Abstract
In this chapter we will consider a conundrum created by the case of the Casimir force in Maxwell’s fish-eye.
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Notes
- 1.
The main results in this chapter were published in [1].
- 2.
Impedance matching is a condition imposed in [2] for securing a virtual geometry for electromagnetic fields.
- 3.
See Appendix and [2].
- 4.
If it is suggested that a force might conceivably act radially outwards, in all directions, we simply remind the reader that the force in physical space must be the same as the force in the virtual free-space, and in virtual space we are dealing with a surface.
- 5.
See the discussion in Sect. 6.4
- 6.
- 7.
The authors of [9], for example, claim ‘Lifshitz theory shows that the self-force is in fact inwardly directed and infinite’.
- 8.
This is simply the inversion in the unit sphere as a mirror transformation of the spectator points. The inversion takes any point \(P\) (other than the origin \(O\)) to its image \(P'\), but also takes \(P'\) back to \(P\), so that the result of applying the same inversion twice is the identity transformation on all the points of the plane other than \(O\). It follows that the inversion of any point inside the reference circle must lie outside it—in this case, beyond the mirror.
- 9.
It is equivalent to a duality transform of the electric Green function, which obeys the same wave equation.
- 10.
This identity has been confirmed to hold for the fish-eye using Mathematica.
- 11.
In Sect. 4.4 we discussed an example of a modification to the Casimir force that would remove divergences in inhomogeneous media, but at the expense of modifying the value of the force itself.
- 12.
This picture of regularisation is owed to a discussion with Simon Horsley, in which he suggested applying Casimir’s interpretation of regularisation to Lifshitz theory too.
- 13.
The inhomogeneous nature of the material cannot simply be removed by a coordinate transformation, as it was in Chap. 6.
- 14.
Of course, in a more realistic theory the particle should not be regarded as a classical object interacting with the quantum vacuum, as in Casimir’s case, but rather as a self-consistent quantum structure.
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Simpson, W.M.R. (2015). The Casimir Force in Maxwell’s Fish-Eye. In: Surprises in Theoretical Casimir Physics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-09315-4_7
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