Vacuum currents in curved tubes

TL;DR

This paper analyzes how spatial curvature and topology influence vacuum currents in charged scalar fields on curved tubes, providing explicit formulas and examining effects for different geometries and radii.

Contribution

It introduces a comprehensive representation of the Hadamard function for curved tubes with topological effects and explores vacuum current behavior across various geometries and curvature scales.

Findings

Vacuum current is periodic with magnetic flux, with a flux quantum period.

Curvature effects are significant for larger tube radii, altering current decay behavior.

Power-law decay of current density occurs for large radii, contrasting with exponential decay in constant radius tubes.

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 rotationally symmetric 2D curved tubes. For a general spatial geometry and for quasiperiodicity condition with a general phase, the representation of the Hadamard function is provided where the topological contribution is explicitly extracted. As an important local characteristic of the vacuum state the expectation value of the current density is studied. The vacuum current is a periodic function of the magnetic flux enclosed by the tube with the period of flux quantum. The general formula is specified for constant radius and conical tubes. As another application, we consider the Hadamard function and the vacuum current density for a scalar field on the Beltrami pseudosphere. Several representations are provided for the corresponding expectation value. For small values of the proper radius of the tube, compared with the curvature radius, the effect of spatial curvature on the vacuum current is weak and the leading term in the corresponding expansion coincides with the current density on a constant radius tube. The effect of curvature is essential for proper radii of the tube larger than the radius of spatial curvature. In this limit the fall-off of the current density, as a function of the proper radius, follows a power-law for both massless and massive fields. This behavior is in clear contrast to the one for a constant radius tube with exponential decay for massive fields. We also compare the vacuum currents on the Beltrami pseudosphere and on locally de Sitter and anti-de Sitter 2D tubes.