Abstract
We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is sufficiently smooth. We point out that, in some particular cases, the Casimir interaction energy for non-planar perfect mirrors can be reproduced by inhomogeneities on planar mirrors.
Highlights
The evaluation of observables in the static Casimir effect presents many interesting challenges, both from the point of view of designing experiments capable of measuring its features with finer resolutions, as well as regarding the calculations involved in the prediction of those features [1]
The derivative expansion (DE) is an approach that may be applied to the case of two smoothly curved surfaces, such that at each point their local separation is smaller than the curvature radii [2,3]
We consider the static Casimir effect for a special class of system, consisting of a quantum real scalar field in the presence of two zero-width, parallel and infinite plates. The latter are not assumed to be homogeneous; rather, each plate will be characterized, regarding its properties, by a real scalar potential defined on the plane. This can be considered as a toy model for the interaction of the electromagnetic field with a thin mirror characterized by position-dependent electromagnetic properties, and generalizes previous works for scalar fields in the presence of homogeneous thin plates, named δ-potentials [6]
Summary
The evaluation of observables in the static Casimir effect presents many interesting challenges, both from the point of view of designing experiments capable of measuring its features with finer resolutions, as well as regarding the calculations involved in the prediction of those features [1]. We consider the static Casimir effect for a special class of system, consisting of a quantum real scalar field in the presence of two zero-width, parallel and infinite plates The latter are not assumed to be homogeneous; rather, each plate will be characterized, regarding its properties, by a real scalar potential defined on the plane. This can be considered as a toy model for the interaction of the electromagnetic field with a thin mirror characterized by position-dependent electromagnetic properties, and generalizes previous works for scalar fields in the presence of homogeneous thin plates, named δ-potentials [6] Under those assumptions about the system, and using the Gelfand-Yaglom’s (GY) theorem [7], we shall obtain a Lifshitz-like formula [8], which yields the Casimir energy as a functional having as arguments the “scalar potentials,” i.e., the two functions which characterize the plates [9].
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