Sunday, February 02, 2020

Does nature have a minimal length?

Molecules are made of atoms. Atomic nuclei are made of neutrons and protons. And the neutrons and protons are made of quarks and gluons. Many physicists think that this is not the end of the story, but that quarks and gluons are made of even smaller things, for example the tiny vibrating strings that string theory is all about. But then what? Are strings made of smaller things again? Or is there a smallest scale beyond which nature just does not have any further structure? Does nature have a minimal length?

This is what we will talk about today.

When physicists talk about a minimal length, they usually mean the Planck length, which is about 10-35 meters. The Planck length is named after Max Planck, who introduced it in 1899. 10-35 meters sounds tiny and indeed it is damned tiny.

To give you an idea, think of the tunnel of the Large Hadron Collider. It’s a ring with a diameter of about 10 kilometers. The Planck length compares to the diameter of a proton as the radius of a proton to the diameter of the Large Hadron Collider.

Currently, the smallest structures that we can study are about ten to the minus nineteen meters. That’s what we can do with the energies produced at the Large Hadron Collider and that is still sixteen orders of magnitude larger than the Planck length.

What’s so special about the Planck length? The Planck length seems to be setting a limit to how small a structure can be so that we can still measure it. That’s because to measure small structures, we need to compress more energy into small volumes of space. That’s basically what we do with particle accelerators. Higher energy allows us to find out what happens on shorter distances. But if you stuff too much energy into a small volume, you will make a black hole.

More concretely, if you have an energy E, that will in the best case allow you to resolve a distance of about ℏc/E. I will call that distance Δx. Here, c is the speed of light and ℏ is a constant of nature, called Planck’s constant. Yes, that’s the same Planck! This relation comes from the uncertainty principle of quantum mechanics. So, higher energies let you resolve smaller structures.

Now you can ask, if I turn up the energy and the size I can resolve gets smaller, when do I get a black hole? Well that happens, if the Schwarzschild radius associated with the energy is similar to the distance you are trying to measure. That’s not difficult to calculate. So let’s do it.

The Schwarzschild radius is approximately M times G/c2 where G is Newton’s constant and M is the mass. We are asking, when is that radius similar to the distance Δx. As you almost certainly know, the mass associated with the energy is E=Mc2. And, as we previously saw, that energy is just ℏcx. You can then solve this equation for Δx. And this is what we call the Planck length. It is associated with an energy called the Planck energy. If you go to higher energies than that, you will just make larger black holes. So the Planck length is the shortest distance you can measure.

Now, this is a neat estimate and it’s not entirely wrong, but it’s not a rigorous derivation. If you start thinking about it, it’s a little handwavy, so let me assure you there are much more rigorous ways to do this calculation, and the conclusion remains basically the same. If you combine quantum mechanics with gravity, then the Planck length seems to set a limit to the resolution of structures. That’s why physicists think nature may have a fundamentally minimal length.

Max Planck by the way did not come up with the Planck length because he thought it was a minimal length. He came up with that simply because it’s the only unit of dimension length you can create from the fundamental constants, c, the speed of light, G, Newton’s constant, and ℏ. He thought that was interesting because, as he wrote in his 1899 paper, these would be natural units that also aliens would use.

The idea that the Planck length is a minimal length only came up after the development of general relativity when physicists started thinking about how to quantize gravity. Today, this idea is supported by attempts to develop a theory of quantum gravity, which I told you about in an earlier video.

In string theory, for example, if you squeeze too much energy into a string it will start spreading out. In Loop Quantum Gravity, the loops themselves have a finite size, given by the Planck length. In Asymptotically Safe Gravity, the gravitational force becomes weaker at high energies, so beyond a certain point you can no longer improve your resolution.

When I speak about a minimal length, a lot of people seem to have a particular image in mind, which is that the minimal length works like a kind of discretization, a pixilation of an photo or something like that. But that is most definitely the wrong image. The minimal length that we are talking about here is more like an unavoidable blur on an image, some kind of fundamental fuzziness that nature has. It may, but does not necessarily come with a discretization.

What does this all mean? Well, it means that we might be close to finding a final theory, one that describes nature at its most fundamental level and there is nothing more beyond that. That is possible, but. Remember that the arguments for the existence of a minimal length rest on extrapolating 16 orders magnitude below the distances what we have tested so far. That’s a lot. That extrapolation might just be wrong. Even though we do not currently have any reason to think that there should be something new on distances even shorter than the Planck length, that situation might change in the future.

Still, I find it intriguing that for all we currently know, it is not necessary to think about distances shorter than the Planck length.


  1. Hi SABINE !!!

    I hope your day is going well.

    I just happened to be up at the moment, and caught your post.

    It was very interesting
    - and thank you for it .
    ... even Plank's comment
    (consideration) of 'aliens'.
    ... interesting.

    My question, though, is
    Why did you include asymtotic safety , when (if you follow it to the end) it becomes readily apparent that - gravity is
    'ruled out'.
    - and you neglected to mention
    Causal Dynamical Triangulation.

    All Love,

    Love Your Work !

    1. I don't know what you even mean by "gravity is ruled out".

      In Causal Dynamical Triangulation the discreteness scale is a regulator and one eventually takes the limit of that scale to zero. Ie, it doesn't have a minimal length by construction, though it may have in less obvious ways, eg because it's ultimately linked to asymptotically safe gravity. In any case, for all I can tell it's presently unclear.

    2. Hi, My Dear.
      Sorry for the late response.
      All Apologies for a personal term 'ruled out'.
      (that's just how I view some incomplete theories)

      As to Asymtotic safe gravity; or ' Asymtotic Safety'. . .

      When you get to the end
      of it,.
      There is a particular
      dynamic arrangement
      of causal sets
      that will tell you.

      - This theory
      does not apply
      to gravity.

      Perhaps I can get
      'all tech on you'
      - some other time.

      Love Your Work!

  2. Did Planck, equal to Einstein, never experimental research?

    So, Planck is, prior to Einstein, the inventor of "Lost in Math"?

    Physics has lost it touch to reality, completely.

    1. weristdas,

      Why don't you at least look at Wikipedia before you bother us with your exceedingly ill-informed opinions?

  3. I largely agree with your discussion here. The Planck length is a scale where the Schwarschild radius r_s = 2GM/c^2 is such that the circumference 2πr_s is for a minimal standing wave so ½ λ = 2πr_s and the wavelength is λ = ħ/Mc. This is the Compton wavelength. This then gives the Planck mass from which we can use the Heisenberg uncertainty principle to get the Planck length ℓ_p = √{8πG/c^3}. I like this with the 8π where this agrees with the string length or Hagedorn temperature of the string.

    The Planck length I think was introduced by Matvei Bronstein in the 1930s. He was in some ways the initial pioneer of work on quantum gravitation. He was arrested in a Stalinist purge in 1937 and executed. Planck introduced the fundamental unit of action h, which we most often express as ħ = h/2π, sometimes called the Dirac constant.

    Happy Groundhog Day.

    1. The Planck length should read above as ℓ_p = √{8πGħ/c^3}.

    2. Here is Bronstein’s argument described by Rovelli.
      Let me emphasize that the Planck length is just a feature of GR and QM, i.e. of classical, non-quantized curved spacetime and QM. No need to quantize spacetime or gravity.
      The big question is how and “when” QM stuff backreacts on the spacetime curvature.

    3. I was going to continue, but I had to pause, so to continue … .

      It is the case the Planck scale is not about some discrete structure to spacetime. It is not the case that spacetime is somehow sliced and diced into chunks. The NASA Fermi and ESA Integral spacecrafts measured the photons that came from burstars billions of lightyears distant. Photon of various wavelengths arrived simultaneously. With the loop quantum gravity paradigm spacetime has a discrete tessellated structure, where there is near the Planck scale a breaking of Lorentz symmetry. This means that a photon with a shorter wavelength couples to these discrete elements of spacetime and the breaking of symmetry is manifested in a longitudinal component to photons that depends on the energy of the photon. This is a sort of induced “photon mass” dependent on energy. There would be a dispersion that depends on the frequency, equivalently wavelength, of a photon that would mean photons have different propagation speeds. None of this was found, and spacetime appears based on these measurements smooth to within 1/50 of the Planck length.

      We might ask what is meant by the Planck length. This is the minimal size, area (pixel) or volume (voxel) where one can isolate a quantum bit of information or qubit. The idea of a discrete structure though may play a role, but not in the way often thought. It is possible to construct spacetime with discrete systems at least for numerical computations. Regge calculus or null-strut Regge calculus are cases, and form ideas of quantum gravitation the dynamic triangulation may also fit in. These discrete structure to spacetime, where if the Planck scale is some minimal length a qubit can be isolate, this is possibly a variant of a form of Weyl chamber called a Kirwan polytope. These are polytopes with a topological index that are obstructions to one type of entanglement transforming into another. The classic case is the distinction between W and GHZ entanglements. A discrete structure in spacetime might then possibly be some underlying structure for how spacetime is built up from entanglements, think of the tessellation as a massive mesh of Kirwan cells. Smooth spacetime is then a fibration over this.

      Spacetime as a fibration is then a sort of epiphenomenology over a more fundamental quantum physics. In a way this means the massive redundancy of description by gauge-like transformations could possibly be removed more completely as something fundamental.

    4. @Reimond: This is essentially what Sabine wrote and characteristic of various arguments. The quanta of black hole comes when one recognizes that this is also a Schwarzschild radius. In order to try to localize a region this small it requires a lot of energy focused in this tiny region. The result is then a quanta of black hole.

  4. An old question -- Does the Lorentz-Fitzgerald contraction become invalid at scales smaller than the Planck length? Is this question the basis of doubly special relativity?

    1. Bill,

      If you think that the minimal length scale is actually a distance in space-time, then the answer is yes. Alas, this is not the case in many examples. Just take Planck's original definition. It's a combination of constants of nature. It's equally constant and does not depend on the observer.

      (This is why it is more accurate to call it a "minimal length scale" rather than just "minimal length", though this distinction gets so easily lost it seems futile to even make it.)

  5. A wonderful overview of a subject so central to the future of modern theory. There is also a paper published Jan. 15th that discusses the 'Physical Significance of Measure'. Perhaps of interest as it goes on to show that discrete measure is also correlated to the phenomenon of gravity, among other phenomena such as quantum entanglement, optics, various classical phenomna and finally an even more fundamental expression than that of Planck's constants: lm=2θt (θ = 3.26239).

    Journal of High Energy Physics, Gravitation and Cosmology

  6. I recall words of John Wheeler, he wrote: "Time is really length, not a new and independent concept according to general relativity. And, mass, too is another way of speaking about length. In brief, classical physics is nothing but lengths." (1962, page 14, Geometrodynamics). If that quote is taken literally, then a minimal length equates to a minimal time and a minimal mass. Of course, Wheeler was speaking only of classical physics. I have yet to see an entirely convincing argument for minimal length in nature. Polchinski's 1998 paper 'Quantum Gravity at the Planck Length' invokes minimal length in a section entitled 'An Alternative to String Theory ? ' His paper is worth reading for other reasons and almost convinces me (quoting Polchinski) "that all good ideas are part of string theory." Needless-to-say, after 40 years of following trends in quantum gravity, it appears to me as if physicists are no closer to that goal, no closer to finding "a final theory." My bet: There is no "final theory."


  7. I don't know much about these things; But I would like to know if I am wrong to think of h as a dynamic magnitude where the energy, length and time are correlated and inseparable?

  8. The quote should not be taken literally — the Planck mass is not the minimum mass (bacteria are much smaller).

    1. Huh? Who said that the Planck mass is a minimum mass? It most definitely is not. It is much larger than the masses of all known elementary particles.

    2. We might think of the Planck mass as a sort of maximal mass. It is the maximal mass that forms a horizon to conceal a single qubit behind an event horizon. A black hole of N Planck masses is the minimal mass that conceals N qubits in the Bekenstein bound. This concealment is then the entropy of the black hole defined by the area of the event horizon.

    3. Sabine,

      two posts higher Gary Alan said "...If that quote is taken literally, then a minimal length equates to a minimal time and a minimal mass..."

    4. stor,

      Thanks for the clarification.

  9. I would suggest that length contraction may just be changing the density structure of an object. Kind of turning the object more into like a balloon, instead of a solid ball.

  10. The Planck length is the shortest distance you can measure, certainly, but is reality always defined by what we can measure? I mean, the Planck length is defined by the distance travelled at "c" in one Planck time, but consider this - what is the distance covered in one Planck time for slower speeds than "c"? It has to be LESS than one Planck length right? So, this shows us that the Planck length is NOT the minimum length in reality and that space is continuous, right down to zero length. What IS constant is the Planck time and THIS is the fundamental unit.

    1. "the Planck length is defined by the distance travelled at "c" in one Planck time"

      The Planck length is defined as the square root of G \hbar/c^3, as I explain in my video. You are assuming incorrectly that it necessarily corresponds to a distance.

  11. I understand the "minimal length" in the same way as the inside of a black hole: Although there might be an "inside", we cannot see that from the outside. There probably is structure below the Plank length, but this structure is invisible from the "outside".

    I have seen several descriptions of our current physics as "emerging" from deeper concepts that also put the cut-off of our current physics knowledge at the Plank scale.

    A parallel with thermodynamics is generally used as an illustration. Thermodynamical concepts, like temperature and pressure, become meaningless when you describe individual atoms. In the same way, QM and GR might become meaningless when you are looking at scales below the Plank scale.

    In this view, the Plank length would not be a minimal length, but just the length below which our current physics becomes useless. More fundamental models could work well below the Plank length, provided distance even makes sense at this level, that is.

  12. Do every measurement have to exchange energy?

  13. Sabine,

    Why did Planck introduce his units? He wanted a system of physical quantities which is independent of a specific definitions of these units.

    From this reason all fundamental physical quantities had to be represented in his new units. Also force was to be represented. Here Planck’s choice was gravity, so the gravitational constant G. Because at that time the gravitational force was understood to be the most fundamental and least questionable force. But this has strongly changed since that time. In 2005 the prior director of the German Einstein Institute, Jürgen Ehlers, made the following statement:

    In former times gravity was taken as the simplest and best understood force. But the situation has changed such that it is now the least understood and most enigmatic force.

    (One could asked these days whether gravity is a force at all; in Einstein’s system it is not.)

    Should we continue the present use of the Planck units in the view of this changed understanding? Gravity is – if seen as a force – an extremely weak force. This influences the Planck units in the way that some values are extremely small and others extremely big – dependent of G being in the nominator or in the enumerator of the particular equation. If the Planck system would be set up again using another force as fundamental like the electric or the strong one, this would change. Change in a way that the spread of the units would be much smaller. And maybe this would fit better in our general understanding of physics.

  14. Black holes are pretty detectable, aren't they?
    We can measure their mass, position and radius, and even capture shadow "image".
    Black holes may collide together and produce some measurable effects.
    Very small black holes are unstable and evaporates quickly releasing all information.
    If black holes are measurable in so many ways, why blame them for distance measure limit?

  15. Has any one considered that most or all the Planck units could be missing a dimensionless number?

  16. how is planck time represented in string theory LQG, AS etc?

    since GR combines space and time, is planck time also a wilson loop in LQG, for example

    1. I don't know about wonderful but it is quite concise and comprehensive :) for completeness sake, there is a broken reference in 3.1.7 on page 21 of the pdf.

  17. 16 orders of magnitude ... a lot of new physics could be hiding there!

    A lot of what's presented in this blogpost rides on extrapolation from the space probed by experiments and observations. Take "G": from memory, the Newtonian inverse square behavior, with a constant that does seem to be constant, has been tested on scales down to mm (or even microns?). We do not yet know if gravity (and hence G) behaves the same way at nanometer scales, let alone at yoctometer (10^-24 m) ones.

    Is G beautiful?

  18. Jean Tate,

    It seems that G is not beautiful.

    In the present understanding of gravity it is not even for sure that G is a constant in the long run. And even now, the different teams measuring G have deviations to each other which are beyond the uncertainty of their measuring methods. So, something doesn't seem to be understood.

    My question again: Why does G have a place in the Planck units?

    1. antooneo,

      This particular blogpost is mostly about implications from theory. As such, the nature of G, and its role in theory (beyond it being a/the "gravity constant"), is beyond scope.

      Now if Sabine were to one day write a blogpost inviting ideas for seemingly feasible fundamental physics experiments a whole lot cheaper than an LHC successor, ... ;-)

    2. Easiest way to think about this: Think about the numerator in Newton's law of gravitation. G*M_P^2 == h-bar c is how we define the Planck mass M_P from G, h-bar and c. So that is a natural mass scale from these three fundamental constants. Planck energy, length, time, temperature etc. follow from there.

  19. I like the comparison, I believe due to Jim Baggott, that the Planck Length is to an atom what an amoeba is to the Milky Way Galaxy. (I checked the math, and it is correct.)

  20. The most salient feature about the Planck length, IMHO, is that Planck energies packed into a Planck length^3 volume also have a Schwarzschild radius which Sabine mentions. This calculates to be 2 times the Planck length as Lawrence notes. But since the Planck vacuum is full of fluctuations this is really a statistical average. J. A. Wheeler pioneered this area with the "geometrodynamic vacuum."

    So, nature certainly does not have a minimum length, but is does have a length at which, if we want to observe anything smaller, we would find ourselves blocked by a black hole horizon according to classical general relativity. But, via Hawking, what we really would run into with quantum theory accounted for, is a blackbody spectrum.

    The question I find of greatest interest, is whether the Planck spectrum we observe from heating a blackbody is in fact just a many-of-orders-of-magnitude-removed sighting of this Planck-scale black hole radiation spectrum. And if the Cosmic Microwave Radiation Background is as cold as observations of this spectrum ever get. If so, then what Planck really put a curve to in 1901 was an empirical manifestation of quantum gravity at the Planck scale.

  21. Sabine, I am perplexed at how Planks original energy relation has now become a metric in meters?
    I have spent some time with eV=vh and find it is Voltage based, with all sources measured using Voltage or Energy sources; even the old books call it a Energy relation?

  22. ##SORRY## moments after I wrote my rant comment I realized my error NOW I see how you get meters (in a co-factor sense with h acting as a multiplicative scalar)! apologies again and delete both notes

  23. I don't know much about these things; But I would like to know if I am wrong to think of h as a dynamic magnitude where the energy, length and time are correlated and inseparable?


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