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The phase-integral and worldline-instanton methods are two widely usedmethods to calculate Schwinger pair-production densities in electric fields offixed direction that depend on just one time or space coordinate in the samefixed plane of the electromagnetic field tensor. We show that for chargedspinless bosons the leading results of the phase-integral method integrated upthrough quadratic momenta are equivalent to those of the worldline-instantonmethod including prefactors. We further apply the phase-integral method tofermion production and time-dependent electric fields parallel to a constantmagnetic field.

Equivalence between the phase-integral and worldline-instanton methods