Scalar vacuum densities on Beltrami pseudosphere

TL;DR

This work analyzes how spatial curvature and a compact azimuthal dimension affect the vacuum of a charged scalar field on the (2+1)-D Beltrami pseudosphere under a constant-phase quasiperiodicity. By constructing Hadamard-function-based representations, the authors separate divergent uncompactified contributions from finite topological parts in the VEVs of $\langle \varphi^{2}\rangle$ and $\langle T_{i}^{\,i}\rangle$, with detailed asymptotics for $r/L$ both small and large. In the small-radius limit, topological terms decay as a power law (except for the conformally coupled massless case where stresses can grow due to logarithmic behavior), while in the large-radius limit they exhibit curvature- and mass-independent power-law growth, mirroring cylindrical-tube results. The analysis also connects the Beltrami pseudosphere to locally Rindler spacetime under conformal transformation, underscoring the role of topology in Casimir-type effects on curved backgrounds and informing potential condensed-matter analogues and holographic studies.

Abstract

We investigate the combined effects of spatial curvature and topology on the properties of the vacuum state for a charged scalar field localized on the (2+1)-dimensional Beltrami pseudosphere, assuming that the field obeys quasiperiodicity condition with constant phase. As important local characteristics of the vacuum state the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor are evaluated. The contributions in the VEVs coming from geometry with an uncompactified azimuthal coordinate are divergent, whereas the compact counterparts are finite and are analysed both numerically and asymptotically. For small values of proper radius of the compactified dimension, the leading terms of topological contributions are independent of the field mass and curvature coupling parameter, increasing by a power-law. In the opposite limit, the VEVs decay following a power-law in the general case. In the special case of a conformally coupled massless field the behavior is different. Unlike the VEV of field squared and vacuum energy density, the radial and azimuthal stresses are increasing by absolute value. As a consequence, the effects of nontrivial topology are strong for the stresses in this case at small values of radial coordinate.

Figures (7)

• Figure 1: The Beltrami pseudosphere embedded in a 3-dimensional Euclidean space.
• Figure 2: The VEV of field squared (left panel) and the azimuthal current density (right panel) as a function of the $L/r$ for conformally ($\xi=1/8$) and minimally ($\xi=0$) coupled massless fields and for fixed $\alpha_{p}=\pi/2$.
• Figure 3: The VEV of field squared (left panel) and the azimuthal current density (right panel) as a function of the mass for conformally ($\xi=1/8$) and minimally ($\xi=0$) coupled fields with $L/r=0.5$ and $\alpha_{p}=2\pi/5$.
• Figure 4: The VEV of field squared (left panel) and the azimuthal current density (right panel) as a function of the phase in the quasiperiodicity condition for conformally ($\xi=1/8$) and minimally ($\xi=0$) coupled fields with $L/r=0.5$ and $ma=0.5$.
• Figure 5: The energy density as a function of the mass (left panel) and of the phase in the quasiperiodicity condition (right panel) for $L/r=0.5$. For the left panel we have taken $\alpha_{p}=2\pi/5$ and for the right panel $ma=0.5$.