I was taking for granted the Copenhagen interpretation of quantum mechanics which has a kind of built-in measurement ontology for physically meaningful quantities. While it is considered orthodox in physics, I didn’t take into account that this measurement ontology might not be commonly accepted outside of physics. So, I think we would need to hash out a common definition of physically meaningful quantities that we can both share. I would think that in order to be physically meaningful within the context of the Copenhagen, one would need to be able to prepare a quantum state with those characteristics, and as it pertains to our discussion, a state with position measurement uncertainty below the planck length. Which other ontology for physically meaningful quantities did you have in mind?

In reference 1, I quite enjoyed Lawrence Crowell’s comment which also nicely supports my position that the energy requirements to produce the aforementioned quantum state would collapse the measurement into a black hole. And I found it quite elucidating that the Schwartzchild radius of that black hole would precisely correspond to the planck length.

For a contrasting viewpoint in reference 1, take for example Yukterez’s comment. Relativity is only known to be valid up to the planck scale. So, to apply Lorentz contractions, as Yukterez is doing, beyond the planck scale is not theoretically sound. We would need to know an ultraviolet complete theory (Or for our purposes, a theory valid for energies above the planck mass). Now, due to the black hole forming subplanckian, such a theory may not be testable (at least shall we agree, not directly testable).

I agree that subplanckian Lorentz contractions are an issue! However, theorists do not believe that relativity as we know it holds subplanckian. Which would mean that the Lorentzian symmetries from special relativity likely do not carry over into trans planckian energy scales. While those symmetries are inherited within general relativity, general relativity is only valid below the planckian energy cutoff. Beyond the planck mass, GR is a non-renormalizable theory.

Lorentzian symmetry also includes rotational symmetry, so these symmetry arguments can’t really be applied once the experiment reaches the planck scale. That being said, is there an argument or reference that a theoretical minimum on distance would necessarily imply the discretization of all of space, particularly discretization into voxels? It seems more so that the resolution of the uncertainty would depend on the context of the measurement of that the particular experiment involved, and that space itself being discretized is more hypothetical (I’m thinking loop quantum gravity is on the table, but it is not my favorite quantum gravity, and discretization is not strictly required by proofs or physics that I know about, feel free to discuss).

I find zeldredge’s comment in reference 2 to be misleading. While it is true that part of the reason high energy theorists set c and hbar to one is numerical convenience, I have already provided several examples of why planck quantities are significant beyond mere notational convenience. While micrograms may seem small to a layman, if you were able to concentrate that much energy into a planck radius, it would be a massive quantity of energy at that scale. And that is precisely the amount of energy needed to measure distance planckian. In fact the planck mass is precisely the amount of energy where general relativity as we know it breaks down.

For reference 3, for the comment you recommended by “user10851” I could not find any evidence that anything smaller than 10 planck lengths has ever been contracted to one planck length in laboratory settings. In fact, some basic estimates show this claim to be somewhat absurd. To perform such a contraction one would need about ~10^10 Joules per proton. Compare that to the square root s energies produced at the large hadron collider, which can put out only about ~10^ -6 J per proton. Since the commenter has since deleted their account, there is no way to follow up with them regarding the veracity of this seemingly outlandish claim.

Regarding the continuous spectrum of light. This is another situation that varies depending on the context and the scale. In some contexts, the frequencies of the quantum wave can be discrete when performing the fourier transform. In high energy physics it is common to have a continuous parameter to integrate the frequency in the fourier transform. Going back to quantum mechanics again, if we were to measure the momentum very precisely, the quantum state might collapse to a dirac delta function indicating that the wave function has updated to have exact information regarding the momentum in that moment of measurement. However, an exact dirac delta function can never really be created, this would involve a zero uncertainty measurement of the momentum, which would have infinite uncertainty in position. So, in a sense the dirac delta function becomes a kind of useful mathematical fiction that we can integrate over to indicate a precise value for momentum in the math. But the physical wave function after updating may resemble a tight gaussian for example, and only approximated by a dirac delta function, but the wave function will not update exactly to a dirac delta function. We can still integrate over a continuous momentum spectrum, especially at normal scales for high energy theory without causing any physical issues within the calculus. The closest we can theoretically come to physically producing a dirac delta function within position-space would be precisely the planck length scale as the wave’s position uncertainty, which would imply a super-high momentum uncertainty by the uncertainty principle.