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Physicist here. You are correct. The uncertainty principle doesnt turn space into some sort of voxel grid. While we still don't know what spacetime looks like

Physicist here. You are correct. The uncertainty principle doesnt turn space into some sort of voxel grid. While we still don't know what spacetime looks like at the smallest scales (we're still waiting on a quantum theory of gravity), quantum field theory measures spacetime using continuous real numbers

naasking 4y ago
> While we still don't know what spacetime looks like at the smallest scales (we're still waiting on a quantum theory of gravity), quantum field theory measures spacetime using continuous real numbers IIRC, the uncertainty principle limits the precision in any possible measurement to something like 60 or 70 digits. So yes, current theories use continuous real numbers, but I wouldn't generalize that to say that's confirmed because we're nowhere near being able to test that level of precision.
- danbruc 4y ago
  I guess you are confusing two things - think of a mark on a real line at x somewhere between 17 mm and 18 mm from the origin. If you measure its location with a ruler, you might only be able to say that it is somewhere between 17 mm and 18 mm from the origin, but this uncertainty in your measurement in no way constraints x to only be located at integer millimeter positions or something like that. There is not even a real limit imposed by the uncertainty principle, you can measure positions as precisely as you want, you just have to pay in momentum uncertainty. Where we really seem to run into a wall is that if you keep making the measurement more and more precise, you need higher and higher energies to achieve ever shorter wavelengths and dumping a lot of energy into a small volume to measure the position really precisely will eventually result in black holes.
  - naasking 4y ago
    > There is not even a real limit imposed by the uncertainty principle, you can measure positions as precisely as you want, you just have to pay in momentum uncertainty. Yes, that's the current continuous model of how this works. That isn't necessarily reflective of reality though. > Where we really seem to run into a wall is that if you keep making the measurement more and more precise, you need higher and higher energies to achieve ever shorter wavelengths and dumping a lot of energy into a small volume to measure the position really precisely will eventually result in black holes. Exactly. In other words, all measurements necessarily have finite precision due to various uncertainties or other physical limits. I didn't pull this limit out of the aether, I came across it from one of Shor's posts [1] where he states that physical constants can't be defined to greater precision than what I specified above. If the physical constants can't have more than 60 digits of precision, then neither can any calculations or measurements based on them. The fact is, we seem to be bounded on all sides to finite precision. [1] https://physics.stackexchange.com/questions/16889/non-computable-numbers-in-physics#comment40236_16893 https://physics.stackexchange.com/questions/16889/non-comput...
    - danbruc 4y ago
      My point is just that limited measurement precision does not imply anything about the underlying structure, could be continuous or not, I am not arguing for one side or the other.
      - naasking 4y ago
        Well then you seem to be agreeing with my original statement: > So yes, current theories use continuous real numbers, but I wouldn't generalize that to say that's confirmed because we're nowhere near being able to test that level of precision.
        danbruc 4y ago
        Yes and no, I guess, depending on what exactly you intended to say. We agree, I guess, that we are currently using real numbers but that does not mean that the universe is actually continuous. Where I am not so sure that we agree is about the experimental side. We could run into some barrier when probing shorter and shorter distances, but this would not necessarily imply that space is not continuous. On the other hand we could also observe effects that clearly indicate a non-continuous structure of space without running into some measurement limit.
        naasking 4y ago
        > Where I am not so sure that we agree is about the experimental side. We could run into some barrier when probing shorter and shorter distances, but this would not necessarily imply that space is not continuous Agreed, though I personally find continuous quantities implausible, despite preferring them at one point. As long as we had a discrete theory with equal predictive and explanatory power and it was equally parsimonious to a continuous theory, I would likely prefer it. I think the next revolution in physics will see an expansion of discretization or other forms of finitism. > On the other hand we could also observe effects that clearly indicate a non-continuous structure of space without running into some measurement limit. Agreed as well.
      - 4y ago
BlueTemplar 4y ago
I want to tone that down somewhat : I think that I misremembered that Planck's length and time were derived from the uncertainty principle... while looking into it, it might not be as obvious ? (So you need both ?) And would you get quantification of mass~energy from that of the impulse (through v having dimensions of space and time), or do you need to use a different approach and assume black holes ? (Or both, and pick the biggest value ?)
- consilient 4y ago
  The Planck length and time don't have any special physical significance, they're just convenient, and coincidentally happen to be roughly the size where we know for sure QFT is insufficient. The "smallest meaningful unit of distance" stuff is nonsense. > And would you get quantification of mass~energy from that of the impulse (through v having dimensions of space and time), or do you need to use a different approach and assume black holes ? (Or both, and pick the biggest value ?) Neither, though the impulse thing is at least in the right neighborhood. (Black holes have absolutely nothing to do with this). You get quantized energy whenever the Hamiltonian of your system has a pure point spectrum. Typically this happens for finite dimensional systems, and infinite dimensional systems with potentials that grow rapidly with increasing distance. Generic infinite dimensional systems will usually have continuous spectra.
  - naasking 4y ago
    > The "smallest meaningful unit of distance" stuff is nonsense. Why?
    - consilient 4y ago
      You get the Planck units by setting c, hbar, G, and Boltzmann's constant to 1. This is convenient for notational purposes but it has no inherent physical significance.
      - naasking 4y ago
        That isn't the only way to argue for a minimal significant distance. Arguably QM sets a strict ~60-70 digit precision limit on physical constants [1], beyond which you arguably can't differentiate between discrete and continuous theories, and so a minimum distance seems like a perfectly sensible way to frame it. [1] https://physics.stackexchange.com/questions/16889/non-computable-numbers-in-physics#comment40236_16893 https://physics.stackexchange.com/questions/16889/non-comput...

naasking 4y ago

> While we still don't know what spacetime looks like at the smallest scales (we're still waiting on a quantum theory of gravity), quantum field theory measures spacetime using continuous real numbers IIRC, the uncertainty principle limits the precision in any possible measurement to something like 60 or 70 digits. So yes, current theories use continuous real numbers, but I wouldn't generalize that to say that's confirmed because we're nowhere near being able to test that level of precision.

danbruc 4y ago
I guess you are confusing two things - think of a mark on a real line at x somewhere between 17 mm and 18 mm from the origin. If you measure its location with a ruler, you might only be able to say that it is somewhere between 17 mm and 18 mm from the origin, but this uncertainty in your measurement in no way constraints x to only be located at integer millimeter positions or something like that. There is not even a real limit imposed by the uncertainty principle, you can measure positions as precisely as you want, you just have to pay in momentum uncertainty. Where we really seem to run into a wall is that if you keep making the measurement more and more precise, you need higher and higher energies to achieve ever shorter wavelengths and dumping a lot of energy into a small volume to measure the position really precisely will eventually result in black holes.
- naasking 4y ago
  > There is not even a real limit imposed by the uncertainty principle, you can measure positions as precisely as you want, you just have to pay in momentum uncertainty. Yes, that's the current continuous model of how this works. That isn't necessarily reflective of reality though. > Where we really seem to run into a wall is that if you keep making the measurement more and more precise, you need higher and higher energies to achieve ever shorter wavelengths and dumping a lot of energy into a small volume to measure the position really precisely will eventually result in black holes. Exactly. In other words, all measurements necessarily have finite precision due to various uncertainties or other physical limits. I didn't pull this limit out of the aether, I came across it from one of Shor's posts [1] where he states that physical constants can't be defined to greater precision than what I specified above. If the physical constants can't have more than 60 digits of precision, then neither can any calculations or measurements based on them. The fact is, we seem to be bounded on all sides to finite precision. [1] https://physics.stackexchange.com/questions/16889/non-computable-numbers-in-physics#comment40236_16893 https://physics.stackexchange.com/questions/16889/non-comput...
  - danbruc 4y ago
    My point is just that limited measurement precision does not imply anything about the underlying structure, could be continuous or not, I am not arguing for one side or the other.
    - naasking 4y ago
      Well then you seem to be agreeing with my original statement: > So yes, current theories use continuous real numbers, but I wouldn't generalize that to say that's confirmed because we're nowhere near being able to test that level of precision.
      - danbruc 4y ago
        Yes and no, I guess, depending on what exactly you intended to say. We agree, I guess, that we are currently using real numbers but that does not mean that the universe is actually continuous. Where I am not so sure that we agree is about the experimental side. We could run into some barrier when probing shorter and shorter distances, but this would not necessarily imply that space is not continuous. On the other hand we could also observe effects that clearly indicate a non-continuous structure of space without running into some measurement limit.
        naasking 4y ago
        > Where I am not so sure that we agree is about the experimental side. We could run into some barrier when probing shorter and shorter distances, but this would not necessarily imply that space is not continuous Agreed, though I personally find continuous quantities implausible, despite preferring them at one point. As long as we had a discrete theory with equal predictive and explanatory power and it was equally parsimonious to a continuous theory, I would likely prefer it. I think the next revolution in physics will see an expansion of discretization or other forms of finitism. > On the other hand we could also observe effects that clearly indicate a non-continuous structure of space without running into some measurement limit. Agreed as well.
    - 4y ago

BlueTemplar 4y ago

I want to tone that down somewhat : I think that I misremembered that Planck's length and time were derived from the uncertainty principle... while looking into it, it might not be as obvious ? (So you need both ?) And would you get quantification of mass~energy from that of the impulse (through v having dimensions of space and time), or do you need to use a different approach and assume black holes ? (Or both, and pick the biggest value ?)

consilient 4y ago
The Planck length and time don't have any special physical significance, they're just convenient, and coincidentally happen to be roughly the size where we know for sure QFT is insufficient. The "smallest meaningful unit of distance" stuff is nonsense. > And would you get quantification of mass~energy from that of the impulse (through v having dimensions of space and time), or do you need to use a different approach and assume black holes ? (Or both, and pick the biggest value ?) Neither, though the impulse thing is at least in the right neighborhood. (Black holes have absolutely nothing to do with this). You get quantized energy whenever the Hamiltonian of your system has a pure point spectrum. Typically this happens for finite dimensional systems, and infinite dimensional systems with potentials that grow rapidly with increasing distance. Generic infinite dimensional systems will usually have continuous spectra.
- naasking 4y ago
  > The "smallest meaningful unit of distance" stuff is nonsense. Why?
  - consilient 4y ago
    You get the Planck units by setting c, hbar, G, and Boltzmann's constant to 1. This is convenient for notational purposes but it has no inherent physical significance.
    - naasking 4y ago
      That isn't the only way to argue for a minimal significant distance. Arguably QM sets a strict ~60-70 digit precision limit on physical constants [1], beyond which you arguably can't differentiate between discrete and continuous theories, and so a minimum distance seems like a perfectly sensible way to frame it. [1] https://physics.stackexchange.com/questions/16889/non-computable-numbers-in-physics#comment40236_16893 https://physics.stackexchange.com/questions/16889/non-comput...