Electromagnetic Interaction

We introduce the coupling to an electromagnetic field to obtain an interpretation of the internal structure of Dirac particles. Consider the interaction of a point charge $e$ with an external electromagnetic field. The 4-potential is $A^\mu = (A_0,\vec{A})$, where $A^\mu$ is a function of $\vec{x}$ only. We make the usual gauge invariant minimal substitution

$$\hat{p}^\mu \rightarrow \hat{p}^\mu -\frac{e}{c} A^\mu ,$$ (5.20)

where $e$ is the magnitude of the charge of an electron. The Dirac equation becomes

$$c\left[ i\hbar\frac{\partial}{\partial(ct)} - \frac{e}{c} A_... ...( \hat{p} - \frac{e}{c}\vec{A}\right) + \beta mc^2\right] \psi$$ (5.21)

or

$$i\hbar\frac{\partial}{\partial t}\psi = \left[ c\vec{\alpha}... ...\frac{e}{c}\vec{A} \right) + eA_0 + \beta mc^2 \right] \psi .$$ (5.22)

This equation contains the interaction with the electromagnetic field:

$$i\hbar \frac{\partial}{\partial t} \psi = \left( \hat{H} + \hat{H}^\prime \right) \psi ,$$ (5.23)

where

$$\hat{H}^\prime = -\frac{e}{c} c\vec{\alpha}\cdot\vec{A} + eA_0 = -\frac{e}{c}\hat{v}\cdot\vec{A} + eA_0 ,$$ (5.24)

and

$$\hat{v} = c\vec{\alpha}$$ (5.25)

is the classical correspondence of the relativistic velocity operator. We can see this by looking at the relativistic extension of the Ehrenfest relation:

$$\frac{d\hat{x}}{dt} = \frac{i}{\hbar} [\hat{H},\hat{x}] = \f... ...ac{ci}{\hbar}\alpha_\mu[\hat{p}^\mu,\hat{x}] = c\vec{\alpha} .$$ (5.26)