Abstract
The Casimir effect arises not only in the presence of material boundaries but also in space with nontrivial topology. In this Letter, 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 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.
Highlights
Since the first work on Casimir effect performed by Casimir [1], it has been extensively studied [2] for more than 60 years
We get the scalar field on a flat manifold with topology of a circle S1
We have investigated the Casimir effect with a helix configuration in two and three dimensions, and it can be generalized to high dimensions
Summary
Since the first work on Casimir effect performed by Casimir [1], it has been extensively studied [2] for more than 60 years. In section we have calculated the Casimir energy and force by imposing the helix boundary conditions and we find that the behavior of the force parallel to the axis of the helix is very much like the force on a spring that obeys the Hooke’s law in mechanics when the r ≪ 1, which is the ratio of the pitch h to the circumference a of the helix. In this case, the force comes from a quantum effect, and so we would like to call the helix structure as a quantum spring. We will give some discussions and conclusions in the last section
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