Schwinger effect in de Sitter space

We consider Schwinger pair production in 1+1 dimensional de Sitter space, filled with a constant electric field E. This can be thought of as a model for describing false vacuum decay beyond the semiclassical approximation, where pairs of a quantum field ϕ of mass m and charge e play the role of vacuum bubbles. We find that the adiabatic ``in" vacuum associated with the flat chart develops a space-like expectation value for the current J, which manifestly breaks the de Sitter invariance of the background fields. We derive a simple expression for J(E), showing that both ``upward" and ``downward" tunneling contribute to the build-up of the current. For heavy fields, with m<sup>2</sup> ≫ eE,H<sup>2</sup>, the current is exponentially suppressed, in agreement with the results of semiclassical instanton methods. Here, H is the inverse de Sitter radius. On the other hand, light fields with m ≪ H lead to a phenomenon of infrared hyperconductivity, where a very small electric field mH≲eE ≪ H<sup>2</sup> leads to a very large current J ∼ H<sup>3</sup>/E. We also show that all Hadamard states for ϕ necessarily break de Sitter invariance. Finally, we comment on the role of initial conditions, and ``persistence of memory" effects.