Is the Planck length the smallest length that exists in the universe or is it the smallest length that can be observed?

Question

I have heard both that Planck length is the smallest length that there is in the universe (whatever this means) and that it is the smallest thing that can be observed because if we wanted to observe something smaller, it  would require so much energy that would create a black hole (or our physics break down). So what is it, if there is a difference at all.

zeldredge · Accepted Answer

Short answer: nobody knows, but the Planck length is more numerology than physics at this point

Long answer: Suppose you are a theoretical physicist. Your work doesn't involve units, just math--you never use the fact that $c = 3 times 10^8 m/s$, but you probably have $c$ pop up in a few different places. Since you never work with actual physical measurements, you decide to work in units with $c = 1$, and then you figure when you get to the end of the equations you'll multiply by/divide by $c$ until you get the right units. So you're doing relativity, you write $E = m$, and when you find that the speed of an object is .5 you realize it must be $.5 c$, etc. You realize that $c$ is in some sense a "natural scale" for lengths, times, speeds, etc. Fast forward, and you start noticing there are a few constants like this that give natural scales for the universe. For instance, $hbar$ tends to characterize when quantum effects start mattering--often people say that the classical limit is the limit where $hbar to 0$, although it can be more subtle than that.

So, anyway, you start figuring out how to construct fundamental units this way. The speed of light gives a speed scale, but how can you get a length scale? Turns out you need to squash it together with a few other fundamental constants, and you get:
$$
ell_p = sqrt{ frac{hbar G}{c^3}}
$$
I encourage you to work it out; it has units of length. So that's cool! Maybe it means something important? It's REALLY small, after all--$approx 10^{-35} m$. Maybe it's the smallest thing there is!

But let's calm down a second. What if I did this for mass, to find the "Planck mass"? I get:
$$
m_p = sqrt{frac{hbar c}{G}} approx 21 mu g
$$

Ok, well, micrograms ain't huge, but to a particle physicist they're enormous. But this is hardly any sort of fundamental limit to anything. It isn't the world's smallest mass. Wikipedia claims that if a charged object had a mass this large, it would collapse--but charged point particles don't have even close to this mass, so that's kind of irrelevant.

It's not that these things are pointless--they do make math easier in a lot of cases, and they tell you how to work in these arbitrary theorists' units. But right now, there isn't a good reason in experiment or in most modern theory to believe that it means very much besides providing a scale.

ACuriousMind · Answer

None of the above. Though there are many speculations about the significance of the Planck length, none is proven in any currently accepted theory.

It is expected, though, that quantum gravity effects become definitely non-neglegible at the energy/distance scale set by the Planck length, so it provides a heuristic scale at which we should not expect our current theories to make accurate prediction.

André · Answer

There is a tiny bit more going on than the otherwise excellent answer by zeldrege suggests. Imagine that you wish to probe an unspecified object to examine its structure. If we use light to look at the structure of an object, we need to have its wavelength smaller than the size of the details we wish to look at. Probing an object that has a (linear) size equal to the Planck length, requires that the energy of the photon be greater than the mass of a black hole of that "size". So, a classical black hole would be formed by our energy probe, thus preventing us to see details inside the object we wish to investigate. We are lead to an apparent contradiction, which suggests an incompatibility between Relativity and Q.M.

Answered by André on November 29, 2020

Árpád Szendrei · Answer

I would only like to add a few things that were not mentioned in the other answers.

We all forget that the Planck-length comes from the Planck-constant, which comes originally from the photoelectric effect, and from Planck's discovery about the correlation between the energy and the frequency of the photon (and originally the difference between the energy levels of the emitting electron).
If we look at QM, we could express all particles as having a wave-function, and all particles having a probability distribution, a frequency and a wavelength as well. But let's just take photons now. If a photon has a certain frequency, it corresponds to it's wavelength too, and the smallest frequency could be (in theory) set to the Planck-length. Why? Because Em waves are information. If you think of a photon as information, then it's frequency can be measured experimentally. And it can be expressed as f=E/h. So if we have a photon (or any particle) with frequency so small, that will correspond to a wavelength that is the scale of the universe, it will not make sense any more. We are simply not able (theoretically we are, but it makes no sense) to express anything with a frequency smaller then that.
If the frequency cannot be smaller then that, then it makes sense to limit it so something. I understand that it is not the exact match with the Planck-length, but still there is a sense in limiting the frequency-minimum (and the wavelength maximum) even if we live in an accelerating expansion.
The reason that I choose photons is that they represent information. And if the wavelength of information itself cannot be larger then the scale of the universe, then it's frequency cannot be smaller then the corresponding frequency. So whether that minimal frequency is exactly the Planck-time (and so corresponds to the Planck-length and the Planck-constant) or not, is another question, especially because it changes as the universe expands.

But since information itself cannot have a smaller frequency then that, I believe it does not make sense to talk about anything (with a frequency of) smaller then that size (and since nothing can have a larger wavelength then the scale of the universe).

Of course this does not correspond to the spatial extension of particles, which, to our knowledge today in some cases are point like (and thus smaller then the Planck-length). I have no information on how we can measure experimentally something's spatial extension if this extension is smaller then the smallest possible frequency of a photon (that has the wavelength of the sale of the universe),  but it would be nice to know.

Is the Planck length the smallest length that exists in the universe or is it the smallest length that can be observed?

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