Is the Planck length the smallest measurable length?

[Danyon]

I read somewhere that you cannot tell the difference in position between a photon (Or any other particle) at some point in time and the same photon (Or other particle) less than a Planck time later, because it would not have traveled further than a Planck length, which apparently is the smallest measurable distance. I don't know how accurate that is, but it seems to lead to a contradiction. Consider a photon at three points in time, called A, B and C. Each point in time is separated by half a Planck time and half a Planck length, According to the theory, you cannot tell the difference between states A and B, nor can you tell the difference between B and C. If there is no measurable difference between states A and B and no measurable difference between B and C then there should be no difference between A and C, but A and C are separated by a Planck time and a Planck length, so a difference in position and time should be possible to measure. So which is it, can a difference be measured or not?
Does this contradiction Imply that distances smaller than a Planck length do not exist, and that space is a discrete grid?

The same paradox occurs when considering photon energies. By measuring a photon's energy you are measuring its wavelength. Consider three photons, the first photon "A", has some arbitrary energy, the second photon "B", has a wavelength of half a Planck length larger than the first photon, the third photon "C", has half a Planck length larger still. The same effect applies to the energy of these three photons. If the Planck length is the smallest measurable length then Photon "A" has indistinguishable energy from photon "B", which has indistinguishable energy from photon "C", but "A" and "C" are separated by a Planck length, which will allegedly allow us to measure a difference in energy. Which is a paradox. Alternatively, If the Planck length is the smallest measurable distance, then you might expect that all photon wavelengths occur in discrete multiples of the Planck length. I don't really know, any ideas?

 

[Vanadium 50]

"I read somewhere" is not a useful source, as nobody can find out what it is you have read. IKn this case, since what you read is wrong, none of the rest of your post is correct either.

 

[Danyon]

"I read somewhere" is not a useful source, as nobody can find out what it is you have read. IKn this case, since what you read is wrong, none of the rest of your post is correct either.

I got the information from the wikipedia article on "Instant" "Within the framework of the laws of physics as they are understood today, for times less than one Planck time apart, we can neither measure nor detect any change" Which probably isn't that good a source, but nevertheless what was exactly wrong about it? Surely if a particle changes position by an unmeasurable amount then you can't tell the difference in its position. Are you saying that particles cannot move by unmeasurable amounts? because I think that would imply that space is a discrete grid, but I'm not quite sure.

The paradox can be stated another way, assuming space is continuous and that particles can travel across smaller distances than a Planck length, and that such actions cannot be measured, then how can any amount of successive unmeasurable changes add up to a measurable change? Now that I mention it, this problem looks exactly like Zeno's paradox. But instead of instants, we have times less than Planck times. Zeno's paradox states that the motion between two points is made up of infinite instants, in each instant the object in motion does not move, Zeno thought that mean't that the object did not actually move.

 

[bhobba]

The paradox can be stated another way, assuming space is continuous and that particles can travel across smaller distances than a Planck length, and that such actions cannot be measured,

You are getting confused between what can be measured and what happens.

First we don't know enough about the Plank scale to say anything for sure - even if it can be measured or not - its pure conjecture. At present space-time is modeled continuously and all our usual axioms of real analysis apply so things like position and time can be any value - there is no limitation.

Thanks
Bill

 

[DrChinese]

I got the information from the wikipedia article on "Instant" "Within the framework of the laws of physics as they are understood today, for times less than one Planck time apart, we can neither measure nor detect any change" Which probably isn't that good a source, but nevertheless what was exactly wrong about it? Surely if a particle changes position by an unmeasurable amount then you can't tell the difference in its position. Are you saying that particles cannot move by unmeasurable amounts? because I think that would imply that space is a discrete grid, but I'm not quite sure.

The paradox can be stated another way, assuming space is continuous and that particles can travel across smaller distances than a Planck length, and that such actions cannot be measured, then how can any amount of successive unmeasurable changes add up to a measurable change? Now that I mention it, this problem looks exactly like Zeno's paradox. But instead of instants, we have times less than Planck times. Zeno's paradox states that the motion between two points is made up of infinite instants, in each instant the object in motion does not move, Zeno thought that mean't that the object did not actually move.

I am sure you are aware that a paradox implies an unwarranted assumption or a tautology (or both). In the case you gave earlier, you talked about "Each point in time is separated by half a Planck time and half a Planck length". Do you see the issue? How could you have that (with certainty) if a Planck length is the smallest measurable unit?

Also, you must be aware that quantum objects do not occupy space time regions like a small billiard ball (which is an assumption of your model). Any free electron can be said to occupy the observable universe as much as it can be said to occupy a point.

 

[Danyon]

You are getting confused between what can be measured and what happens.

First we don't know enough about the Plank scale to say anything for sure - even if it can be measured or not - its pure conjecture. At present space-time is modeled continuously and all our usual axioms of real analysis apply so things like position and time can be any value - there is no limitation.

Thanks
Bill

I'm aware of our lack of understanding of the Planck scale, but I still thought the paradox was interesting enough to warrant discussion. I'm curious as to what you think of the three photons of different energies, because space-time is modeled continuously I assume that means that the wavelength of a photon can be any length, and if that's so then the photons described in my post would exist. If you where able to measure the exact energy difference between "A" and "B" (From my example) or "B" and "C", you would be able to say that photon "B"s wavelength is half a Planck length larger than photon "A"s and smaller than "C"s and that the half a Planck length difference results in meaningful differences between the photons.
If the conjecture that the Planck length is the smallest measurable length holds true, then that would imply that there should be no measurable difference between the energy of photon "A" and photon "B" and no measurable difference between "B" and "C". But there should be a difference between "A" and "C". This is clearly contradictory. On top of that photon "B" appears to have an energy apparently indistinguishable from two photons of measurably different energies. which again is contradictory. I take this to mean that either space is quantised, or there is in fact a measurable difference between distances smaller than a Planck length.

 

[bhobba]

If the conjecture that the Planck length is the smallest measurable length holds true, then that would imply that there should be no measurable difference between the energy of photon "A" and photon "B" and no measurable difference between "B" and "C". But there should be a difference between "A" and "C". This is clearly contradictory.

You realize everything you measure with has a certain tolerance? That has zero implication for what can theoretically occur.

Our knowledge of the plank scale is so non existent what's going on is simply guessing of zero actual value.

Thanks
Bill

 

[Danyon]

You realize everything you measure with has a certain tolerance? That has zero implication for what can theoretically occur.

Our knowledge of the plank scale is so non existent what's going on is simply guessing of zero actual value.

Thanks
Bill

I'm aware that measuring devices have a certain tolerance, and I'm aware that Planck scale is not proven to be physically significant, but that doesn't mean discussing hypotheticals about it can't be interesting, especially when lots of people seem to claim that the Planck scale is physically meaningful. String theory, black hole physics and loop quantum gravity all use Planck scale units. It seems to me that regardless of whether or not you measure the energy of photons A and B or not, the photons still had some definite actual energy(Assuming their energy is discrete). That energy will be different for each photon or it will be the same. The question is whether or not the universe creates a difference in energy between A and B, does the universe distinguish between A and B, and B and C regardless of human measurement, you don't have to answer that, it was rhetorical.

 

[Danyon]

I am sure you are aware that a paradox implies an unwarranted assumption or a tautology (or both). In the case you gave earlier, you talked about "Each point in time is separated by half a Planck time and half a Planck length". Do you see the issue? How could you have that (with certainty) if a Planck length is the smallest measurable unit?

Also, you must be aware that quantum objects do not occupy space time regions like a small billiard ball (which is an assumption of your model). Any free electron can be said to occupy the observable universe as much as it can be said to occupy a point.

While it is allegedly the smallest measurable length, it may not be the smallest actual length. I started the thought experiment assuming that space is continuous (Though I did not write that) and that particles move through it continuously, I assumed that the Planck length is just the smallest measurable length. My assumptions should have been written. Yes, I did understand that paradoxes imply unwarranted assumptions, I reasoned out that the unwarranted assumption was either that space was continuous, or that the Planck length is the smallest measurable length. It doesn't seem like it could be both as that would lead to a contradiction. If space was not continuous, and therefore was discrete, then you couldn't measure anything smaller than a Planck length.

Would you agree that the wavelength of a discrete photon is defined by the distance between two points? Specifically, each point will be placed on two adjacent crests, Each point acting like a tiny billiard ball with a specific location in space. These points define the energy of the photon. So at least the photon example should still apply (I think, maybe). Also, on the topic of paradoxes, I thought of a paradox a few months ago that I thought was pretty interesting, it doesn't seem to have unwarranted assumptions or tautologies however. It goes as following. "A man suddenly develops the belief that he has a delusion" Is the man deluded? If the man is deluded then his belief is correct and therefore he is not deluded, which means he is deluded and so on. If the man if not delusional, then the man's belief is wrong and therefore he is delusional which means he isn't and so on. It's a variant of "This statement is false"

 

[Nugatory]

Also, on the topic of paradoxes, I thought of a paradox a few months ago that I thought was pretty interesting, it doesn't seem to have unwarranted assumptions or tautologies however. It goes as following. "A man suddenly develops the belief that he has a delusion" Is the man deluded? If the man is deluded then his belief is correct and therefore he is not deluded, which means he is deluded and so on. If the man if not delusional, then the man's belief is wrong and therefore he is delusional which means he isn't and so on. It's a variant of "This statement is false"

You might want to google for "Russell's Paradox". However, that's a completely different class of "paradox" than you're raising in this thread.

The apparent paradox that started this thread is based on a misunderstanding of what the Planck units are and what the wavelength of a photon is, so the thread can be closed. You will find a bunch of good threads here about what a photon is and is not - the picture most people have based on the pop-sci press is very misleading.