Lifshitz formula by spectral summation method

TL;DR

This paper presents a rigorous spectral summation method to derive the Lifshitz formula for Casimir energy, addressing complex frequency plane issues and unifying mode summation with scattering formalism.

Contribution

The paper introduces a novel contour integral approach that fully resolves the complex frequency plane cuts in deriving the Lifshitz formula.

Findings

Successfully transitions to imaginary frequencies considering complex roots and cuts.

Clarifies contributions of different electromagnetic modes to Casimir energy.

Identifies and corrects inconsistencies in previous derivations.

Abstract

The Lifshitz formula is derived by making use of the spectral summation method which is a mathematically rigorous simultaneous application of both the mode-by-mode summation technique and scattering formalism. The contributions to the Casimir energy of electromagnetic excitations of different types (surface modes, waveguide modes, and photonic modes) are clearly retraced. A correct transition to imaginary frequencies is accomplished with allowance for all the peculiarities of the frequency equations and pertinent scattering data in the complex $\omega $ plane, including, in particular, the cuts connecting the branch points and complex roots of the frequency equations (quasi-normal modes). The principal novelty of our approach is a special choice of appropriate passes in the contour integrals, which are used for transition to imaginary frequencies. As a result, the long standing problem of cuts in the complex $\omega $ plane is solved completely. Inconsistencies of some previous derivations of the Lifshitz formula are traced briefly. For completeness of the presentation, the necessary mathematical facts are also stated, namely, solution of the Maxwell equations for configurations under consideration, scattering formalism for parallel plane interfaces, determination of the frequency equation roots, and others.