The Casimir effect with finite-mass photons

Abstract

Using the Proca equations, which are appropriate when the photon has a finite mass m, the force between two perfectly conducting parallel slabs, each of width N, and separated by a distance 2 L, is calculated. The approach is through the L-dependence of the combined quantum zero-point energies (ZPE) of all the normal modes of the system. The general results are evaluated in a regime with λ = mL ⪡ 1 and ν = mN ⪡ 1. The two leading finite-mass corrections are of relative order λ 2 and λ 4 log λ; both stem from essentially kinematic corrections to the modes already present in the Maxwell case m = 0, and having a discrete spectrum between the slabs. There are further corrections of relative order λ 4 and (for N ⪢ L) λ 4 log( N L ) , the last stemming from dynamically new (penetrating) modes present only for m ≠ 0, to which even perfect conductors are almost transparent, and which possess a continuous spectrum. The calculation has byproducts which may prove more fruitful than the results themselves. These are: (i) A complete analysis of the Proca normal-mode structure for parallel-plane geometry, the first such complete analysis, as far as it is known, for any system. A special role is played by the component A 2 of the vector potential normal to the slabs, which is unique amongst the potentials and fields in that not only A 2 but also ∂A 2 ∂z are continuous across the surfaces; this governs the classification of the modes, and effectively reduces the analysis of the penetrating modes to that for a scalar field. (ii) A clearer understanding of the way in which the total ZPE of continuum modes varies with system parameters like L and N. (iii) Requisite for (ii), a statement of Levinson's theorem in one dimension, which for even-parity modes displays unfamiliar features.