In-in worldline formalism in pair creating fields

An in-in framework under Schwinger pair creating fields in strong-field quantum electrodynamics is formulated using in-out propagators in coordinate space, that have first-quantized or worldline representation. The framework is derived to all orders in the background field coupling from both the Bogoliubov coefficient method and Schwinger-Keldysh closed-time path formalism. In-out matrix elements in pair creating fields are readily handled using first-quantized methods, and the approach we develop serves to facilitate the evaluation of in-in observables in pair creating backgrounds. We find that in-in augmentations to the in-out partition function and or propagator amount to the insertion of a non-local interaction term that sandwiches a function that serves to enclose singularities in complex Schwinger propertime. Furthermore, we show the resummation of the in-in partition function leading to vacuum non-persistence that en-route gives an exact first-quantized definition of creating $N$-pairs.