We compute the influence of an external magnetic field on the Casimir energy of a massive charged scalar field confined between two parallel infinite plates. For this case the obtained result shows that the magnetic field inhibits the Casimir effect.The Casimir effect can be generally defined as the effect of a non-trivial space topology on the vacuum fluctuations of relativistic quantum fields [1,2]. The corresponding change in the vacuum fluctuations appears as a shift in the vacuum energy and an associated vacuum pressure. This shift is known as the Casimir energy of the field due to the given space constraints. The original Casimir effect [3] is the attraction of two neutral perfectly conducting parallel plates placed in vacuum. The boundary conditions imposed by the metallic plates confine the vacuum fluctuations of the quantum electromagnetic field in the space between the plates. The effect of the boundary conditions can be viewed as a departure from the trivial topology of lR 3 to the topology of lR 2 × [0, a], where a is the distance between the plates. The resulting shift in the vacuum energy of the quantum electromagnetic field was computed by Casimir and is given by [3]:

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We show how to obtain the first multipole contributions to the electromagnetic radiation emitted by an arbitrary localized source directly from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field. This procedure avoids the unnecessary calculation of the electromagnetic potentials.

We consider the quantization of a scalar κ-deformed field up to the point of obtaining an expression for its vacuum energy. The expression is given by the half sum of the field frequencies, as in the nondeformed case, but with the frequencies obeying the κ-deformed dispersion relation. We consider a set of κ-deformed Maxwell equations and show that for the purpose of calculating the Casimir energy the Maxwell field, as in the non-deformed case, behaves as a pair of scalar fields. Those results provide a foundation for computing the Casimir energy starting from the the half sum of field frequencies. A method of calculation starting from this expression is briefly described.

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We compute the influence of boundary conditions on the Euler-Heisenberg effective Lagrangian scalar QED scalar for the case of a pure magnetic field. The boundary conditions constrain the quantum scalar field to vanish on two parallel planes separeted by a distance a and the magnetic field is assumed to be constant, uniform and perpendicular to the planes. The effective Lagrangian is obtained using Schwinger's proper-time representation and exhibits new contributions generated by the boundary condition much in the same way as a material pressed between two plates exhibits new magnetic properties. The confined bosonic vacuum presents the expected diamagnetic properties and besides the new non-linear a-dependent contributions to the susceptibility we show that there exists also a new a-dependent contribution for the vacuum permeability in the linear approximation. *

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