Equivalent Sets of Solutions of the Klein–Gordon Equation with a Constant Electric Field

We argue extensively in favor of our earlier choice of the in and out states (among the solutions of a wave equation with one-dimensional potential). In this connection, we study the nonstationary and “stationary” families of complete sets of solutions of the Klein–Gordon equation with a constant electric field. A 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 the quantum number $p_3$ by a boost along the $x_3$ axis (in the direction of the electric field) with the velocity $-v$. By changing the gauge, we can bring the solutions in all sets to the same potential without changing quantum numbers. Then the transformations of solutions in one set (with the quantum number $p_v$) to the solutions in another set (with the quantum number $p_{v'}$) have group properties. The stationary solutions and sets have the same properties as the nonstationary ones and are obtainable from stationary solutions with the quantum number $p^0$ by the same boost. It turns out that each set can be obtained from any other by gauge manipulations. All sets are therefore equivalent, and the classification (i.e., assigning the frequency sign and the in and out indices) in any set is determined by the classification in the set $\psi_{p_3}$, where it is obvious.