Equivalent sets of solutions of the Klein-Gordon equation with a constant electric field

Abstract

In connection with the problem of choosing the in- and out-states among the solutions of a wave equation with one-dimensional potential we study nonstationary and "stationary" families of complete sets. A nonstationary set consists of the solutions with the quantum number $p_v=p^0v-p_3.$ It can be obtained from the nonstationary set with quantum number $p_3$ by a boost along $x_3$-axis (along the direction of the electric field) with velocity $-v$. By changing the gauge the solutions in all sets can be brought to one and the same potential without changing quantum numbers. Then the transformations of solutions of one set (with quantum number $p_v$) to the solutions in another set (with quantum number $p_{v'}$) have the group properties. The "stationary" solutions and sets possess the same properties as the nonstationary ones and are obtainable from stationary solutions with quantum number $p^0$ by the same boost. It turns out that any set can be obtained from any other by gauge manipulations. So all 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 $p_3$-set, where it is evident.