Equivalent sets of solutions of the Dirac equation with a constant electric field

Abstract

Two families of sets, nonstationary and stationary, are obtained. Each nonstationary set $ψ_{p_v}$ consists of the solutions with the quantum number $p_v=p^0v-p_3.$ It can be obtained from the nonstationary set $ψ_{p_3}$ with quantum number $p_3$ by a boost along $x_3$-axis (along the direction of the electric field) with velocity $-v$. Similarly, any stationary set of solutions characterized by a quantum number $p_s=p^0-sp_3$ can be obtained from stationary solutions with quantum number $p^0$ by the same boost with velocity $-s$. All these sets are equivalent and the classification (i.e. ascribing the frequency sign and in-, out- indexes) in any set is determined by the classification in $ψ_{p_3}$-set, where it is beyond doubt.