5 ms·

> The key argument that finite volume contains finite information seems potentially true but suspect. Case in point, take a unit square, it will have a diagonal

> The key argument that finite volume contains finite information seems potentially true but suspect. Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's very finite (ie., one number!) I share the suspicion you have, but I think your counter example is not valid because there is no unit square in (physical) reality. If you would try to measure the diagonal of the unit square you would at some (finite) point run into trouble with the uncertainty principle.

posix_me_less 8y ago
All measurements have limited precision (number of significant decimal digits gained), so the amount of data that can be gained indeed is limited. But there is no general law preventing increasing the precision with better instruments and better methods, so as measurement gets more advanced, more data (such as more decimal digits) can be obtained. The uncertainty principle postulates that uncertainties of coordinate and its conjugated momentum cannot be simultaneously arbitrarily low. It is not clear to me how that postulate would prevent arbitrary increase in precision of position measurements. In theory one could have uncertainty 10E-100m in position of an electron, if uncertainty in its momentum is of the order 10E66 or higher.
- spiralx 8y ago
  > But there is no general law preventing increasing the precision with better instruments and better methods, so as measurement gets more advanced, more data (such as more decimal digits) can be obtained. That's not true, as at the Planck length both quantum and general relativistic effects are equally important, preventing arbitrary measurements over smaller scales - that's the case even for a continuously-divisible universe. In most quantum gravitational theories today the Planck length truly is a minimum possible distance in the same way the Planck time is the minimum possible duration, representing a quantized space-time with hard limits to the precision of measurements.
  - posix_me_less 8y ago
    Quantum gravity "theories" are speculative extrapolations of current knowledge for extreme situations, not accepted general laws of physics. They have problems with consistency (discretization of lengths and other quantities breaks relativity postulates) and even bigger problem with lack of experience with phenomena that they describe (minimum possible lengths/times/energy/etc).
    - deleted 8y ago
      [deleted]
    - spiralx 8y ago
      Yes I know, that's why I made the general point first, then talked about how quantum gravity theories impose even harder limits if true.
- edna314 8y ago
  > In theory one could have uncertainty 10E-100m in position of an electron, if uncertainty in its momentum is of the order 10E66 or higher. I'm not an expert in quantum mechanics, but what I was thinking was that to measure the diagonal of a physical unit square you would have to physically build something like a unit frame from some metal atoms for example. Would this still be an intact metal frame if the uncertainty of electron momentum would be on the order of 10E66 or higher? Naively I would think if the electron could potentially have a momentum of 10E66 due to uncertainty it is very quickly anywhere but not part of the metal frame anymore. Edit: That part aside if you go to even smaller length scales wouldn't the Planck length be a lower bound for how precise a measurement could be?
  - posix_me_less 8y ago
    Of course it is difficult to do, but the point was that uncertainty principle by itself does not limit precision of single measurement; there are other reasons, but those are practical (we do not have the technology and understanding to build such a setup). > Naively I would think if the electron could potentially have a momentum of 10E66 due to uncertainty it is very quickly anywhere but not part of the metal frame anymore. The electron presumably cannot go faster than light, so if the measurement is done fast, the electron won't move much in that time, even with immense momentum. Also, big uncertainty in momentum does not mean the electron will have large momentum for a long time. If there is some containment device, the electron can have immense momentum for a short time that however changes direction so quickly that the electron does not escape the small containment. For example, atoms are such containment devices; the more the atom has protons, the smaller the region the electron is contained in.

posix_me_less 8y ago

All measurements have limited precision (number of significant decimal digits gained), so the amount of data that can be gained indeed is limited. But there is no general law preventing increasing the precision with better instruments and better methods, so as measurement gets more advanced, more data (such as more decimal digits) can be obtained. The uncertainty principle postulates that uncertainties of coordinate and its conjugated momentum cannot be simultaneously arbitrarily low. It is not clear to me how that postulate would prevent arbitrary increase in precision of position measurements. In theory one could have uncertainty 10E-100m in position of an electron, if uncertainty in its momentum is of the order 10E66 or higher.

spiralx 8y ago
> But there is no general law preventing increasing the precision with better instruments and better methods, so as measurement gets more advanced, more data (such as more decimal digits) can be obtained. That's not true, as at the Planck length both quantum and general relativistic effects are equally important, preventing arbitrary measurements over smaller scales - that's the case even for a continuously-divisible universe. In most quantum gravitational theories today the Planck length truly is a minimum possible distance in the same way the Planck time is the minimum possible duration, representing a quantized space-time with hard limits to the precision of measurements.
- posix_me_less 8y ago
  Quantum gravity "theories" are speculative extrapolations of current knowledge for extreme situations, not accepted general laws of physics. They have problems with consistency (discretization of lengths and other quantities breaks relativity postulates) and even bigger problem with lack of experience with phenomena that they describe (minimum possible lengths/times/energy/etc).
  - deleted 8y ago
    [deleted]
  - spiralx 8y ago
    Yes I know, that's why I made the general point first, then talked about how quantum gravity theories impose even harder limits if true.
edna314 8y ago
> In theory one could have uncertainty 10E-100m in position of an electron, if uncertainty in its momentum is of the order 10E66 or higher. I'm not an expert in quantum mechanics, but what I was thinking was that to measure the diagonal of a physical unit square you would have to physically build something like a unit frame from some metal atoms for example. Would this still be an intact metal frame if the uncertainty of electron momentum would be on the order of 10E66 or higher? Naively I would think if the electron could potentially have a momentum of 10E66 due to uncertainty it is very quickly anywhere but not part of the metal frame anymore. Edit: That part aside if you go to even smaller length scales wouldn't the Planck length be a lower bound for how precise a measurement could be?
- posix_me_less 8y ago
  Of course it is difficult to do, but the point was that uncertainty principle by itself does not limit precision of single measurement; there are other reasons, but those are practical (we do not have the technology and understanding to build such a setup). > Naively I would think if the electron could potentially have a momentum of 10E66 due to uncertainty it is very quickly anywhere but not part of the metal frame anymore. The electron presumably cannot go faster than light, so if the measurement is done fast, the electron won't move much in that time, even with immense momentum. Also, big uncertainty in momentum does not mean the electron will have large momentum for a long time. If there is some containment device, the electron can have immense momentum for a short time that however changes direction so quickly that the electron does not escape the small containment. For example, atoms are such containment devices; the more the atom has protons, the smaller the region the electron is contained in.