Is there a fully quantum-field theoretic treatise of Planck's Law for black-body radiation?

Question

I recall from my undergraduate statistical mechanics and QM classes that Planck's Law may be derived fairly straight-forwardly by considering the density of states of EM radiation in thermal equilibrium with a particle inside a cubic box. Indeed the derivation on this wiki page is just the same and appears (IMO) to be fairly concise and accurate.

I wondered therefore, that if we invoke all the magic of QFT (such as the fields being promoted to operators and higher (than tree-level) order particle interactions taken into account,) can a similar derivation still be carried out? (Or indeed a dissimilar one?)

Associated to which, would such a derivation (if it is even possible) require consideration of QED alone or would the effects of the broken electro-weak symmetry have a significant effect on recreating the correct radiation spectrum?

Answer 1

The derivation of the Planck distribution takes into account the quantization of the electromagnetic field: any particular wave-vector of the field can have quantized energies, which are interpreted as the number of photons times the energy of each one of them. At the time Planck derived his formula, this is all he needed. He had no formulation of a physical theory in which the electromagnetic field was quantized, but he showed that when you manage to get one, that quantization alone will be sufficient.

In quantum field theory, this is built-in as the starting point: you promote each field to an operator, and when you "second quantize"* you satisfy the requirement for the deriving the Planck spectrumm - the energy in each mode of the electromagnetic field is quantized. I believe second quantization of the Maxwell field (no need to add electrons to get QED) can be found in every text on the subject. After you do that, the derivation of the Planck spectrum is not all that different from Planck's.

This is only the starting point though. Once you have QFT you can add interactions and calculate corrections to the spectrum and any other thermodynamic quantity - due to QED, electroweak symmetry breaking, and what have you. Those will be normally small corrections**.

For the spectrum, the corrections comes effectively from adding interactions between the photons. The single photon stays massless (because of gauge invariance) and therefore its spectrum does not get modified. But the energy of two photons is no longer the sum of their individual energies. This should correct the measured spectrum in a small way.

(As a side comment: those calculations in thermal field theory are not that simple, as compared to usual loop integrals you get at zero temperature.)

• Just for the record, I strongly dislike this term for other reasons, but here is seems appropriate.

** It is interesting whether or not such corrections have been measured - thermal effects of QED. I don't know the answer to that.