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Why can’t you cut a Planck unit in half?
The why is not really known. But we simply cannot. There is not line where one particle ends and another particle begins. The best you can do is give a probability distribution, but some of the particles will be in places where they're not really supposed to be. This is actually what drives the fusion processes in stars. The nuclei don't actually have enough kinetic energy to fuse--but she is the protons in one hydrogen nucleus just magically appear in the nucleus of a neighboring hydrogen atom.
You literally can't have distances that are smaller than these probability distributions.
Wikipedia's description quotes Bernard Carr and Steven Giddings as saying that any attempt to investigate the possible existence of shorter distances [via particle accelerator] would result in black holes rather than smaller objects
Why, though?
You have probably heard of the Heisenberg uncertainty principle? It's the one about how you can't both know the position and the speed of an electron or photon, because the observation itself changes the outcome of the other.
Something similar exists for length. If we try to observe things at Plancks length, we introduce issues about whether the thing or space even exists there. The observation of infinitely small space requires infinitely large energy in this space causing a black hole or something. I'm not really sure I get it.
There are several good YouTubes on it, but this video sort of made sense to me: https://youtu.be/snp-GvNgUt4
this was one of the better descriptions for why nothing smaller than that can be measured, but I'm aware that my pop-sci joke post is starting to annoy actual students of physics - so who knows if this discussion stays up.