The specific heat of an EM wave in classical physics

Question

I'm reading Dirac’s Principles of Quantum Mechanics and, in the first chapter, he states:

… There must certainly be some internal motion in an atom to account for its spectrum, but the internal degrees of freedom, for some classically inexplicable reason, do not contribute to the specific heat. A similar clash is found in connexion with the energy of oscillation of the electromagnetic field in a vacuum. Classical mechanics requires the specific heat corresponding to this energy to be infinite, but it is observed to be quite finite.

I assume by an ‘oscillation of the electromagnetic field in a vacuum’ he means an EM wave in the vacuum, but I’m unsure what he means by the specific heat corresponding to the energy of the EM wave.

I suppose the point is that it is not a well-defined concept in classical physics (since it is infinite/divergent), but I’m unsure how you’d even try to calculate this? Is it just that $Q = mc\Delta T$ and for light $m = 0 \therefore c \to \infty$? That seems unsatisfying. Given his previous reference to internal degrees of freedom I assume it has something to do with the degrees of freedom of the classical EM field? Further, what does he mean by ‘it is observed to be finite’? How would you have done an experiment to measure this in the 1910s?

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Full context from the book for reference:

For example, if an atomic system has its equilibrium disturbed in any way and is then left alone,itwill beset in oscillation and the oscillations will get impressed on the surrounding electromagnetic field, so that their frequencies may be observed with a spectroscope. Now whatever the laws of force governing the equilibrium, one would expect to be able to include the various frequencies in a scheme comprising certain fundamental frequencies and their harmonics. This is not observed to be the case. Instead, there is observed a new and unexpected connexion between the frequencies, called Ritz's Combination Law of Spectroscopy, according to which all the frequencies can be expressed as differences between certain terms, the number of terms being much less than the number of frequencies. This law is quite unintelligible from the classical standpoint.

One might try to get over the difficulty without departing from classical mechanics by assuming each of the spectroscopically observed frequencies to be a fundamental frequency with its own degree of freedom, the laws of force being such that the harmonic vibrations do not occur. Such a theory will not do, however, even apart from the fact that it would give no explanation of the Combination Law, since it would immediately bring one into conflict with the experimental evidence on specific heats. Classical statistical mechanics enables one to establish a general connexion between the total number of degrees of freedom of an assembly of vibrating systems and its specific heat. If one assumes all the spectroscopic frequencies of an atom to correspond to different degrees of freedom, one would get a specific heat for any kind of matter very much greater than the observed value. In fact the observed specific heats at ordinary temperatures are given fairly well by a theory that takes into account merely the motion of each atom as a whole and assigns no internal motion to it at all.

This leads us to a new clash between classical mechanics and the results of experiment. There must certainly be some internal motion in an atom to account for its spectrum, but the internal degrees of freedom, for some classically inexplicable reason, do not contribute to the specific heat. A similar clash is found in connexion with the energy of oscillation of the electromagnetic field in a vacuum. Classical mechanics requires the specific heat corresponding to this energy to be infinite, but it is observed to be quite finite. A general conclusion from experimental results is that oscillations of high frequency do not contribute their classical quota to the specific heat.

Dirac, P. A. M. (1930) The Principles of Quantum Mechanics. 4th edn, OUP, p. 2.

Answer 1

Dirac is essentially alluding to the ultraviolet catastrophe. According to classical statistical mechanics, you have the equipartition theorem which states that each harmonic oscillator contributes an additional $k_B$ term to heat capacity.

In the case of an atom, if you want to represent it as a large number of harmonic oscillators to capture the numerous emission lines, it would predict a very large heat capacity which is not measured.

Similarly, for the EM field, each mode acts as a harmonic oscillator. By Fourier transform, there is an infinite number of them, so similarly, you'd get an infinite heat capacity. As the term ultraviolet catastrophe implies, the issue is with the increasing high wavenumbers, which are arbitrarily large due to the continuum of space.

Experimentally, you can confront this predicted infinite heat capacity with the Stefan-Boltzmann law. Indeed, you can accurately measure internal energy $U$ of the EM field and then deduce the heat capacity by derivation: $$ C = \frac{\partial U}{\partial T} $$ Technically, you do not measure directly internal energy $U$, but rather thermal radiation. By enclosing the radiator in a cavity, you can let the system reach equilibrium and therefore recover internal energy of the EM field in the cavity by measuring the emitted radiation through a small hole. This is how the blackbody spectrum was later measured in the early 1900s (conceptually, this is highly simplified).