Topological Blocking of the Schwinger Effect in the Salpeter Equation: A Lefschetz Thimble Analysis

Abstract

We present a comprehensive Lefschetz thimble analysis of the one-dimensional Salpeter equation under a strong electric field. By treating the non-local square-root operator within the framework of algebraic analysis, we construct the full solution space, which includes relativistic generalizations of the Airy Ai and Bi functions and their negative-energy counterparts. Through a direct comparison with the Dirac and Klein-Gordon equations, we provide a geometric explanation for the absence of Klein paradox and the Schwinger effect in the Salpeter equation. Furthermore, our findings establish a unified geometric interpretation of the Schwinger effect across different relativistic wave equations.