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I read recently that to consider the Planck length of less than 1.6 x 10 exp -34 meters the smallest unit that can manifest 4 coordinate space time to be incorrect in its interpretation. The reason was that it violated a requirement of Lorentz invariance. I am familiar with the invariant interval between two coordinate sytems (S exp2) and am wondering how we can say that a sub planck scale may or may not preserve this invariance if we can not measure things below this scale. Is that a purely mathematical reasoning based on the structure built by special relativity?

Qmechanic
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J. Gray
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  • Related: https://physics.stackexchange.com/q/9720/2451 , https://physics.stackexchange.com/q/185939/2451 , https://physics.stackexchange.com/q/33273/2451 and links therein. – Qmechanic Dec 13 '18 at 22:30

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I would say all of space and time, and many other things we discuss in physics, are a purely mathematical construct. We attach meaning to them as we see fit. The space of our experience is the space in our mind, we interpolate and extrapolate to convince ourselves that space and time form a continuum. But Lorentz invariance is present in QFT and works, so destroying it destroys everything. The possible issue related to a quantized view of spatial and temporal intervals challenge even our ability to write a PDE in any area of physics. However I am not convinced of the meaning of the Plank scale and its interpretation. I think that this will be resolved once we have a good language for quantum gravity or a new paradigm for gravity. But that's just opinion. If you would like to read more on the philosophical considerations of space and time I would recommend reading A. N. Whitehead's Concept of Nature. What he describes is an experiential meaning of open set topology and then reasons that this mandates that this topology is a natural representation of space and time at the topological level (I don't think he ever uses the term topology, that is my inference). I'm not saying that Whitehead's writings are without flaw. But it's interesting to see what people have considered. Whitehead was one member of the team Russell and Whitehead who wrote Principa Mathematica in the early 1900's. Big names in logic, linguistics, and mathematical philosophy.