# Is photon energy formula correct for wavelengths comparable to Planck's lengths?

Photon energy formula states that energy of photon is $h\nu$ where $h$ is Planck's constant and $\nu$ is photon frequency. But at the same time AFAIK it is accepted that known physical interactions (electromagnetism, quantum fields, etc.) are non-existent in scales smaller than Planck's units.

Therefore, I'm asking if photons' wave lengths can be smaller than Planck's units. Is given formula correct or does energy of photon reaches infinity as it's wavelength approaches Planck's length?

Given both claims, namely photon energy formula and laws at Planck's scales, are correct, I see this as a contradiction in a theory. How is it resolved if it even is?

• Because we don't know limitations of Planck units... – Jon Custer Jun 27 '18 at 19:14
• The formula is correct, the equations of motion are just more messy and apparently are unknown so far. – Vladimir Kalitvianski Jun 27 '18 at 19:15

Well, this is the very definition of the Planck length, $L_P\sim 1.6 \cdot 10^{-35}$m. It is the minimum of max(λ,1/λ).

In words, and in natural Planck units, a photon of energy E ~ 1/λ will have a Schwarzschild radius r ~ E ~ 1/λ. Increasing λ will make the photon retreat inside its own Schwarzschild horizon, leaving us outside with no meaningful appreciation of its behavior.

Never reach for a contradiction when ignorance will suffice.

• A contradiction is that how would a photon acquire a wavelength smaller than Planck's length. To me it is like for a massive object acquiring speed greater than $c$. But, of course, this interpretation may be wrong. – user168013 Jun 27 '18 at 22:22
• Once the photon hides inside its own BH horizon, you cannot talk about its wavelength. You know nothing about it. Not something, nothing. – Cosmas Zachos Jun 27 '18 at 22:24
• This is similar to what would happen to object once it is consumed by a black hole. You can say nothing about it after that happened. But the analogy is clear: such event never occurs, since it requires infinite time (energy?). – user168013 Jun 27 '18 at 22:28
• ? I don't see any infinite energy anywhere. Nor infinite time, for us outside the BH. The definition says any particle with energy more of that of a Planck mass is something we, poor Quantum Gravity ignoramuses, should not perorate about. – Cosmas Zachos Jun 27 '18 at 22:33
• What if I'm traveling at a high speed so that the photon in my reference frame is red-shifted so that the wavelength becomes larger than the Planck length? Then it would not form a BH. So somewhere, something is not right. – flippiefanus Jun 29 '18 at 7:40

The energy of such a photon is about $40~J$ or $2.5 \cdot 10^{11}$ proton masses. That is a lot of energy for a single photon, but not infinite. It likely explodes in a burst of particles.

• I believe no one was even close to creating such photon. Then while I could calculate energy of such photon based on given formula myself, you say such photon will likely to explode. That it's probably impossible to test if formula is really true at such wavelengths, unlike the case of formula of force, impulse and energy which in their newtonian forms were proved to be wrong at relativistic scales. – user168013 Jun 27 '18 at 20:18
• "It likely explodes in a burst of particles." I don't believe you can make that conserve four-momentum if the photon in question is on-shell. but the whole point of the various Planck scales is that they represent situations where for know that current theories will fail us and therefore out-limits for the search for new physics. You can't talk sensible about a photon with the Planck energy because we don't have any theories that are correct for processes involving such a thing. – dmckee --- ex-moderator kitten Jun 27 '18 at 20:56
• @dmckee if you assume that this rather outrageous photon is entirely isolated indeed it can not desintegrate. I do not make such an assumption. – my2cts Jun 27 '18 at 23:04
• Not to be a pedant, but the Planck mass is about $10^{19}$ proton masses, and about $10^9$J... the kinetic energy of a big ship, no? – Cosmas Zachos Jun 28 '18 at 14:19
• Maybe I should add a factor of c then. – my2cts Jun 28 '18 at 16:18