On an alternative, implicit renormalization procedure for the Casimir energy

Abstract

We propose a procedure for the renormalization of Casimir energy that is based on the implicit versions of standard steps of renormalization procedure — regularization, subtraction and removing the regularization. The proposed procedure is based on the calculation of a set of convergent sums, which are related to the original divergent sum for the non-renormalized Casimir energy. Then, we construct a linear equation system that connects this set of convergent sums with the renormalized Casimir energy, which is a solution to this system of equations. This procedure slightly reduces the indeterminacy that arises in the standard procedure when we choose the particular regularization and the explicit form of the counterterm. The proposed procedure can be applied not only to systems with the explicit transcendental equation for the spectrum but also to systems with the spectrum that can be obtained only numerically. However, to perform the proposed procedure, we need a parameter of the system that satisfies two conditions: (i) we can obtain explicit analytical expressions (as a function of our parameter) for coefficients for all divergent and unphysical terms in divergent sum for Casimir energy; (ii) infinite value of this parameter should be the “natural” renormalization point, i.e. Casimir energy must tend to zero when parameter tends to infinity.