Mode summation approach to Casimir effect between two objects

TL;DR

This paper develops a mode summation approach to derive explicit Casimir energy formulas for various object configurations, extending the TGTG method to new scenarios with different boundary conditions and object types.

Contribution

It introduces a mode summation perspective to derive and extend TGTG formulas for Casimir energies involving diverse geometries and boundary conditions.

Findings

Derived explicit T-matrices for planes, spheres, and cylinders under various boundary conditions.

Formulated translation matrices for different object configurations.

Extended TGTG formulas to new scenarios not previously considered.

Abstract

In this paper, we explore the TGTG formula from the perspective of mode summation approach. Both scalar fields and electromagnetic fields are considered. In this approach, one has to first solve the equation of motion to find a wave basis for each object. The two T's in the TGTG formula are T-matrices representing the Lippmann-Schwinger T-operators, one for each of the objects. The two G's in the TGTG formula are the translation matrices, relating the wave basis of an object to the wave basis of the other object. After discussing the general theory, we apply the prescription to derive the explicit formulas for the Casimir energies for the sphere-sphere, sphere-plane, cylinder-cylinder and cylinder-plane interactions. First the T-matrices for a plane, a sphere and a cylinder are derived for the following cases: the object is imposed with general Robin boundary conditions; the object is semitransparent; and the object is magnetodielectric. Then the operator approach is used to derive the translation matrices. From these, the explicit TGTG formula for each of the scenarios can be written down. Besides summarizing all the TGTG formulas that have been derived so far, we also provide the TGTG formulas for some scenarios that have not been considered before.