Exact one-loop QED actions in global $\mathrm{(A)dS}_2$

Abstract

Using the in-out formalism, we derive the exact one-loop QED effective actions for spinor field in a uniform electric field in two-dimensional global (anti-)de Sitter (A)dS$_2$ spacetime. The one-loop effective action probed by a scalar or spinor field is determined by the scattering matrix relating the out-vacuum to the in-vacuum, which is in turn fixed by the Bogoliubov coefficients of the corresponding Klein-Gordon or Dirac equation in the presence of both a gauge field and curved spacetime. Remarkably, the vacuum persistence amplitude -- twice the imaginary part of the one-loop effective action -- is related, via the Bogoliubov relations, to the mean number of particle-antiparticle pairs spontaneously produced by the background fields. The Bogoliubov coefficients or mean number of pair-production for charged scalar and spinor fields in global (A)dS$_2$ lead to QED effective actions expressed in terms of both proper-time integrals and Hurwitz zeta functions. These effective actions reveal a strong interplay between the electric field and spacetime curvature and correctly reproduce the limiting cases of pure (A)dS$_2$ spacetime and a uniform electric field in Minkowski space. We further discuss the physical implications of the resulting QED effective actions in (A)dS$_2$.

Figures (5)

• Figure 1: The contour for the integrations \ref{['eq_ConI']} and \ref{['eq_ConII']}.
• Figure 2: The plots of real part (a) and imaginary part (b) of $\mathcal{W}^{\rm (sp)}_{\rm dS}$ against $E$ and $H$ for spinor QED in dS$_2$ in unit of $m = 1$ and $q = 1$. [Left panel] The one-loop effective action increases from $\mathrm{Re} \mathcal{W}^{\rm (sp)}_{\rm dS}(E=0, H=0) = 0$ as $E$ and $H$ increase. The increase is more significant for $H$ than $E$, but the action almost saturates for large $H$ regardless of $E$. The curvature effect ($R =2 H^2$) is more noticeable for weak field than for strong electric field. [Right panel] The vacuum decays significantly as $H$ increases and gently as $E$ increases.
• Figure 3: The plots of real part (a) and imaginary part (b) of $\mathcal{W}^{\rm (sp)}_{\rm AdS}$ against $E$ and $H$ for spinor QED in AdS$_2$ in unit of $m = 1$ and $q = 1$. The necessary condition for pair production is $q E > m H$, violation of BF bound. [Left panel] The effective action increases more significantly as $H$ increases than as $E$ increases. Except for the small curvature region and Maxwell scalar, the curvature effect dominates over the effect of the Maxwell scalar in the region that violates the BF bound. [Right panel] The vacuum decays more rapidly for large curvature than for large Maxwell scalar.
• Figure 4: Connection of QED actions between dS$_2$ and AdS$_2$ and reduction to QED action in Minkowski spacetime and pure dS$_2$.
• Figure 5: The plots show the difference between real parts (a) [left panel] and imaginary parts (b) [right panel] of $\mathcal{W}^{\rm (sp)}_{\rm dS}$ and $\mathcal{W}^{\rm (sp)}_{\rm AdS}$ against $E$ and $H$ for spinor QED in dS$_2$ and AdS$_2$ in units of $m = 1$ and $q = 1$. In both plots the domains are constrained by the necessary condition for pair production in AdS$_2$ that is $q E > m H$, violation of BF bound.