Generalized quantum spring

Abstract

Recently, it was found that after imposing a helix boundary condition on a scalar field, the Casimir force coming from the quantum effect is linearly proportional to r, which is the ratio of the pitch to the circumference of the helix. This linear behavior of the Casimir force is just like that of the force obeying the Hooke's law on a spring. In this paper, inspiring by some complex structures that lives in the cells of human body like DNA, protein, collagen etc., we generalize the helix boundary condition to a more general one, in which the helix consists of a tiny helix structure, and makes up a hierarchy of helix. After imposing this kind of boundary condition on a massless and a massive scalar, we calculate the Casimir energy and force by using the so-called zeta function regularization method. We find that the Hooke's law with the generalized helix boundary condition is not exactly the same as usual one. In this case, the force is proportional to the cube of r instead. So we regard it as a generalized Hooke's law, which is complied by a generalized quantum spring. PACS numbers: 03.70.+k, 11.10.-z I. INTRODUCTIONWith some boundary conditions, the dynamics and spectrum of a quantum field will be changed, and then it could lead to an observable effect. One of such a kind of phenomena is called the Casimir effect [1][2]. In the classical electrodynamics, the force acting between two planes doesn't exist. However, due to the quantum effect, there exists the zero-point fluctuation of the vacuum and then its spectral density changes with time. As a result, once there are two infinitely large perfectly parallel conducting planes placed in the vacuum, there will be an attractive force between these two planes.The Casimir effect has been studied a lot due to the development of much more precise measurements [3] and technological advancements since the last decade. The study on the Casimir force indeed gives some enlightenment to the nanotechnology such as the actuation of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS) [4]. However the same force could be generated by the stiction of devices, so there still need further works to enhance the strength and also sign of the Casimir force. It should be noticed that a repulsive force would provide an anti-"stiction" effect.From the theoretical aspect, the boundary conditions, material properties, temperature, and geometry that makes the Casimir force different have been studied a lot. And there are also some new methods that have been developed for computing the Casimir force between a finite number of some compact objects [5], see also [6][7] for the case that inside a rectangular box or cavity. When the helix boundary condition was imposed on a scalar field in a flat spacetime, and when the pitch of the helix is smaller than its circumference, the Casimir force is linearly proportional to the ratio r of the pitch to the circumference, which is just like the Hooke's law that governs the force on a spring, so the authors call it quantum spring [8] ...