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On the Wikipedia page for Gauge Theory, it is mentioned that

Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson.

Later in the same Wikipedia page, a detailed mathematical explanation is given regarding the electron field and the "four-vector potential of electromagnetic field".

I am a college student who has only taken a course in classical electromagnetism and Maxwell's equations; I am not very well-versed in quantum field theory and advanced physics.

But, I want to understand the "physical intuition" behind this Gauge theory point of view. Am I correct in thinking that:

  • We can derive all of classical electrodynamics (Maxwell's equations) simply from the $U(1)$ symmetry of the electron field?
  • We can derive all of quantum electrodynamics (Photons) simply by quantizing the electromagnetic four-potential?
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    $\begingroup$ What do you mean by "electronic process"? Do you mean processes involving electrons? Because if you mean "electromagnetic process" it makes your question self-answering and redundant. Remember, that moving charges produces a magnetic field which qualifies as electromagnetic phenomena, and electrons are not the only charged particles that exist. Protons in a proton beam, for example. $\endgroup$
    – DKNguyen
    May 9, 2021 at 8:54
  • $\begingroup$ Hi and welcome to Physics SE! I think this question could be better if separated into two parts – the question on what a photon is could well stand on its own. $\endgroup$
    – Jonas
    May 9, 2021 at 9:23
  • $\begingroup$ @DKNguyen Definitely, Maxwell's equations apply to protons as well as electrons. But what I am asking is even more fundamental- do Maxwell's equations themselves arise from the U(1) symmetries of the Electron Field? So, electrons not only interact with electric and magnetic fields, but the very existence of EM fields is caused by Electron fields themselves. Protons also interact with EM fields, but there is no such thing as a "Proton field" responsible for breathing fire into Maxwell's equations themselves! $\endgroup$
    – Anuj Manoj Shah
    May 9, 2021 at 9:25
  • $\begingroup$ In quantum field theory, the electromagnetic field exists without requiring the electron/positron field to exist too. $\endgroup$
    – Jeanbaptiste Roux
    May 9, 2021 at 9:29

2 Answers 2

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We can derive all of classical electrodynamics (Maxwell's equations) simply from the $U(1)$ symmetry of the electron field?

Maxwell's electromagnetism doesn't require an electron field, it can exist independently of electrons in the vacuum, or it can even be coupled to other fields (e.g. charged scalar fields, gravity, etc.). The choice of $U(1)$ selects Maxwell's theory from a more general class of gauge theories called Yang-Mills theories, which are parameterized by Lie groups/algebras.

We can derive all of quantum electrodynamics (Photons) simply by quantizing the electromagnetic four-potential?

Yes, quantum electromagnetism can be obtained from classical electromagnetism (Maxwell's equations) by applying canonical quantization. There are extra complications due to gauge invariance, you'll need the theory of quantization of constrained systems to consistently quantize Maxwell's theory. But you will in the end obtain a quantum theory of noninteracting photons.

QED (Quantum Electrodynamics) is a bit of a misnomer, because apart from Maxwell field / photons, it also includes a completely different matter field which is the Dirac field, with particles called electrons. These two fields / particles interact with each other in a certain way which is dictated by the gauge symmetry. QED is therefore different from quantized electromagnetism, which can be thought of as an approximation to QED that is valid in situations where the electron field is negligible.

Also note that even though in almost all cases quantum theories are obtained from classical theories by canonical quantization, this procedure is heuristic and has no physical meaning. In reality, only the quantum theory exists. The classical theory is simply an approximation. So to really understand what's going on, one needs to postulate the quantum theory, and then prove that the classical theory holds in a certain regime that corresponds to setting $\hbar \rightarrow 0$ in all mathematical expressions.

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There are other charged particles besides electrons, such as protons, pions, W bosons. These can all contribute to electromagnetic phenomena.

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