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

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Casimir Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 834))

Abstract

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, \(\sum\frac{1}{ 2}\hbar\omega,\) seems manifestly divergent. And local energy densities, obtained from the vacuum expectation value of the energy-momentum tensor, \(\langle T_{00}\rangle ,\) 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 example, for the Boyer sphere. It is known that such local divergences can have no effect on macroscopic causality.

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Notes

  1. 1.

    In general, this need not be the case. For example, Romeo and Saharian [54] show that with mixed boundary conditions the surface divergences need not vanish for parallel plates. For additional work on local effects with mixed (Robin) boundary conditions, applied to spheres and cylinders, and corresponding global effects, see [55–57, 50]. See also Sect. 3.2.2 and [51, 53].

  2. 2.

    The first steps have been made for calculating the Casimir energy for an ellipsoidal boundary [34, 35], but only for scalar fields since the vector Helmholtz equation is not separable in the exterior region.

  3. 3.

    Note that the corresponding TE contribution the electromagnetic Casimir pressure would not be zero, for there the sum starts from \(l=1\).

  4. 4.

    Note there is a sign error in (4.8) of [74].

  5. 5.

    This argument is a bit suspect, since the analytic continuation that defines the integrals has no common region of existence. Thus the argument in the following subsection may be preferable. However, since that term is properly a contact term, it should in any event be spurious.

  6. 6.

    Note that in previous works, such as [45, 46], the surface term was included, because the integration was carried out only over the interior and exterior regions. Here we integrate over the surface as well, so the additional so-called surface energy is automatically included. This is described in the argument leading to (3.20a). Note, however, if (3.226) is integrated over a small interval enclosing the \(\delta\)-function potential,

    $$ \int\limits_{\xi_1-\epsilon}^{\xi_1+\epsilon} \hbox{d}\xi \xi f_\xi=-\xi_1\Updelta T^{\xi\xi}, $$

    where \(\Updelta T^{\xi\xi}\) is the discontinuity in the normal-normal component of the stress density. Dividing this expression by \(\xi_1\) gives the usual expression for the force on the plate.

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Acknowledgements

I thank the US Department of Energy and the US National Science Foundation for partial support of this work. I thank my many collaborators, including Carl Bender, Iver Brevik, Inés Cavero-Peláez, Lester DeRaad, Steve Fulling, Ron Kantowski, Klaus Kirsten, Vladimir Nesterenko, Prachi Parashar, August Romeo, K.V. Shajesh, and Jef Wagner, for their contributions to the work described here.

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  1. Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK, 73019, USA

    Kimball A. Milton

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  1. Kimball A. Milton
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Correspondence to Kimball A. Milton .

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  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    Diego Dalvit

  2. , Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    Peter Milonni

  3. Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    David Roberts

  4. , Theoretical Division, Los Alamos National Laboratory, Los Alamos, 87545, New Mexico, USA

    Felipe da Rosa

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© 2011 Springer-Verlag Berlin Heidelberg

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Milton, K.A. (2011). Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity. In: Dalvit, D., Milonni, P., Roberts, D., da Rosa, F. (eds) Casimir Physics. Lecture Notes in Physics, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20288-9_3

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  • DOI: https://doi.org/10.1007/978-3-642-20288-9_3

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