Reciprocal relation of Schwinger pair production between $\textrm{dS}_2$ and $\textrm{AdS}_2$

TL;DR

This paper explores the symmetry and reciprocal relations of Schwinger pair production in two-dimensional de Sitter and anti-de Sitter spaces, revealing how electric fields influence particle creation differently in these curved spacetimes.

Contribution

It demonstrates a reciprocal relation of pair production between dS$_2$ and AdS$_2$, and connects their behaviors through analytical continuation of spacetime curvature.

Findings

Electric fields enhance pair production in dS space.

Weak electric fields below the BF bound suppress pair production in AdS space.

The mean number of produced pairs relates to contour integrals and residue sums.

Abstract

The Klein-Gordon and Dirac equation for a massive charged field in a uniform electric field has a symmetry of two-dimensional global de Sitter (dS) and anti-de Sitter (AdS) space. In the in-out formalism the mean numbers of spinors (spin-1/2 fermions) and scalars (spin-0 bosons) spontaneously produced by the uniform electric field are exactly found from the Bogoliubov relations both in the global and planar coordinates of (A)dS$_2$ space. We show that the uniform electric field enhances the production of charged spinor and scalar pairs in the planar and global dS space while the AdS space reduces the pair production in which weak electric fields below the Breitenlohner-Freedman (BF) bound prohibits pair production. The leading Boltzmann factor in dS space can be written as the Gibbons-Hawking radiation or Schwinger effect enhanced by e-folding factors less than one that give the QED effect or the curvature effect. We observe that dS$_2$ and AdS$_2$ spaces are connected by QED, such as a reciprocal relation between the mean number of spinors and scalars provided that the spacetime curvature is analytically continued. The leading behavior of the mean numbers for spinors and scalars is explained as a residue sum of contour integrals of the frequency or momentum in the phase-integral formulation.