The Casimir force of Quantum Spring in the (D+1)-dimensional spacetime

TL;DR

This paper investigates the Casimir effect for scalar fields with helix boundary conditions, called quantum spring, in (D+1)-dimensional spacetime, revealing exact energy and force expressions, symmetry properties, and mass dependence.

Contribution

It provides exact formulas for Casimir energy and force in arbitrary dimensions, highlighting differences between odd and even dimensions and the impact of mass.

Findings

Casimir force is attractive in all cases.

Exact expressions involve Bernoulli numbers and zeta functions.

Force depends on spacetime dimensions and scalar field mass.

Abstract

The Casimir effect for a massless scalar field on the helix boundary condition which is named as quantum spring is studied in our recent paper\cite{Feng}. In this paper, the Casimir effect of the quantum spring is investigated in $(D+1)$-dimensional spacetime for the massless and massive scalar fields by using the zeta function techniques. We obtain the exact results of the Casimir energy and Casimir force for any $D$, which indicate a $Z_2$ symmetry of the two space dimensions. The Casimir energy and Casimir force have different expressions for odd and even dimensional space in the massless case but in both cases the force is attractive. In the case of odd-dimensional space, the Casimir energy density can be expressed by the Bernoulli numbers, while in the even case it can be expressed by the $\zeta$-function. And we also show that the Casimir force has a maximum value which depends on the spacetime dimensions. In particular, for a massive scalar field, we found that the Casimir force varies as the mass of the field changes.