Surface divergences and boundary energies in the Casimir effect

Milton, Kimball A.; Cavero-Peláez, Inés; Wagner, Jef

doi:10.1088/0305-4470/39/21/s52

Cited by 13 publications

(19 citation statements)

References 23 publications

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“…The fact that the integral over all space of the total energy density (electric plus magnetic) vanishes, as shown by Eq. (19), is consistent with the vanishing value of the field energy as obtained by taking the average value of the field Hamiltonian in (11) on the dressed ground state (12). If the singular terms in r = 0 in (17) and (18) containing the delta function and its derivatives were neglected (that is, considering only the usual r −7 terms), the space integral of the field energy would have been diverging due to its behavior at r = 0; in such a case, a sharp inconsistency between local and global self-energy would have appeared.…”

Section: Singular Behaviour Of the Field Energy Density Near A Psupporting

confidence: 85%

Vacuum Casimir energy densities and field divergences at boundaries

Bartolo¹,

Butera²,

Lattuca³

et al. 2015

J. Phys.: Condens. Matter

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We consider and review the emergence of singular field fluctuations or energy densities at sharp boundaries or point-like field sources in the vacuum. The presence of singular energy densities of a field may be relevant from a conceptual point of view, because they contribute to the self-energy of the system. They could also generate significant gravitational effects. We first consider the case of the interface between a metallic boundary and the vacuum, and obtain the structure of the singular electric and magnetic energy densities at the interface through an appropriate limit from a dielectric to an ideal conductor. Then, we consider the case of a nondispersive and nondissipative point-like source of the electromagnetic field, described by its polarizability, and show that also in this case the electric and magnetic energy densities show a singular structure at the source position. We discuss how, in both cases, these singularities give an essential contribution to the electromagnetic self-energy of the system; moreover, they solve an apparent inconsistency between the space integral of the field energy density and the average value of the field Hamiltonian. The singular behavior we find is softened, or even eliminated, for boundaries fluctuating in space and for extended field sources. We discuss in detail the case in which a reflecting boundary is not fixed in space but is allowed to move around an equilibrium position, under the effect of quantum fluctuations of its position. Specifically, we consider the simple case of a one-dimensional massless scalar field in a cavity with one fixed and one mobile wall described quantum-mechanically. We investigate how the possible motion of the wall changes the vacuum fluctuations and the energy density of the field, compared with the fixed-wall case. Also, we explicitly show how the fluctuating motion of the wall smears out the singular behaviour of the field energy density at the boundary.

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Section: Singular Behaviour Of the Field Energy Density Near A Psupporting

confidence: 85%

Vacuum Casimir energy densities and field divergences at boundaries

Bartolo¹,

Butera²,

Lattuca³

et al. 2015

J. Phys.: Condens. Matter

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“…These divergences are a consequence of the oversimplification of a model where the physical interactions are replaced by the imposition of boundary conditions for all modes of a fluctuating quantum field. Of course, this is an idealization, as real physical systems cannot constrain all the modes (for a discussion of surface divergences and their physical interpretation see [1][2][3][4][5][6][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94] and references therein). The appearance of divergences in the VEVs of physical quantities indicates that a more realistic physical model should be employed for their evaluation on the boundaries.…”

Section: Vacuum Currents In the Geometry Of A Single Platementioning

confidence: 99%

Casimir effect for scalar current densities in topologically nontrivial spaces

2015

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We evaluate the Hadamard function and the vacuum expectation value (VEV) of the current density for a charged scalar field, induced by flat boundaries in spacetimes with an arbitrary number of toroidally compactified spatial dimensions. The field operator obeys the Robin conditions on the boundaries and quasiperiodicity conditions with general phases along compact dimensions. In addition, the presence of a constant gauge field is assumed. The latter induces Aharonov-Bohm-type effect on the VEVs. There is a region in the space of the parameters in Robin boundary conditions where the vacuum state becomes unstable. The stability condition depends on the lengths of compact dimensions and is less restrictive than that for background with trivial topology. The vacuum current density is a periodic function of the magnetic flux, enclosed by compact dimensions, with the period equal to the flux quantum. It is explicitly decomposed into the boundary-free and boundary-induced contributions. In sharp contrast to the VEVs of the field squared and the energy-momentum tensor, the current density does not contain surface divergences. Moreover, for Dirichlet condition it vanishes on the boundaries. The normal derivative of the current density on the boundaries vanish for both Dirichlet and Neumann conditions and is nonzero for general Robin conditions. When the separation between the plates is smaller than other length scales, the behavior of the current density is essentially different for non-Neumann and Neumann boundary conditions. In the former case, the total current density in the region between the plates tends to zero. For Neumann boundary condition on both plates, the current density is dominated by the interference part and is inversely proportional to the separation. a

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“…A controversial issue concerns with the appearance of surface divergences (and their cut-off dependence) in the calculation of field energy densities, in the presence of metallic boundary conditions [17][18][19][20][21]. The physical origin of these divergences has been recently questioned in the literature and the possibility of removing them through a suitable regularization procedure has been discussed in the case of a scalar field [19,21].…”

Section: Introductionmentioning

confidence: 99%

Vacuum local and global electromagnetic self-energies for a point-like and an extended field source

Passante

Rizzuto²,

Spagnolo³

2013

Eur. Phys. J. C

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We consider the electric and magnetic energy densities (or equivalently field fluctuations) in the space around a point-like field source in its ground state, after having subtracted the spatially uniform zero-point energy terms, and discuss the problem of their singular behavior at the source's position. We show that the assumption of a point-like source leads, for a simple Hamiltonian model of the interaction of the source with the electromagnetic radiation field, to a divergence of the renormalized electric and magnetic energy density at the position of the source. We analyze in detail the mathematical structure of such singularity in terms of a delta function and its derivatives. We also show that an appropriate consideration of these singular terms solves an apparent inconsistency between the total field energy and the space integral of its density. Thus the finite field energy stored in these singular terms gives an important contribution to the self-energy of the source. We then consider the case of an extended source, smeared out over a finite volume and described by an appropriate form factor. We show that in this case all divergences in local quantities such as the electric and the magnetic energy density, as well as any inconsistency between global and spaceintegrated local self-energies, disappear.

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Surface divergences and boundary energies in the Casimir effect

Cited by 13 publications

References 23 publications

Vacuum Casimir energy densities and field divergences at boundaries

Vacuum Casimir energy densities and field divergences at boundaries

Casimir effect for scalar current densities in topologically nontrivial spaces

Vacuum local and global electromagnetic self-energies for a point-like and an extended field source

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