Lifshitz formula for the Casimir force and the Gelfand-Yaglom theorem

Abstract

We provide a Quantum Field Theory derivation of Lifshitz formula for the Casimir force due to a fluctuating real scalar field in $d+1$ dimensions. The field is coupled to two imperfect, thick, plane mirrors, which are modeled by background potentials localized on their positions. The derivation proceeds from the calculation of the vacuum energy in the Euclidean version of the system, reducing the problem to the evaluation of a functional determinant. The latter is written, via Gelfand-Yaglom's formula, in terms of functions depending on the structure of the potential describing each mirror; those functions encode the properties which are relevant to the Casimir force and are the reflection coefficients evaluated at imaginary frequencies.

13 pages, 1 figure

References in corpus (4)

• The Casimir force and the quantum theory of lossy optical cavities
• Pistons modeled by potentials
• Casimir effect with dynamical matter on thin mirrors
• Casimir energies with finite-width mirrors

Cited by in corpus (8)

• Lifshitz formula by spectral summation method
• Casimir Force for Absorbing Media in an Open Quantum System Framework: Scalar Model
• Casimir Effect in the Presence of External Fields
• On the Derivative Expansion for the Electromagnetic Casimir Free Energy at High Temperatures
• Casimir energy due to inhomogeneous thin plates
• On the static Casimir effect with parity-breaking mirrors
• Domain wall interactions due to vacuum Dirac field fluctuations in 2+1 dimensions
• New asymptotic techniques for the partial wave cut-off method for calculating the QED one loop effective action