A classical treatment of blackbody radiation
T. W. Marshall
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In one page
This is one of the founding papers of stochastic electrodynamics, the programme that asks how much of quantum behaviour classical physics can recover if you grant it one thing: that empty space is filled with a real random electromagnetic field. Trevor Marshall had already shown that an ordinary charged spring, shaken by a random field whose energy rises with the cube of frequency, settles into a state with many of the properties of the quantum ground state. Here he extends that to space with objects in it — surfaces where boundary conditions apply. He shows that for the radiation field between two conducting plates the resemblance to quantum electrodynamics survives, and he then uses the result to calculate the attraction between two parallel metal plates entirely classically. That attraction is the Casimir force. Lifshitz and his co-workers had reached similar ground with a hybrid method, quantum inside, macroscopic boundaries outside; Marshall shows the calculation can be made consistently classical throughout.
Why it matters hereChapter two argues that the vacuum is a real structured medium rather than an accounting device, and this paper is the earliest clean demonstration that the most famous piece of vacuum evidence — the pull between two mirrors — falls out of a classical random field with the right spectrum. That makes it a chapter one paper too: it is a worked example of the same measurement being explained twice, from two directions, which is how confidence in a physical picture is actually built.
What it claims
01A charged classical harmonic oscillator placed in a random electromagnetic field with energy spectrum four pi h nu cubed over c cubed reproduces many properties of the quantum-mechanical ground state.Summary, first sentence
Published and peer-reviewed02The work is extended to the case where other material bodies are present — that is, to surfaces on which macroscopic boundary conditions are imposed.Summary, second sentence
Published and peer-reviewed03In the case of the radiation field between two conducting plates, the resemblance to quantum electrodynamics is maintained.Summary, third sentence
Published and peer-reviewed04The result is used to carry out an entirely classical calculation of the attraction between two parallel metal plates due to dispersion forces — the Casimir effect.Summary, fourth sentence
Settled physics05Considerable progress on problems of this type had been made by Lifshitz and co-workers by imposing macroscopic boundary conditions inside a quantum-mechanical calculation, but the exact status of such a hybrid approach had never been quite clear; the present calculation shows how the treatment may be made consistently classical.Summary, fifth and sixth sentences
Published and peer-reviewed06There seems no reason why the method should not be extended to treat, for instance, the attraction between dielectric plates.Summary, closing clause
What to watch
The way in
https://doi.org/10.1007/BF02750449SOURCE NOT REACHED IN FULL. Published as Il Nuovo Cimento volume 38, issue 1, pages 206 to 215, July 1965. The article is closed access and carries no Creative Commons statement; OpenAlex and Unpaywall both return closed with no repository copy, Springer serves an authorisation redirect rather than the article, the paper predates arXiv, and INSPIRE and Europe PMC hold no record of it — all checked 2026-09-08. No text of the paper is reproduced here. The summary and the claims are written from the publisher’s own printed summary for the article, which Crossref does not carry but which the CoLab index reproduces in full from the Springer record; the locators cite that summary sentence by sentence, and one obvious scanning artefact in it, ‘mateiial’ for ‘material’, is read as the word intended. AUTHOR. T. W. Marshall is Trevor W. Marshall, whose 1963 Proceedings of the Royal Society paper opened this line of work; the registry record carries only the initials as the journal prints them. REGISTRY NOTE. This work is filed in the batch alongside Timothy Boyer’s papers because it is the same research programme, but it is Marshall’s paper, not Boyer’s. COMPANION READING. Boyer’s own review of what became of this programme is in this library under an open licence and can be read in full at /library/stm-1011f1af4f.
How to cite it
T. W. Marshall (1965) A classical treatment of blackbody radiation. doi:10.1007/BF02750449
Where it sits in the curriculum