Quantum Spring from the Casimir Effect

TL;DR

This paper explores the quantum spring effect arising from the Casimir force in a space with helix topology, revealing spring-like behavior at small ratios and gravity-like behavior at large ratios, using zeta function techniques.

Contribution

It introduces the concept of a quantum spring from the Casimir effect in helix topology and analyzes its behavior in two and three dimensions.

Findings

Casimir force acts like a spring at small pitch-to-circumference ratio

Force behaves like Newtonian gravity at large ratio

Both forces are attractive and consistent across dimensions

Abstract

The Casimir effect arises not only in the presence of material boundaries but also in space with nontrivial topology. In this paper, we choose a topology of the flat $(D+1)$-dimensional spacetime, which causes the helix boundary condition for a Hermitian massless scalar field. Especially, Casimir effect for a massless scalar field on the helix boundary condition is investigated in two and three dimensions by using the zeta function techniques. The Casimir force parallel to the axis of the helix behaves very much like the force on a spring that obeys the Hooke's law when the ratio $r$ of the pitch to the circumference of the helix is small, but in this case, the force comes from a quantum effect, so we would like to call it \textit{quantum spring}. When $r$ is large, this force behaves like the Newton's law of universal gravitation in the leading order. On the other hand, the force perpendicular to the axis decreases monotonously with the increasing of the ratio $r$. Both forces are attractive and their behaviors are the same in two and three dimensions.