In-in worldline formalism in pair creating fields

TL;DR

The authors formulate a real-time in-in framework for Schwinger pair production in strong-field QED by recasting in-in observables in terms of in-out propagators with a worldline representation. They establish two equivalent derivations—via Bogoliubov coefficients and via the Schwinger-Keldysh closed-time path—showing that in-in augmentations correspond to inserting a non-local interaction that captures vacuum instability, allowing all-orders resummation. The resulting first-quantized structures yield a compact expression for the N-pair production probability through Bell polynomials and determinants, and reproduce familiar Schwinger results in explicit backgrounds. The work thus extends the worldline formalism to real-time, non-equilibrium settings and provides a foundation for further extensions to non-Abelian, curved, or open-line worldline configurations.

Abstract

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.

Figures (2)

• Figure 1: (a) Contour for the Schwinger propertime kernel given for the anti-commutation function, $G(x,y)$, in Eq. \ref{['eq:anticommutationWL']}. (b) Contour for the density function serving to extract singularities associated with Schwinger pair production in the imaginary Schwinger propertime plane. The contour of $\rho_h(x,y)$, in Eq. \ref{['eq:rhoh']}, is shown in red. The closed contour includes the contour in black, which is permissible for fields in the absence of branch cuts that might otherwise forbid the closure.
• Figure 2: SK closed-time contour $\mathcal{C}$. Causal half, $+$, extends from $x_0^\text{in}\to-\infty$ to $x_0^\text{out}\to\infty$, and anti-causal half, $-$, in the opposite direction. Both $\pm$ paths are non-trivially connected at $x_0^\text{out}$, leading to for example for Dirac operators the following BC: