Schwinger's Result On Particle Production From Complex Paths WKB Approximation

TL;DR

This paper derives Schwinger's gauge invariant particle production rate in a uniform electric field using the complex trajectory WKB approximation, clarifying previous discrepancies and establishing the method's correctness.

Contribution

It introduces and validates the complex trajectory WKB approximation for calculating particle production, providing a gauge invariant derivation of Schwinger's result.

Findings

Derived the imaginary part of the effective Lagrangian using CWKB.

Clarified and confirmed the correctness of CWKB over other methods.

Established gauge invariance in the particle production calculation.

Abstract

This paper presents the derivation of Schwinger's gauge invariant result of $Im \cal{L}_{eff}$ upto one loop approximation, for particle production in an uniform electric field through the method of complex trajectory WKB approximation (CWKB). The CWKB proposed by one of the author's \cite{bis:ijtp} looks upon particle production as due to the motion of a particle in complex space-time plane, thereby requiring tunneling paths both in space and time. Recently \cite{srini:iucaa,srini1:iucaa} there have been some efforts to calculate the reflection and transmission co-efficients for particle production in uniform electric field that differ from our expressions for the same. In this paper we clarify the confusion in this regard and establish the correctness of CWKB.