Comment on the sign of the Casimir force

I show that reflection positivity implies that the force between any mirror pair of charge-conjugate probes of the quantum vacuum is attractive. This generalizes a recent theorem of Kenneth and Klich [9] to interacting quantum fields, to arbitrary semiclassical bodies, and to quantized probes with nonoverlapping wavefunctions. I also prove that the torques on charge-conjugate probes tend always to rotate them into a mirror-symmetric position. The Casimir force [1] and related electromagnetic interactions [2, 3] are the dominant forces between neutral non-magnetic objects at scales ranging from a few nanometers to hundreds of microns. These forces play therefore an important role both in mesoscopic gravity experiments [4], and in the design and fabrication of microand nanoelectromechanical systems [5, 6]. Of particular interest is the possibility of repulsive interactions, since a frequent cause of malfunction of miniature devices is the collapse and subsequent permanent adhesion of mechanical elements, known as ‘stiction’ [7, 8]. It is therefore important to understand under what conditions such residual electromagnetic forces may change sign. Motivated by this question Kenneth and Klich have recently shown that a mirror pair of dielectric bodies always attract [9] . Their proof is based on a detailed study of the determinant formula that enters in the leading-order semiclassical calculation of the force. Here I want to ∗Unité mixte de recherche (UMR 8549) du CNRS et de l’ENS, associée à l’Université Pierre et Marie Curie et aux fédérations de recherche FR684 et FR2687 .