Dirac Electron in External Electromagnetic Potentials

The covariant form of the Dirac equation of a freely moving particle, Eq. (5.54), allows us to incorporate arbitrary external electromagnetic fields. These fields can then be used to describe the interaction of electrons with light. But it must be noted that the treatment of light is then purely classical. A fully quantized description of light and matter on an equal footing is introduced only in quantum electrodynamics and discussed in chapter 7. 

For the Dirac equation with external electromagnetic fields in covariant form it is appropriate to define suitable 4-quantities, In this way, time and spa- 

The components of the 4-potential are given by AT (, A). Note that the vector potential A = A, A, A ) contains the contravariant components of the 4-potential. According to what follows Eq. (5.54), we need to add a Lorentz scalar to the (scalar) Dirac Hamiltonian in order to preserve Lorentz covariance. This Lorentz scalar shall depend on the 4-potential. The simplest choice is a linear dependence on the 4-potential and by multiplication with 7H we obtain the desired Lorentz scalar. Minimal coupling thus means the following substitution for the 4-momentum operator 

we have a unified substitution pattern at hand, which also comprises the time-like coordinates. Substitution of Eq. (5.116) in the field-free Dirac equation as written in Eq. (5.54) yields the covariant form of the Dirac equation with external electromagnetic fields. 

This equation can be transformed to a more familiar form by separating spatial and time coordinates 

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