Casimir Energy

The Casimir effect for the electromagnetic field between parallel metallic plates can be obtained from Eq. (23) the Casimir energy and pressure are... 

Similarly, from Eq. (24), we find the Casimir energy and pressure for the Dirac field confined between parallel plates, with anti-periodic boundary conditions, as ... 

To show how powerful our method is, let us consider fermions confined in a tridimensional box at finite temperature. The energy-momentum tensor is a long expression for the general case of a parallelepiped box, but it follows from Eq. (29) that the Casimir energy for a cubic box of edge L is given by... 

Again, as (3 —> 0, the Stefan-Boltzmann term dominates, while in the limit (5 —> oo, we find the Casimir energy at zero temperature, the second term in the right hand side of Eq. (38). 

This effect resembles the traditional Casimir effect, which describes the attraction between two parallel metallic mirrors in vacuum. Here, however, the fluctuating (bosonic) electromagnetic fields are replaced by fermionic matter fields. Furthermore, the Casimir energy is inferred from the geometry-dependent part of the density of states, and its sign is not fixed, but oscillates according to the relative arrangement and distances of the cavities. 

Let us now invert the logic and define the Casimir energy as the energy resulting from the geometry-dependent part of the density of states (d.o.s.) - a concept that is closely related to the shell correction energy in nuclear physics ... 

Now the Casimir energy can be extracted from the geometry-dependent part of the density of states as a simple integral... 

Note that the Casimir calculation under the presence of fermionic on-relativistic) matter fields simplifies enormously since the presence of the second scale, the chemical potential p,=h2k2F/2rn or the Fermi momentum kF, provides for a natural UV-cxAoii, Any = p, and kuv = kF- Therefore the Casimir energy for fermions between two impenetrable (parallel) planes at a distance L is simply given as... 

Therefore the Casimir energy for the two spherical cavities inside a non-relativistic non-interacting fermion background can be approximated in terms of a spherical Bessel function j as... 

Abstract. Casimir energy calculations are presented for the massless scalar field for pyramidal and conical wedges and cavities. 

In the coming sections we present the Casimir energy calculations for a pyramidal wedge and then for a pyramidal cavity and a conical cavity. We give the details for the former one, and for the conical cavity we simply present the results. 

Since we have a compact region (all the momenta are now discrete) it is more convenient to employ the energy spectrum formulation to obtain the Casimir energy. The wave function for the massless scalar field in the cavity is... 

Here Ei, E2 and 3 are the Casimir energies for the cube with sides a, for the rectangle with sides a, and for the one dimensional system of length a (Mostepanenko and Trunov, 1997) ... 

The total Casimir energy for the massless scalar field in this conical cavity is again positive ... 

We still do not know what is the connection between the sign and the magnitude of Casimir energy and the shapes and the dimension of the cavity or the space. 

Ahmedov, H. and Duru, I. H. Casimir Energy for a Wedge with Three Surfaces and for a Pyramidal Region, math-ph/0407030, in J. Math. Phys Birrel, N. D. and Davies, P. C.V. Quantum Fields in Curved Spaces. Cambridge University Press, (1982). 

The first three terms represent the Casimir energy [31], that is the energy shift induced in the vacuum by the presence of the RKS-potential. In addition, the... 

If K —> 1, Wmn —> nip1 l)/24. Adding to this expression the initial Casimir energy (159), we obtain the total asymptotical minimum value [164]... 

Although the results of this chapter, which were obtained in the framework of simplified one-dimensional and three-dimensional models, cannot be applied directly to the analysis of the sonoluminescence problem, they are not in favor of Schwinger s hypothesis. The main difficulty is connected with quite different timescales of the phenomena. The accumulation of the dynamical Casimir energy is a very slow process, which needs a great number of wall oscillations, whereas the sonoluminescence pulses (containing up to 107 photons) have the... 

This statement implies that not only the Coulomb interaction is included in Er and Exc but also the (retarded) Breit interaction. It thus points at the fact that a consistent and complete discussion of many-electron systems and consequently of RDFT must start from quantum electrodynamics (QED). RDFT necessruily has to reflect the various features of QED, both on the formal level and in the derivation of explicit functionals. The most important differences to the noiu-elativistic situation arise from the presence of infinite zero point energies and ultraviolet divergencies. In addition, finite vacuum corrections (vacuum polarization, Casimir energy) show up in both fundamental quantities of RDFT, the four current and the total energy. These issues have to be dealt with by a suitable renormalization procedure which ultimately relies on the renormalization of the vacuum Greens functions of QED. The first attempt to take... 

Unfortunately, there is a price to be paid for this universal definition of the energy scale. NHiile the expectation values of (175) are automatically finite, the same is not true for (179). In order to understand the mechanism which leads to divergencies let us consider the energy of the perturbed vacuum with respect to the homogeneous vacuum (often called Casimir energy) within perturbation theory. The basic elements of the perturbation expansion are the Greens function of the perturbed vacuum. 

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