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

TL;DR

This work derives exact Schwinger pair production rates for charged scalars and spinors in uniform electric fields on ${\rm dS}_2$ and ${\rm AdS}_2$, using the $SO(2,1)/SO(1,2)$ symmetry to obtain closed-form Bogoliubov coefficients in both global and planar coordinates. The mean numbers of produced pairs are expressed in simple hyperbolic-function forms and are shown to satisfy a reciprocal relation between dS$_2$ and AdS$_2$ under analytic continuation of curvature, illuminating a deep link between QED effects and spacetime geometry. The analysis employs both exact mode solutions (via hypergeometric/Whittaker functions) and phase-integral methods to interpret the leading Boltzmann factors as contour-residue contributions, clarifying how the Maxwell field and curvature jointly govern pair production. The results unify the treatment across coordinate patches and geometries, quantify BF-type constraints in AdS, and suggest a curvature-analytic continuation perspective on QED in curved backgrounds with potential implications for the one-loop effective action in (A)dS spaces.

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.