But... How could space-time not be quantized? That would imply the existence of infinities in the structure of the universe. It is like the ultraviolet catastrophe but in space-time.
1. Something split up into discrete chunks (like integers vs real numbers)
2. Something which acts quantum mechanically (i.e. with superposition and entanglement and all that stuff)
In this context "space-time is also quantized" means space-time is acting like a quantum system.
Whether space-time is discrete or continuous is a wholly seperate question. There is no particular evidence that it is discrete, but if you make the discretization scale small enough it theoretically could be. Most things work more nicely if it is continuous though.
An alternate hypothesis, that a particle is "virtual" and its position/momentum "vector" contains a finite amount of information, is actually both extremely plausible and explains odd paradoxes like the Heisenberg uncertainty principle. That would result in "quantized" spacetime for the same reason floating point numbers are imprecise.
Also if anyone who knows wants to explain why the Bekenstein bound is even a thing, I'd love to hear that too.
So, I don't understand any of the math in that paper but is there any easy/intuitive way to explain why a higher information density would require negative free energy?
I guess I need to understand the relationship between information density and free energy first. Hmm.
Not really, no. If you don't have the equivalent of a good undergraduate education in physics very little about QFT is going to be accessible to you.
Just like the ultraviolet catastrophe was solved by quantizing the photons, not the energy levels.
(What I don't know is if there is some space-space interaction that can't be quantized at the "interaction" level instead of the "space" one.)
Detecting (something about) the particle through gravity would at best remove the interference, not create an \infty somewhere in the model, implying that the model is self inconsistent.
The black hole information paradox from that article presumably fits. The conclusion from GR is that no information can escape, which is ultimately incompatible with QM, and that conclusion ultimately depends on the infinite density of the singularity.
I think the more charitable reading is that we'll find situations where either no sensible calculation can be done, or the sensible calculations we do churn out nonsense. Divergence in the UV catastrophe was an example of that, and Baez covered more here:
Struggles with the Continuum, https://arxiv.org/abs/1609.01421
1 Newtonian gravity
2 Quantum mechanics of charged particles
3 Classical electrodynamics of point particles
They explicitly mention they use point particles. Problem in 1 Newtonian gravity is especially egregious example. They model planets(!) as material points and see how they come infinitely close to each other. The author does mention these problems are pure math unrelated to reality.
EDIT: Hmm, so doesn't the fact that this particular value makes the math simplified not also imply some kind of meaning?
No. There are plenty of other sets of constants you can choose to set to 1, inducing other length scales. Only dimensionless constants have physical meaning in isolation; dimensioned quantities are meaningful only with respect to each other.
The oddity we see with things at relativistic energies is simply because - as with all discretised representations of continuous-time systems - things get a bit fucky close to Nyquist, and the maths breaks down.
You're thinking of the Planck Length, which is the result of combining the Planck Constant with the speed of light. It also happens to be the wavelength of a photon so short that its energy makes it a black hole, connecting it to gravity.
It's often presented as if it were the fundamental unit of length, but that's a guess at best. There's no specific reason to believe it, other than the fact that it happens to be a way to connect length, charge, and gravity. That's evocative, but hardly proof. There's neither evidence for it nor a well-supported theory behind it.