Local Casimir energies for a thin spherical shell

Rectangular cavities are solvable models that nevertheless touch on many of the controversial or mysterious aspects of the vacuum energy of quantum fields. This paper is a thorough study of the two-dimensional scalar field in a rectangle by the method of images, or closed classical (or optical) paths, which is exact in this case. For each point r and each specularly reflecting path beginning and ending at r, we provide formulas for all components of the stress tensor T µν (r), for all values of the curvature coupling constant ξ and all values of an ultraviolet cutoff parameter. Arbitrary combinations of Dirichlet and Neumann conditions on the four sides can be treated. The total energy is also investigated, path by path. These results are used in an attempt to clarify the physical reality of the repulsive (outward) force on the sides of the box predicted by calculations that neglect both boundary divergences and the exterior of the box. Previous authors have studied "piston" geometries that avoid these problems and have found the force to be attractive. We consider a "pistol" geometry that comes closer to the original problem of a box with a movable lid. We find again an attractive force, although its origin and detailed behavior are somewhat different from the piston case. However, the pistol (and the piston) model can be criticized for extending idealized boundary conditions into short distances where they are physically implausible. Therefore, it is of interest to see whether leaving the ultraviolet cutoff finite yields results that are more plausible. We then find that the force depends strongly on a geometrical parameter; it can be made repulsive, but only by forcing that parameter into the regime where the model is least convincing physically.

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“…This theme has been repeatedly visited since then [22,23,15,24,25,26] and will play a major role in the present work.…”

Section: History and Motivationmentioning

confidence: 89%

Vacuum stress and closed paths in rectangles, pistons and pistols

Fulling

Kaplan

Kirsten

et al. 2009

J. Phys. A: Math. Theor.

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“…In fact, the result may be obtained from the reduced Green's function given in Ref. 29 by an evident substitution. Here, we content ourselves by stating the result for the Green's function in the region of the annulus, a − < r, r ′ < a + :…”

Section: Cylindrical Shell Of Finite Thicknessmentioning

confidence: 99%

“…Remarkably, this is exactly one-half the result found in the same weak-coupling expansion for the leading conformal divergence outside a sphere. 29 Therefore, like the strong-coupling result, this limit is universal, depending on the sum of the principal curvatures of the interface. Note this vanishes for n = 1, so in every case this divergence is integrable.…”

Section: Conformal Weak Couplingmentioning

confidence: 99%

“…In the limit as δ → 0 we recover the δ-function potential. As for the sphere 29 it is straightforward to find the Green's function for this potential. In fact, the result may be obtained from the reduced Green's function given in Ref.…”

Section: Cylindrical Shell Of Finite Thicknessmentioning

confidence: 99%

“…29 It is utterly without physical significance (in the absence of gravity), and may be eliminated with the conformal choice for the parameter ξ, ξ = 1/6.…”

mentioning

confidence: 99%

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Local and Global Casimir Energies in a Green's Function Approach

Milton¹,

Cavero-Peláez²,

Kirsten³

2008

The Eleventh Marcel Grossmann Meeting

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The effects of quantum fluctuations in fields confined by background configurations may be simply and transparently computed using the Green's function approach pioneered by Schwinger. Not only can total energies and surface forces be computed in this way, but local energy densities, and in general, all components of the vacuum expectation value of the energy-momentum tensor may be calculated. For simple geometries this approach may be carried out exactly, which yields insight into what happens in less tractable situations. In this talk I will concentrate on the example of a scalar field in a circular cylindrical delta-function background. This situation is quite similar to that of a spherical delta-function background. The local energy density in these cases diverges as the surface of the background is approached, but these divergences are integrable. The total energy is finite in strong coupling, but in weak coupling a divergence occurs in third order. This universal feature is shown to reflect a divergence in the energy associated with the surface, the integrated local energy density within the shell itself, which surface energy should be removable by a process of renormalization.

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Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity

Milton

2011

Casimir Physics

Self Cite

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From the beginning of the subject, calculations of quantum vacuum energies or Casimir energies have been plagued with two types of divergences: The total energy, which may be thought of as some sort of regularization of the zero-point energy, ∑ 1 2h ω, seems manifestly divergent. And local energy densities, obtained from the vacuum expectation value of the energy-momentum tensor, T 00 , typically diverge near boundaries. These two types of divergences have little to do with each other. The energy of interaction between distinct rigid bodies of whatever type is finite, corresponding to observable forces and torques between the bodies, which can be unambiguously calculated. The divergent local energy densities near surfaces do not change when the relative position of the rigid bodies is altered. The self-energy of a body is less well-defined, and suffers divergences which may or may not be removable. Some examples where a unique total self-stress may be evaluated include the perfectly conducting spherical shell first considered by Boyer, a perfectly conducting cylindrical shell, and dilute dielectric balls and cylinders. In these cases the finite part is unique, yet there are divergent contributions which may be subsumed in some sort of renormalization of physical parameters. The finiteness of self-energies is separate from the issue of the physical observability of the effect. The divergences that occur in the local energy-momentum tensor near surfaces are distinct from the divergences in the total energy, which are often associated with energy located exactly on the surfaces. However, the local energy-momentum tensor couples to gravity, so what is the significance of infinite quantities here? For the classic situation of parallel plates there are indications that the divergences in the local energy density are consistent with divergences in Einstein's equations; correspondingly, it has been shown that divergences in the total Casimir energy serve to precisely renormalize the masses of the plates, in accordance with the equivalence principle. This should be a general property, but has not yet been established, for 1 2 Kimball A. Milton example, for the Boyer sphere. It is known that such local divergences can have no effect on macroscopic causality.

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Local Casimir energies for a thin spherical shell

Cited by 17 publications

References 30 publications

Vacuum stress and closed paths in rectangles, pistons and pistols

Vacuum stress and closed paths in rectangles, pistons and pistols

Local and Global Casimir Energies in a Green's Function Approach

Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity

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