The Casimir effect within scattering theory
Astrid Lambrecht · Paulo A Maia Neto · Serge Reynaud
Abstract and summary · read the original at the source
In one page
Astrid Lambrecht, Paulo Maia Neto and Serge Reynaud review how the Casimir force — the attraction between two mirrors produced by the vacuum’s own electromagnetic fluctuations — is calculated when the mirrors are real rather than ideal. Their tool is scattering theory: describe each mirror by how it reflects light, and the whole force follows from those reflection amplitudes. Casimir’s textbook formula assumes perfect reflection at every frequency, so it depends only on the gap, the area and two constants of nature. Real metal mirrors reflect well only below their plasma frequency, so at separations of a few hundred nanometres the finite conductivity of gold or aluminium matters, and so does surface roughness. The authors show that the widely used proximity force approximation, which treats a bumpy surface as a stack of flat ones, systematically underestimates the roughness correction — by about sixty per cent at a two-hundred-nanometre gap. They propose the sideways Casimir force between corrugated plates as the cleanest test, and name the force’s temperature dependence as the open experimental question.
Why it matters hereChapter 2 rests on the Casimir force being the vacuum’s most accessible macroscopic effect, and this is the paper that says exactly how accurately we can predict it: a single formula that turns any mirror’s measured reflectivity into a force. That is the calculation any vacuum-energy device — Moddel’s resonator, White’s Casimir cell — has to be designed against, and it is also the calculation that sets how tightly a Casimir experiment can bound new short-range forces on the evidence ladder of chapter 1.
What it claims
01Casimir’s 1948 result for two perfectly reflecting parallel plates depends only on the separation, the area and two constants of nature, the speed of light and Planck’s constant — about 0.1 micronewton of attraction for one square centimetre at one micrometre — and it is independent of the fine-structure constant, because perfect reflection is a saturated response to the field.Section I, equation 1
Settled physics02Written in scattering form, the Casimir force between two mirrors is an integral over imaginary frequencies of the mirrors’ own reflection amplitudes; this reproduces the Lifshitz expression as the special case of thick bulk mirrors and stays valid for slabs, multilayers and any mirror whose reflection differs from that model.Section II, equations 10 to 13
Settled physics03No real mirror reflects perfectly at all frequencies: metals reflect well only below a plasma frequency whose wavelength is about 107 nanometres for aluminium and 137 nanometres for gold, so at gaps of that order the finite conductivity changes the force substantially, and the residual theoretical uncertainty comes from not knowing the optical properties of the particular samples used.Section II A, finite conductivity correction
Settled physics04A non-specular scattering formula, in which the reflection amplitudes become matrices that mix wavevectors and polarisations, extends the calculation to rough and corrugated surfaces; against it the proximity force approximation systematically underestimates the roughness correction, by roughly sixty per cent for a two-hundred-nanometre gap and a three-hundred-nanometre roughness wavelength.Section III A, equation 28 and figure 8
Published and peer-reviewed05For the corrugated plane-and-sphere experiment of Chen and colleagues the full calculation gives a lateral Casimir force of 0.20 piconewton where the proximity force approximation gives 0.28 piconewton; measuring that difference would be the first unambiguous evidence of the approximation’s limits and the first observation of a non-trivial geometry effect on the Casimir force.Section III C, figure 12
What to watch06The temperature dependence of the Casimir force between dissipative mirrors remains a matter of debate in the literature, and the authors state that the thermal effect has not been unambiguously proven in experiment — its observation is one of the most urgent challenges for experimenters.Abstract and Section IV, conclusion
What to watch
Read it · abstract
Abstract
We review the theory of the Casimir effect using scattering techniques. After years of theoretical efforts, this formalism is now largely mastered so that the accuracy of theory-experiment comparisons is determined by the level of precision and pertinence of the description of experimental conditions. Due to an imperfect knowledge of the optical properties of real mirrors used in the experiment, the effect of imperfect reflection remains a source of uncertainty in theory-experiment comparisons. For the same reason, the temperature dependence of the Casimir force between dissipative mirrors remains a matter of debate. We also emphasize that real mirrors do not obey exactly the assumption of specular reflection, which is used in nearly all calculations of material and temperature corrections. This difficulty may be solved by using a more general scattering formalism accounting for non-specular reflection with wavevectors and field polarizations mixed. This general formalism has already been fruitfully used for evaluating the effect of roughness on the Casimir force as well as the lateral Casimir force appearing between corrugated surfaces. The commonly used ‘proximity force approximation’ turns out to lead to inaccuracies in the description of these two effects.
The way in
https://doi.org/10.1088/1367-2630/8/10/243LICENCE — DOWNGRADED from the skeleton’s open-licence status. Unpaywall, OpenAlex and Semantic Scholar all label this DOI cc-by, but that is the journal-level label for New Journal of Physics rather than a statement carried by this 2006 article. The publisher’s own page for the version of record — New Journal of Physics volume 8, article 243, published 20 October 2006, in the Focus on Casimir Forces collection — reads only ‘Published under licence by IOP Publishing Ltd’, with no Creative Commons statement anywhere on it; IOPscience answers automated requests with a bot interstitial, so the page was read from the Internet Archive capture of 8 July 2026 and checked again on 2026-09-08. The author copy on arXiv, quant-ph/0611103 version 1 of 9 November 2006, is posted under arXiv’s default non-exclusive distribution licence, which is not a Creative Commons licence. The same determination was reached on this site for the sister article New Journal of Physics 8 (2006) 237. New Journal of Physics is free to read, which is what the gold open-access status records, but free to read is not a licence to republish. TEXT — the body carries the abstract alone, as it appears on the arXiv copy, which matches the published abstract. The summary and the claims below were written from the complete author’s copy, which was read in order to write them accurately and is not reproduced here; the full text is free to read at the publisher and on arXiv.
How to cite it
Astrid Lambrecht, Paulo A Maia Neto, Serge Reynaud (2006) The Casimir effect within scattering theory. doi:10.1088/1367-2630/8/10/243
Where it sits in the curriculum
What the vacuum isInertia and gravity from the vacuumThe evidence ladder