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

TL;DR

This paper uses Lefschetz thimble analysis to explore the Salpeter equation under strong electric fields, revealing topological effects that suppress the Schwinger effect and Klein paradox.

Contribution

It provides a geometric interpretation of the Schwinger effect and Klein paradox suppression in the Salpeter equation using advanced algebraic and topological methods.

Findings

Constructed the full solution space including relativistic Airy functions.

Provided a geometric explanation for the absence of Klein paradox.

Unified the interpretation of the Schwinger effect across relativistic wave equations.

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.