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Reciprocal relation of Schwinger pair production between dS2 and AdS2

Chiang-Mei Chen1,2,*, Chun-Chih Huang1,†, Sang Pyo Kim3,2,4,5,‡, and Kuan-Yen Lin1,§

  • *Contact author: cmchen@phy.ncu.edu.tw
  • †Contact author: makedate0809@gmail.com
  • ‡Contact author: sangkim@kunsan.ac.kr
  • §Contact author: kowal2144@gmail.com
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Phys. Rev. D 112, 125028 – Published 24 December, 2025

DOI: https://doi.org/10.1103/nn9m-x8cf

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)dS2 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 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 dS2 and AdS2 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.

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Phys. Rev. D 112, 125028– Published 24 December, 2025
  • Received 21 July 2025
  • Accepted 5 December 2025
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