Rederivation of the Casimir force under the completeness relation of continuum operator

Abstract

Casimir effects manifests that, the two closely paralleled plates, generally produce a macroscopic attractive force due to the quantum vacuum fluctuations of the electromagnetic fields. The derivation of the force requires an {\it artificial} regulator by removing the divergent summation. By including naturally a spectrum density factor, based on the observation that an incomplete eigenvectors of observable, such as the eigenstates for the photons in the free field, can form a complete set of eigenvectors by introducing a unique spectrum transformation, an alternative way is presented to rederive the force, without using a regulator. As a result, the Casimir forces are obtained with the first term $-\pi^2 \hbar c/(240 a^4)$ attractive, and the second one, $-\pi^4 \hbar c^3 \sigma^2/(1008 a^6)$, also attractive but smaller, with $a$ the plate separation, and $\sigma$ a to-be-determined small constant number in the spectrum density factor.