Induced energy-momentum tensor of the scalar field in 3D de Sitter QED

TL;DR

We compute the renormalized energy–momentum tensor $\\langle T_{\\mu\\nu}\\rangle_{\\mathrm{ren}}$ for a charged scalar field in three-dimensional de Sitter spacetime with a uniform electric field, using adiabatic regularization. By constructing exact in-vacuum mode functions in terms of Whittaker functions and subtracting second-order adiabatic counterterms, we obtain finite, covariantly conserved expressions for $T_{00}$, $T_{11}$, and $T_{22}$, with $T_{01}$ vanishing after renormalization. The strong-field regime shows Schwinger-like growth with the field parameter $\\lambda$, while the infrared regime reveals mass-dependent infrared divergences as $m\to 0$, and the trace $T^{\\mu}_{\\mu}$ vanishes in the massless, conformally coupled limit, confirming no genuine Weyl anomaly in odd dimensions. Overall, the results illuminate vacuum polarization and backreaction in a tractable $dS_3$ setting and connect nonperturbative pair production to curved-spacetime quantum effects.

Abstract

In this work, we investigate the renormalized energy--momentum tensor of a quantized charged scalar field in three-dimensional de Sitter spacetime $\mathrm{dS}_{3}$ under the influence of a uniform electric field. Using the adiabatic regularization method, we systematically remove ultraviolet divergences and obtain explicit finite expressions for the components of the induced energy--momentum tensor. The numerical analysis demonstrates that the renormalized tensor behaves smoothly with respect to the parameters of the system and exhibits physically consistent limits in both the strong-field and infrared regimes. The induced energy density grows with the field strength and follows a quadratic behavior, which is consistent with the Schwinger mechanism in three dimension. In the opposite infrared regime, the tensor components display inverse-mass dependence, revealing infrared divergences typical of nearly massless scalar fields in curved space. Finally, we evaluate the trace of the renormalized tensor and show that for a massless, conformally coupled scalar field the trace anomaly vanishes, confirming the absence of a genuine Weyl anomaly in odd-dimensional spacetimes. These results provide a consistent and covariant description of quantum vacuum polarization and backreaction effects in three-dimensional de Sitter geometry.

Figures (3)

• Figure 1: Plot of the absolute value of the $T_{00}$ component of the induced energy-momentum tensor is shown for the electric field parameter $\lambda=-eE/H^{2}$. The solid line $\xi=\frac{1}{8}$ and the dashed line $\xi=0$ correspond to different values of the mass parameter $\lambda_{m}=m/H$, as indicated. The scales on both axes are logarithmic.
• Figure 2: Plot of the absolute value of the $T_{11}$ component of the induced energy-momentum tensor is shown for the electric field parameter $\lambda=-eE/H^{2}$. The solid line $\xi=\frac{1}{8}$ and the dashed line $\xi=0$ correspond to different values of the mass parameter $\lambda_{m}=m/H$, as indicated. The scales on both axes are logarithmic.
• Figure 3: Plot of the absolute value of the $T_{22}$ component of the induced energy-momentum tensor is shown for the electric field parameter $\lambda=-eE/H^{2}$. The solid line $\xi=\frac{1}{8}$ and the dashed line $\xi=0$ correspond to different values of the mass parameter $\lambda_{m}=m/H$, as indicated. The scales on both axes are logarithmic.