The In uence of an External Magnetic Field on the Fermionic Casimir E ect

where ` is the area of each plate, a is the separation between them and the relation a ` is assumed in order to implement the condition of small separation. As a consequence of this shift there is an attractive force on the plates which was measured by Sparnaay in 1958 [2] and more recently with high accuracy by Lamoreaux [3] and by Mohideen and Roy [4, 5]. This shift in energy is known as the Casimir energy and belongs together with its consequences and related phenomena to the realm of the so called Casimir e ect [6, ?, 8]. Generally, the Casimir e ect can be de ned as the e ect of non-trivial space topology on the vacuum uctuations of any relativistic quantum eld (cf., e.g., [6, 7]). The source of the non-triviality of the space topology is provided by several kinds of boundary conditions, background elds and constraints. Under this general de nition the e ect discovered by Casimir and given by the energy (1) can be viewed as a consequence of passing from the trivial topology of lR 3 to the topology of lR 2 [0; a], due to the boundary conditions imposed on the electromagnetic eld by the metallic plates with separation a. The fermionic Casimir e ect is of particular importance due to the fundamental role played by the electron in QED and the quarks in QCD. In the case of quarks we have a boundary condition of con nement given by nature, which makes the Casimir energy a natural ingredient in the hadron structure. The fermionic Casimir energy was rst computed by Johnson [9] in the context of the MIT-bag model [10] for a massless Dirac quantum eld con ned between parallel planes with separation a. A more realistic description of quarks and gluons inside a hadron requires the con ning boundary conditions to be imposed on a spherical surface. The Casimir e ect in spherical geometry for massive elds is a much more complicated problem and has only recently been completely solved for massive fermionic [11] and scalar [12] elds. In the case of con ning planes the fermionic Casimir energy E(a) obtained by Johnson [9] is given by: