I am currently learning the basics of quantum electrodynamics. As I was reading through my book of choice on the subject (The Quantum Theory of Light, by Rodney Loudon), it occurred to me that it would be hard to describe the Doppler effect in a QED framework, because the frequencies of individual modes are guaranteed to be a multiple of $\hbar$, but the continuous transformation on frequency imposed by the Doppler effect could shift these frequencies into a non-multiple of $\hbar$. Does anyone know of a mathematical basis for describing a quantized electromagnetic field that is moving towards or away from the observer? Please keep in mind that I am a relative newbie to this subject!
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1$\begingroup$ Where does your book claim that the frequencies are multiples of ℏ???? That statement doesn't even withstand dimensional analysis. $\endgroup$– FlatterMannCommented Apr 12, 2023 at 16:31
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1$\begingroup$ @FlatterMann the proportionality constant is $1/E$. Of course $E$ is continuous. $\endgroup$– JEBCommented Apr 12, 2023 at 19:15
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1$\begingroup$ @JEB Exactly. I think that's one of those basic misunderstandings about quantum mechanics. Energy is not quantized. Angular momentum is. Of course energy comes in discreet quanta, but that is a trivial consequence of the fact that we do discrete measurements. It's already like that in classical mechanics if we are talking about individual measurements. $\endgroup$– FlatterMannCommented Apr 12, 2023 at 20:53
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The Doppler effect is included in the Lorentz transformation.
Since the relativistic QFT is invariant under Lorentz transformations, describing the theory in a boosted-coordinate from the original one gives equivalent results to the original theory.
However, if we observe from this coordinate system, the time and energy of the photon should take values that are apparently different from the energy and momentum of the original coordinate system. But they are related by Lorentz transformation and thus follow the Doppler effect.
Thus, if you want to see where the Doppler effect appears in the relativistic QFT, for example, let’s expand the free photon field by plane waves $e^{ik\cdot x}$, and apply the boost transformation to that field. Before and after the boost transformation, the four-momentum $k$ of the plane wave would change from, say, $k$ to $k'$, and the Doppler effect is naturally included in the transformation law of the $k$ and $k’$.
