I don’t really doubt that you’re right. Most everything I read on the subject agrees with or is consistant with what you’re saying. But the idea is still very confusing to me, so I appreciate your explanations. Let me try to make my troubles more clear.
So far as I understand it, a Planck time is a minimum because that’s the time it takes the fastest possible thing to pass through the minimum possible length. If something were going 99% the speed of light, or 75% or any percentage other than 100%, 50%, 25%, 12.5% etc. then it would travel through the Planck length in a non-whole number of Planck times. So something traveling at 75% the speed of light would travel through the Planck length at 1.5 Planck times. Maybe we can’t measure this. That’s fine. But say something were to travel at a constant velocity through two Planck lengths in three Planck times. Wouldn’t it just follow that it went through each Planck length in 1.5 Planck times? It may be that we can’t measure anything with precision greater than whole numbers of Planck times, but in this scenario it wouldn’t follow from that that time is discontinuous.
Mathematically speaking, you can say “in average it travelled for 1 Planck length in 1.5 Planck time”. But physically speaking, it doesn’t mean anything. Quantum mechanics works with wavefunction. Objects don’t have an absolutely precise position. To know where the object is, you need to interact with it. To interact with it, you need something to happen. Due to Heinsenberg’s Uncertainity Principle (even if you consider it as a “certainity principle” as Eliezer does), you just can’t locate something more precisely in space than a Planck length, nor more precisely in time than a Planck time. Done at quantum level, objects don’t have a precise position and speed. So saying “it moves at 0.75c so it crosses 1 Planck length in 1.5 Planck time” doesn’t hold. It can only hold as an average once the object evolved for many Planck times (and moved many Planck length).
Mathematically speaking, you can say “in average it travelled for 1 Planck length in 1.5 Planck time”. But physically speaking, it doesn’t mean anything. Quantum mechanics works with wavefunction.
I see. But this raises again my original worry: does QM’s claim about Planck times actually say anything about the continuity of time? Or just something about the theoretical structure of QM? Or just something about the greatest possible experimental precision? Does a limit on the precision of time at this level imply that these are actual indivisible and discontinuous units?
Maybe I’m just too steeped in pragmatism to notice, but it seems your question has already been answered. For example:
Does a limit on the precision of time at this level imply that these are actual indivisible and discontinuous units?
No, a limit on precision tells you that it’s not meaningful to ask whether or not there are actual indivisible and discontinuous units. There’s no experiment that could tell the difference.
I think pragmatism is a fine approach here, but could you clarify for me what your think the answer to my question is exactly? If it’s not meaningful to ask whether or not there are indivisible and discontinuous units, then is the answer to my question “Does QM’s claims about Planck time imply that time is discontinuous?” simply “No” because QM says nothing meaningful about the question one way or the other?
In ‘pure’ QM (without gravity), the Planck length has no special significance, and spacetime is assumed to be continuous. But we know that QM as we know it must be an approximation because it disagrees with GR (and/or vice versa), and the ‘correct’ theory of quantum gravity might predict weird things at the Planck scale. So far, most proposed theories of quantum gravity have little more predictive power than “The woman down the street is a witch; she did it”, though some do predict stuff such as the dispersion of gamma rays I’ve mentioned elsewhere.
is the answer to my question “Does QM’s claims about Planck time imply that time is discontinuous?” simply “No” because QM says nothing meaningful about the question one way or the other?
We’re trying to dissolve the question by pointing out that there exists a third option besides “continuous” or “discontinuous”. So the answer to “Does QM’s claims about Planck time imply that time is discontinuous?” would be “No, but neither is it continuous, but a third thing that tends to confuse people.”
Edit: retracted because I don’t think this is helpful.
I don’t really doubt that you’re right. Most everything I read on the subject agrees with or is consistant with what you’re saying. But the idea is still very confusing to me, so I appreciate your explanations. Let me try to make my troubles more clear.
So far as I understand it, a Planck time is a minimum because that’s the time it takes the fastest possible thing to pass through the minimum possible length. If something were going 99% the speed of light, or 75% or any percentage other than 100%, 50%, 25%, 12.5% etc. then it would travel through the Planck length in a non-whole number of Planck times. So something traveling at 75% the speed of light would travel through the Planck length at 1.5 Planck times. Maybe we can’t measure this. That’s fine. But say something were to travel at a constant velocity through two Planck lengths in three Planck times. Wouldn’t it just follow that it went through each Planck length in 1.5 Planck times? It may be that we can’t measure anything with precision greater than whole numbers of Planck times, but in this scenario it wouldn’t follow from that that time is discontinuous.
Mathematically speaking, you can say “in average it travelled for 1 Planck length in 1.5 Planck time”. But physically speaking, it doesn’t mean anything. Quantum mechanics works with wavefunction. Objects don’t have an absolutely precise position. To know where the object is, you need to interact with it. To interact with it, you need something to happen. Due to Heinsenberg’s Uncertainity Principle (even if you consider it as a “certainity principle” as Eliezer does), you just can’t locate something more precisely in space than a Planck length, nor more precisely in time than a Planck time. Done at quantum level, objects don’t have a precise position and speed. So saying “it moves at 0.75c so it crosses 1 Planck length in 1.5 Planck time” doesn’t hold. It can only hold as an average once the object evolved for many Planck times (and moved many Planck length).
I see. But this raises again my original worry: does QM’s claim about Planck times actually say anything about the continuity of time? Or just something about the theoretical structure of QM? Or just something about the greatest possible experimental precision? Does a limit on the precision of time at this level imply that these are actual indivisible and discontinuous units?
Maybe I’m just too steeped in pragmatism to notice, but it seems your question has already been answered. For example:
No, a limit on precision tells you that it’s not meaningful to ask whether or not there are actual indivisible and discontinuous units. There’s no experiment that could tell the difference.
I think pragmatism is a fine approach here, but could you clarify for me what your think the answer to my question is exactly? If it’s not meaningful to ask whether or not there are indivisible and discontinuous units, then is the answer to my question “Does QM’s claims about Planck time imply that time is discontinuous?” simply “No” because QM says nothing meaningful about the question one way or the other?
In ‘pure’ QM (without gravity), the Planck length has no special significance, and spacetime is assumed to be continuous. But we know that QM as we know it must be an approximation because it disagrees with GR (and/or vice versa), and the ‘correct’ theory of quantum gravity might predict weird things at the Planck scale. So far, most proposed theories of quantum gravity have little more predictive power than “The woman down the street is a witch; she did it”, though some do predict stuff such as the dispersion of gamma rays I’ve mentioned elsewhere.
We’re trying to dissolve the question by pointing out that there exists a third option besides “continuous” or “discontinuous”. So the answer to “Does QM’s claims about Planck time imply that time is discontinuous?” would be “No, but neither is it continuous, but a third thing that tends to confuse people.”
Edit: retracted because I don’t think this is helpful.