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Perhaps another way to put it is, what exactly does it mean to quantize the EM field and why is it necessary? What mathematical properties does the quantized version of the field have that the classical version doesn't and vice versa?

For context, I was reading a thread about where specifically classical E&M fails and we hence need QED to get accurate predictions, and stuff like the ultraviolet catastrophe and the photoelectric effect were mentioned. I get why those things necessitated the introduction of quantum mechanics and especially modeling photons as wavefunctions. What I don't get is why we can't just use "regular" relativistic QM with Maxwell's equations and the same mathematical E and B fields (which together form the EM field) from classical E&M. What goes wrong if we try to do this and how exactly does QED fix it? Or, to put it another way, what properties does a field modeling photons as wavefunctions need to have that the classical EM field doesn't have?

Qmechanic
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Mikayla Eckel Cifrese
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2 Answers2

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Maxwell equations cannot explain (among others) the electric photo effect. This is the reason why the electromagnetic field has to be quantized.

Second Quantization formally overcomes the difficulty to explain the quantum character of the EM-field.

For more details I recommend to read books on Quantum field theory. Although it is a bit old-fashioned you could read Landau/Lifshitz's book on Relativistic Quantum Theory (volume 4). It starts off right away with quantization of the electromagnetic field.

Lagrangian
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Questions of this nature is rather badly treated in the literature. People continue to quote bad answers, even though Wikipedia correctly points out that you can have classical EM field and quantum electrons to explain the photoelectric effect. (Oh, it no longer says so. Alas, one cannot hope for Wiki to always stay comprehensively correct.)

However, the same Wiki page would fail to point out that you still need a quantum description of light, because even if the photoelectric effect of electron ejection only using high frequency classical light, it cannot explain that the electrons get emitted right away. The classical description would still have the time delay as the classical wave slowly concentrates enough energy on the metallic surface to eject electron quanta. Instead, experiments show that there is negligible time lag between irradiation of the metallic surface, and the ejection of electrons, which is only explained by quantising light.

Needless to say, we also have plenty of experimental evidence today that photons, the quantum of light, behave totally different from classical predictions, in complicated scenarios undreamt of in classical viewpoints. Even if you insist to describe laser light classically, you cannot explain the really quantum behaviour, the corelations, the Bell's type experiments.

The essential difference is that the vector potential A field, from which you derive the E and B fields, is upgraded from a classical field function, into a quantum operator on the underlying photon field (this is quantum field as in QFT). This means that when you create or destroy a photon, it is changing by a discrete amount. You cannot reduce the amplitude of the oscillation below the basic quantum. And because it is not localised, when you destroy a photon, you are removing the quanta from everywhere in the universe at once. This is the non-locality that Einstein was so concerned about in the 1927 Solvay conference, that nobody understood what he was talking about.

naturallyInconsistent
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