Are there any theoretical limits on the energy of a photon?

Question

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 1

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.

Answer 2

If a photon is too energetic then it might create a black hole according to general relativity. This poses a serious problem when trying to define the process of measuring something with arbitrary accuracy. Some heuristic considerations, based on putting the Heinsenberg's uncertainty principle together with classical General Relativity lead to an estimate of the accuracy with which you can measure some of the coordinates of an event in space-time by sending, say, photons with a certain energy distribution. After this threshold an event horizon is formed and no light can get back to the observer, so the measuring process loses its operational meaning. This has led Doplicher, Fredenhagen and Roberts to postulate that the coordinates of local orthogonal frames should be replaced by (unbounded) self-adjoint operators. For more details on this, a nice review on the subject of Quantum Space-time has just appeared on the arxiv. Briefly, commutation relations among the coordinates are assumed, $$[q^\mu,q^\nu] = i\lambda_P^2 Q^{\mu\nu}$$ and they depend on a characteristic scale given by Planck's length $\lambda_P$, and this captures the above considerations, since one can then retrieve an uncertainty principle for coordinates which is in good accordance with what it is expected from the above heuristic considerations, that is not all coordinates of an event in space-time can be measured simultaneously with arbitrary accuracy.

Answer 3

Edited because I can't read.

The Greisen–Zatsepin–Kuzmin (GZK) limit is one particular theoretical limit involving high-energy protons. Beyond this energy limit, protons scatter too much with the cosmic microwave background radiation to travel any significant distance towards Earth.

Now, in principle, photons should also interact with the CMB, scattering of two photons has already been experimentally confirmed. There are reference frames in which CMB photons have (nearly) arbitrarily high energies, and so one would expect pair production from extremely high energy gamma rays. According to this source the threshold energy is about $10^{15}\,\mathrm{eV}$. A quick search for the term "gamma ray opacity" brings up a lot of interesting papers also, including this one by Dweck and Krennrich , which deals with gamma ray scattering off of extragalactic background light.

Answer 4

I would say no, since the energy of a photon depends on the reference frame, so if a photon beam is directed toward you, the energy of the photons in your reference frame will depends on the relative speed between you and the photon source. When this speed tend toward $c$ then the energy of each photon will tend toward $+\infty$

Answer 5

Yes, you could say a photon could gain enough energy to be bound by its own gravitational field, this class of particle is known as the Geon, invented by John A. Wheeler. That energy would correspond to the Planck energy - but its very speculative and such a particle is probably not stable.