Abstract
Recalculating the Bogoliubov coefficients from the solutions in Kim and Page (2008), we obtain the mean number of boson pairs in a uniform electric field in the global coordinates of the dS2 and AdS2 spaces, which have the correct zero-field and zero-curvature limits. We then study the vacuum persistence, twice of the imaginary part of the effective action, at the one-loop level. The mean number in the AdS2 space gives the lowest limit to the Breitenloher–Freedman bound in the uniform electric field within which the AdS space is stable against the Schwinger pair production. Remarkably the mean numbers in the dS2 and AdS2 spaces satisfy the reciprocal relation NdS(R,E)NAdS(R,E)=1 under an analytical continuation of the scalar curvature R.
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