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

TL;DR

The paper demonstrates that various families of solutions to the Dirac equation in a constant electric field are equivalent, with classifications derived through Lorentz boosts along the field direction, unifying stationary and nonstationary solution sets.

Contribution

It introduces a unified framework showing that different solution sets of the Dirac equation are related via Lorentz boosts, clarifying their equivalence and classification.

Findings

All solution sets are equivalent through Lorentz transformations.

Classification of solutions is consistent across different sets.

The approach simplifies understanding of Dirac solutions in electric fields.

Abstract

Two families of sets, nonstationary and stationary, are obtained. Each nonstationary set $\psi_{p_v}$ consists of the solutions with the quantum number $p_v=p^0v-p_3.$ It can be obtained from the nonstationary set $\psi_{p_3}$ 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 $\psi_{p_3}$-set, where it is beyond doubt.