Mode summation approach to Casimir effect between two objects

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.

42 pages

References in corpus (42)

• New Developments in the Casimir Effect
• The Casimir force between real materials: experiment and theory
• The `Friction' of Vacuum, and other Fluctuation-Induced Forces
• The Casimir effect: Recent controversies and progress
• Casimir forces between arbitrary compact objects
• Scattering Theory Approach to Electrodynamic Casimir Forces
• Opposites Attract - A Theorem About The Casimir Force
• Casimir interaction between a plate and a cylinder
• Casimir forces in a T operator approach
• The Casimir effect for a sphere and a cylinder in front of plane and corrections to the proximity force theorem
• Casimir effect for curved geometries: PFA validity limits
• Scalar Casimir effect between Dirichlet spheres or a plate and a sphere
• Worldline algorithms for Casimir configurations
• Casimir Forces between Compact Objects: I. The Scalar Case
• The Casimir Effect and Geometric Optics
• Casimir edge effects
• Exact Casimir interaction between eccentric cylinders
• Casimir Effects: An Optical Approach I. Foundations and Examples
• Casimir forces between cylinders and plates
• Multiple Scattering Methods in Casimir Calculations
• Casimir interaction between plane and spherical metallic surfaces
• Thermal Casimir Effect in the Plane-Sphere Geometry
• Exact zero-point interaction energy between cylinders
• Casimir force for a sphere in front of a plane beyond Proximity Force Approximation
• Thermal Casimir effect for Drude metals in the plane-sphere geometry
• Casimir Effects: an Optical Approach II. Local Observables and Thermal Corrections
• Generalized Lifshitz formula for a cylindrical plasma sheet in front of a plane beyond proximity force approximation
• Comments on the Sign and Other Aspects of Semiclassical Casimir Energies
• Casimir forces between arbitrary compact objects: Scalar and electromagnetic field
• Vacuum energy between a sphere and a plane at finite temperature
• On the corrections beyond proximity force approximation (PFA)
• Casimir interaction between a cylinder and a plate at finite temperature: Exact results and comparison to proximity force approximation
• Exact Casimir Interaction Between Semitransparent Spheres and Cylinders
• Casimir effect between two spheres at small separations
• First analytic correction beyond PFA for the electromagnetic field in sphere-plane geometry
• Quantum energies with worldline numerics
• First analytic correction to the proximity force approximation in the Casimir effect between two parallel cylinders
• Casimir Energy and Entropy in the Sphere--Sphere Geometry
• Universality versus material dependence of fluctuation forces between metallic wires
• Casimir interactions of an object inside a spherical metal shell
• Casimir energy between media-separated cylinders: the scalar case
• Casimir potential of a compact object enclosed by a spherical cavity

Cited by in corpus (14)

• Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation
• Casimir Energy for concentric $\delta$-${\delta }^{\prime }$ spheres
• Regularization versus renormalization: Why are Casimir energy differences so often finite?
• Casimir interaction between a sphere and a cylinder
• Sphere-plate Casimir interaction in (D+1)-dimensional spacetime
• Exact classical sphere-plate Casimir interaction in (D+1)-dimensional spacetime
• Ferminoic Casimir effect between spheres
• Finite temperature Casimir interaction between spheres in $(D+1)$-dimensional spacetime: Exact computations and asymptotic expansions
• Repulsive Casimir-Lifshitz pressure in closed cavities
• On the Difference Between the Vacuum Casimir Energies for Grounded and Isolated Conductors
• Zero and finite temperature Casimir effect of massive vector field between real metals
• Casimir interaction between spherical and planar plasma sheets
• On the Shape Dependence of the Tangential Casimir Force
• Massive scalar Casimir interaction beyond proximity force approximation