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An accelerated electron can emit a photon, but if I consider the reference system on where the electron is at rest, it doesn't have energy to emit any photon.

Abel Gutiérrez
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    For the electron to be accelerating some external agent must be doing work on it. The energy to create the photon comes from that external agent. – John Rennie Nov 09 '23 at 13:59
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    An accelerated electron cannot be at rest in an inertial frame. – mmesser314 Nov 09 '23 at 14:08
  • @JohnRennie why does the energy go to the photon and not to the electron directly? Is it because to accelerate the electron you need an electric field? In that case I understand that that photon is mediating between the electron and the source of the field to conserve momentum. – Abel Gutiérrez Nov 09 '23 at 14:18
  • @AbelGutiérrez when studying photon and electron interactions, the main stream physics theory is quantum mechanics. In order to be accelerated , it must get a dp/dt either from a photon on mass shell (photons have zero mass) or a virtual photon from a charged particle. See for photon electron https://en.wikipedia.org/wiki/Compton_scattering – anna v Nov 09 '23 at 14:34
  • @mmesser314 yes it can. At rest here may mean with zero instantaneous velocity, not necessarily with zero instantaneous acceleration, or zero velocity for a non-zero time interval. – Ján Lalinský Nov 21 '23 at 10:31
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    Does this answer your question? Why does accelerating electron emits photons? – Rococo Dec 02 '23 at 23:08

2 Answers2

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Either the acceleration is low, energy of emitted photon is low, and time when the photon is emitted is uncertain, and the speed at the time of the event of emitting is uncertain,

or the acceleration is high, energy of emitted photon is high, and time when the photon is emitted has a low uncertainty, and the speed at the time of the emitting event is still uncertain, because the speed changes a lot during a short time.

stuffu
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"Emission of a photon" is a figure of speech used either to refer to integral terms/diagrams in mathematical calculations in QED, or to refer to a simplified and flawed/inaccurate mechanical model of interaction of matter and EM field, where light is imagined to be made of localized particles, a "photon bullet" or "shot" model. This kind of mechanical model is often used with success to analyze e.g. the Compton scattering, or absorption/emission of light due to composite system such as atom or molecule, but its problems show up quickly when one tries to analyze interaction of a point charged particle with static external field, such as the example you point out. Then, instantaneous emission of radiation energy quantum $hf$, without some pre-existing radiation energy quantum (so the situation is different from the one in the mechanical analysis of the Compton scattering), does not conform to ideas of local conservation of energy. That's one of the reasons for why we believe this kind of mechanical model of light is an inaccurate, wrong model for understanding the EM interaction between charged particles. One cannot replace Maxwell's equations with a mechanical model of photon bullets.

In the best theory we have, field theory, emission of EM radiation is not some instantaneous process that would take zero time or short clear-cut time. Emission of EM radiation, as described by mathematics of a field theory, is a continuous process, with no sharp start or end, and "emission of photon" is a manner of speech used to described either the terms in mathematics, or to refer to what the complicated interaction between the matter and the EM field results in when state of things is checked by some measurement (that result can be that some material system such as molecule has absorbed an energy quantum $hf$).

EM radiation produced by accelerated electron cannot be always assigned nice photon energy $hf$, because such field need not be in simple Fock state corresponding to a well-defined photon number, especially in the time interval when the emission is happening. It may be in superposition of various different photon numbers instead. Only when the radiation is absorbed in detector and state of detector is checked, energy $hf$ for some $f$ turns out to be consistent with what happened. But by that time, the radiation has already interacted with the detector, and where exactly has the energy flowed in from is hard to map.

The question you ask can also be asked in classical theory: where does the point electron, which has no internal energy available for losing, get energy to radiate energy away when accelerated? The answer in classical theory is that it only radiates Poynting energy, which is already infinite in the particle, and is not the correct expression of EM energy in theory of point particles. So even though it radiates Poynting energy, this is not the correct "real" EM energy.

There is a consistent theory of point particles due to Jakov Frenkel (somewhat similar to Tetrode's theory of direct inter-particle interaction, in that self-action is not present), where EM energy is given by a different formula in terms of individual particle fields. Here, in contrast to the Poynting formula, EM field/radiation of a single isolated particle does not carry EM energy. Only overlapping fields due to two or more point charged particles can have non-zero EM energy.

Alternatively, we can regard electron as composite charged body where the Poynting formula for energy is correct; then the energy lost to Poynting energy of radiation may come from the source accelerating the body (when accelerated by a working external force), or from stored EM energy in this composite body (when accelerating due to magnetic Lorentz force) - in the latter case, the body has to be able to lose or gain some internal energy.

There is some indication change of internal energy might happen to electron, because we know there are heavier variants of it - muon and tauon. But this happens rarely, during energetic collisions of electrons or other particles. In low-energy experiments where no new particles emerge it does not happen, so there the point particle model of electron, with no internal degrees of freedom, is appropriate.

Ján Lalinský
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