Schwinger's result on particle production from complex paths WKB approximation

This paper presents the derivation of Schwinger's gauge-invariant result of Im L-eff up to 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 authors(1) looks upon particle production as being due to the motion of a particle in complex space-time plane, thereby requiring tunneling paths both in space and time. Recently(2,3) there have been some efforts to calculate the reflection and the transmission coefficients for particle production in an 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.