In-in formalism with resummation in a constant electric field: propagators including nontrivial boundary wavefunctions

A bstract We present the derivation of an alternative representation of the real-time in-in formalism under a spatially homogeneous and time independent electric field. Because the system exhibits instability associated with pair production of particles and antiparticles, the perturbation theory should be reorganized depending on the choice of the reference vacuum. We recast the boundary wavefunctions into the quadratic self-energy-like terms in the functional integration formalism. The resulting generating functional in the modified in-in formalism leads to the propagators that resum infinite diagrams necessary to capture the vacuum-instability effects. The proper-time representations of the propagators reproduce the known expressions from the canonical operator formalism, but our derivation based on the generating functional along the closed-time path clarifies the origin of the additional proper-time contour and provides a better physical understanding. Finally, as a concrete example of the application, we compute the in-in expectation value of the vector current in a constant electric field, and find that the simple one-loop calculation captures the pair production effect.