible to separate the wavefunction into a gauge- and spin-dependent factor and a reduced wavefunction that satisfies a reduced form of the Dirac equation that is independent of the choice of gauge and spin operator. In generalizing QED to include the magnetic field exactly, the conventional momentum representation for the Feynman amplitudes is not available, because momentum perpendicular to the magnetic field is not conserved. However, the separation of the wavefunction allows an analogous separation of the electron propagator and the vertex functions, with the gauge-independent part closely analogous to the momentum-space representation in the unmagnetized case. The gauge-dependent part (partially) describes the location of the center of gyration of the electron, and how it changes in a QED interaction, and such information is rarely of interest, and is simply ignored when using the reduced theory.
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