Is there any lower or upper limit on the energy of a photon? i.e. does the mathematical framework we currently use to study photons blow up when a photon surpasses a certain upper limit of energy? (or the same on the opposite side?)
My thoughts: If I let the energy of a photon tend to infinity, its wavelength would be tending to zero, and since it is thought that we cannot really distinguish things when they are on the scale of a Planck Length, would the photon have its maximum energy when its wavelength is equal to the Planck length? (Feel free to correct me, I feel I might not be.) and for the opposite end of the spectrum a photon with the least energy would have zero wavelength, implying no photon, which is a trivial case.
Answer
One of the main problems of Quantum Gravity is that Quantum Mechanics (in broad sense, including QFT) holds for arbitrary energies, i.e. there is no structural inner bounds to its validity nor there are known systems for which QM fails. In other terms, in the framework of QM, a collison between particles having Planckian energy has nothing special. On the other hand, at very high energy General Relativity holds and, as far it is known, its consequences must be taken in account. In particular, it is a theorem by Schoen and Yau that "[w]hen enough matter is condensed in a small region, gravitational effects will be strong enough to cause collapse and a black hole will be formed" (quoting the abstract of their article). So, there is a limit on the energy of a photon? Strictly speaking, I don't know. There is no physical evidence, as far as I know, for saying yes or no. If the answer was in the affermative however, QM has to be modified, in order to include an "inner blow up" at very high energies. Notice that one can't say for sure that a QG theory exists nor even that it is a physical necessity. The only persuading physical reason I know of to say that a more comprehensive theory ought to exist is that the cosmic microwave background, a specific GR-object, follows a black-body curve, a specific QM-behaviour.
At the opposite extreme, "soft" photons would not create any difficulty of this sort (provided that GR is "switched off", i.e., has no effects, otherwise picking sufficiently small regions the situation is in principle the same than above); interestingly, this limit is "fussy" for electrons, in the sense that, since electrons are very light but not massless, it is difficult to succeed in keeping them at nonrelativistic regime and check whether they behave according to nonrelativistic QM. Obviously, photons of too high energy are rather unuseful as probes, since annoying effects such as couple creation arise and most measurements become unsensible.
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