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It makes sense that there could be an upper limit to the frequency/energy for individual photons if the universe as we know it is quantized.

But, the highest energy photons I've heard about have a frequency between $10^{20} \ \text{Hz}$ to $10^{30} \ \text{Hz}$. A Planck photon however would have around $10^{35} \ \text{Hz}$. If we assume a cosmological model wherein space-time is quantized to a fundamental metric such as a Planck length, is there any reason to think the highest possible energy of a photon would be higher or lower than that?

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    $\begingroup$ The energy of a photon depends on the chosen reference frame, so it is not clear what you are asking. $\endgroup$
    – A.V.S.
    Commented Jun 21, 2019 at 5:00
  • $\begingroup$ Physics is invariant of the frame of reference though, so regardless of how you are moving, a photon will appear to move at the same speed, and space-time would still be quantized. The question is fundamentally related to what we assume as true of physics, so it doesn't really matter. Pick a frame of reference you feel comfortable with and use the same frame in both the Planck result and the other theoretical limitation, unless you have an argument for why none would exist. $\endgroup$
    – askmathquestions
    Commented Jun 21, 2019 at 5:16
  • $\begingroup$ A.V.S is talking about Doppler shift. $\endgroup$
    – wcc
    Commented Jun 22, 2019 at 14:47

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There are two issues here. Physics does not set an upper limit to photon energy, although at high energy particle antiparticle pair creation will in practice be limiting. If the universe is finite, then it has a finite energy that obviously cannot be exceeded by a photon.

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  • $\begingroup$ current physics does not set an upper bound. What I am asking is what happens when you make the assumption that one exists. $\endgroup$
    – askmathquestions
    Commented Jun 23, 2019 at 17:07
  • $\begingroup$ Then you should edit your question. $\endgroup$
    – my2cts
    Commented Jun 23, 2019 at 20:09
  • $\begingroup$ My question explicitly states I am making the assumption. $\endgroup$
    – askmathquestions
    Commented Jun 23, 2019 at 20:46
  • $\begingroup$ You should edit your question if you want that to be clear. $\endgroup$
    – my2cts
    Commented Jun 23, 2019 at 21:53
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You seem to refer to in your question to quantum foam/quantum gas models which posit that QFT is an epiphenomenon of an underlying physics which is discrete and/or statistical-mechanical at the Planck scale, or below.

In such models the foam/gas is usually assumed to be randomly distributed (e.g. Poisson distributed), because Poisson distribution provides Lorenz invariance to a high accuracy over large scales (not necessarily absolutely, but exceeding the current bounds of our ability to measure).

In discrete fluid models (e.g. gas-like "hard sphere" models) we should expect fundamental limit to the wavelength of a photon in the same way as there is an upper frequency limit (lower wavelength limit) to sound waves in a gas. As the wavelength approaches the mean free path (in this case usually assumed to be around the Planck scale) we would expect "something different" to happen.

What exactly will happen will depend very much on the properties of the fluid.

By way of example, consider a classical ideal gas in which the wave velocity (speed of sound/light) is $c$ and the mean free path is $\ell_P$ (the Planck length).

In monatomic gases such as helium, neon, xenon etc attenuation is insignificant at low frequencies, but rises with the square of the frequency. At some point where the wavelength approaches the mean free path length (average distance between collisions) the contribution to the kinetic energy of atoms due to the wave becomes much larger than variance in the distribution of kinetic energy due to the thermal motion of the atoms. Therefore we expect that the random thermal motion of the atoms will tend to quickly disperse the wave energy as heat.

Very roughly, if we take the "square of frequency" seriously, we might say that a wavelength of $\lambda = \ell_P $ will attenuate in about $1 \ell_P $, and a wavelength of $2 \ell_P $ will attenuate in about $4 \ell_P $. Again roughly, the attenuation distance will be proportional to the $\lambda^2/\ell_P$ . To put it another way, the attenuation distance is roughly proportional to $\lambda \frac{\lambda}{\ell_P}$ (i.e. the wavelength, multiplied by the ratio of the wavelength to the mean free path length, the Planck length in this example).

Some rough examples follow:

  • For a 1m radio wave, the vacuum attenuation distance would be proportional to $10^{35}$ metres, or a trillion times the diameter of the observable universe.

  • For the CMBR, about 1mm wavelength, the vacuum attenuation distance would be proportional to $10^{-6}/10^{-35} = 10^{29}$ metres, about a million times the diameter of the observable universe.

  • 1000nm infra-red light (wavelength $10^{-6}$ metres) would attenuate over $10^{-12}/10^{-35} = 10^{23}$ metres, or about the diameter of the universe or so.

  • A 0.1nm x-ray laser (10^-10 metres) would attenuate over $10^{-20}/10^{-35} = 10^{15}$ metres or 0.1 light year.

  • A $10^{20}$Hz photon has a wavelength about $10^{-13}$ and would attenuate after about $10^9$ metres - a million kilometres, or about 3 seconds.

  • A $10^{30}$Hz photon has a wavelength about $10^{-23}$ metres and would attenuate after about $10^{-12}$ metres ($10^{-21}$ seconds).

  • Highest energy photon candidate observed according to https://arxiv.org/abs/0903.1127 is on the order of $10^{19}ev$, or $10^{-25}$ metres and would attenuate after $10^{-50}/10^{-35} = 10^{-15}$ metres, or about the diameter of a proton.

Very probably I have made several errors above. The general point is when the frequency is small (wavelength is large), attenuation is close to zero. As frequency rises, attenuation will at some point become observable and this will cast light on the underlying physics (if any).

TL;DR: There are many theories for what the New Physics will turn out to be. Good search terms include "Stochastic quantization", "Spin foam", "zitterbewegung".

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Due to Lorentz invariance, the spectrum of photon energy does not cut off, but goes all the way to infinity. So, photons much higher than the Planck energy would be possible.

Update: So, the notion of quantized spacetime is of course speculative at this point. It implies that there may be different ways in which one can try to do it. However, it needs to be consistent with our current understanding, which has been tested and shown to be working. One of these requirements is Lorentz invariance. So, unless we through out Lorentz invariance, any scheme to quantize spacetime that implies a violation of Lorentz invariance would not be correct.

Considering the role of photons in mediating force among particles, one can use the tried-and-tested quantum field theory calculations. Here one finds that integrals over what effectively comes down to the photon energy need to extend to infinity. If one were to introduce a cut-off for such an integral, the cut-off scale would appear explicitly in the result. If the cut-off is the Planck scale, the contribution of the Planck scale may be so small that it would not make an observable difference in observations that we are currently able to make. However, at the same time it also shows that there is no reason to introduce such a scale in the first place to match experimental observations. By implication, the Planck scale is at present only a hypothetical scale, one that cannot at present be observed in any experiments.

As a result, it becomes rather difficult to predict what the effect of some hypothetical scheme to quantized spacetime would be at some hypothetical scale. Our current understanding does not allow arbitrary modifications to the theories, unless we suspect that something is wrong somewhere.

Not sure if this addresses your concern.

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    $\begingroup$ According to very liberal classical models you can make those assumptions, but you're still ignoring the premise of the problem. Suppose there does exist a finite upper bound on photon energy. Then, as force exchanges are propagated by photon exchange of quantized energies over quantized space and time, there is also a contained upper bound on the doppler shift in a given frame. The premise of the problem is the assumption that space-time is quantized, so by default, any model in physics must also be bound by the same restriction as we cannot make observations outside of space-time. $\endgroup$
    – askmathquestions
    Commented Jun 22, 2019 at 16:31
  • $\begingroup$ Is it possible that Lorentz invariance could be violated at values of $\gamma$ not yet achieved in controlled experiments? $\endgroup$
    – garyp
    Commented Jun 23, 2019 at 11:28
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    $\begingroup$ Lorentz invariance doesn't need to be thrown out at all. All it would simply mean is that the transformations in vector space that model Lorentz invariance have to be quantized as well. These quantinizations can still be mapped asymptotically. $\endgroup$
    – askmathquestions
    Commented Jun 23, 2019 at 17:13
  • $\begingroup$ @garyp: Yes, it is possible I presume, but we would not be able to know about that unless we can actually make observation to see it. So, that means it would currently amount to mere speculations and as a result, we cannot give a definitive answer based on our current understanding. $\endgroup$
    – flippiefanus
    Commented Jun 23, 2019 at 17:36
  • $\begingroup$ @askmathquestions: well would you agree that even if the transformations are quantized in the way (assuming I understand what you mean by that), it would not allow one to limit the energy in that way, because even the quantized transformation would allow arbitrary large energies. $\endgroup$
    – flippiefanus
    Commented Jun 23, 2019 at 17:39

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