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We demonstrate that the positive frequency modes for a complex scalar fieldin a constant electric field (Schwinger modes), in three different gauges, canbe represented as exact Lorentzian worldline path integral amplitudes. Althoughthe mathematical forms of the mode functions differ in each gauge, we show thata simple prescription for Lorentzian worldlines' boundary conditions dispensesthe Schwinger modes in all three gauges (that we considered) in a unifiedmanner. Following that, using our formalism, we derive the exact Bogoliubovcoefficients and, hence, the particle number, \textit{without} appealing to thewell-known connection formulas for parabolic cylinder functions. This result isespecially relevant in view of the fact that in a general electromagnetic fieldconfiguration, one does not have the luxury of closed-form solutions. We arguethat the real time worldline path integral approach may be a promisingalternative in such non-trivial cases. We also demonstrate, usingPicard-Lefschetz theory, how the so-called worldline instantons emergenaturally from relevant saddle points that are complex.

Lorentzian worldline path integral approach to Schwinger effect