Casimir type effects for scalar fields interacting with material slabs

TL;DR

This paper develops a rigorous method to calculate Casimir effects for scalar fields interacting with finite-width material slabs, deriving explicit formulas and connecting to known results like the Lifshitz formula.

Contribution

It introduces a new calculation approach for Casimir energies involving inhomogeneous scalar fields and slabs, including a procedure to obtain finite self-energies without renormalization.

Findings

Derived explicit propagator and S-matrix for scalar fields with slabs.

Calculated Casimir self-energy and energy between two slabs.

Connected results to the Lifshitz formula in specific parameter regimes.

Abstract

We study the field theoretical model of a scalar field in presence of spacial inhomogeneities in form of one and two finite width mirrors (material slabs). The interaction of the scalar field with the defect is described with position-dependent mass term. For the single layer system we develop a rigorous calculation method and derive explicitly the propagator of the theory, S-matrix elements and the Casimir self-energy of the slab. Detailed investigation of particular limits of self-energy is presented, and connection to know cases is discussed. The calculation method is found applicable to the two mirrors case as well. By means of it we derive the corresponding Casimir energy and analyze it. For particular values of the parameters of the model the obtained results recover the Lifshitz formula. We also propose a procedure to obtain unambiguously the finite Casimir \textit{self}-energy of a single slab without reference to any renormalizations. We hope that our approach can be applied to calculation of Casimir self-energies in other demanded cases (such as dielectric ball, etc.)