This question came from Brian Clegg who has written a stack of popular science books and who is also to be found on twitter and uses the same blogger template as I.“Given that the Planck mass is nothing like a minimum possible mass, why do some people assume that the Planck length/time are the minimum possible length and time? If we have no problem with there being masses that are (much) smaller than the Planck mass, why do we assume that the Planck length and time are the fundamental units of quantisation? Why could there not be smaller distances and times, just as there are smaller masses?”

Dear Brian,

Before I answer your question, I first have to clarify the relation between the scales of energy and distance. We test structures by studying their interaction with particles, for example in a light microscopes. In this case the shortest structures we can resolve are of the order of the wavelength of the light. This means if we want to probe increasingly shorter distances, we need shorter wavelengths. The wavelength of light is inversely proportional to the light’s energy, so we have to reach higher energies to probe shorter distances. An x-ray microscope for example can resolve finer structures than a microscope that uses light in the visible range.

That the resolution of shorter distances requires higher energies isn’t only so for measurements with light. Quantum mechanics tells us that all particles have a wavelength and the same applies for measurements made by interactions with electrons, protons, and all other particles. Indeed using heavier particles is of advantage to get short wavelengths, which is why electron microscopes reveal more details than light microscopes. Besides this, electrically charged particles are easier to speed up to high energies than neutral ones.The reason we build particle colliders is because of this simple relation: larger energies resolve shorter distances. Your final question thus should have been “Why could there not be smaller distances [than the Planck length], just as there are

*larger*masses?”

One obtains the Planck mass, length, and time when one estimates the scale at which quantum gravity becomes important, when the quantum uncertainty in the curvature of space-time would noticeably distort the measurement of distances. One then obtains a typical curvature, and a related energy-density, at which quantum gravity becomes relevant. And this scale is given by the Planck scale, that is an energy of about 10

^{15}times that of the LHC, or 10

^{-35}meter respectively. This is such a tiny distance/high energy that we cannot reach it with particle colliders, not now and not anytime in the soon future.

“Why do some people assume that the Planck length/time are the minimum possible length and time?”

It isn’t so much assumed as that it is a hint we take from estimates, thought experiments, and existing approaches to quantum gravity. It seems that the very definition of the Planck-scale – it being where quantum gravity becomes relevant – prevents us from resolving structures any shorter than this. Quantum gravity implies that space and time themselves are subject to quantum uncertainty and fluctuations, and once you reach the Planck scale these uncertainties of space-time add to the quantum uncertainties of the matter in space-time. But since gravity becomes stronger at higher energy density, going to even higher energies does no longer improve the resolution, it just causes stronger quantum gravitational effects.

You have probably heard that nobody yet knows how the complete theory of quantum gravity looks like because quantizing it can’t be done the same way as one quantizes the other interactions – this just leads to meaningless infinite results. However, a common way to tame infinities is to cut off integrals or to discretize the space one integrates over. In both cases this can be done by introducing a minimal length or a maximal energy scale. And since the Planck length and Planck energy are the only scales that we know of for what quantum gravity is concerned, it is often assumed that they play the role of this “regulator” that deals with the infinities.

More important than this general expectation though is that indeed almost all existing approaches to quantum gravity introduce a short-distance regulator in one way or the other. String theory resolves short distances by getting rid of point particles and instead deals only with extended objects. In Loop Quantum Gravity, one has found a smallest unit of area and of volume. Causal Dynamical Triangulation is built on the very idea of discretizing space-time in finite units. In Asymptotically Safe Gravity, the resolution of distances below the Planck length seems prohibited because the strength of gravity depends on the resolution. Moreover, the deep connections that have been discovered between thermodynamics and gravity also indicate that Planck length areas are smallest units in which information can be stored.

“Why do we assume that the Planck length and time are the fundamental units of quantisation?”

This isn’t commonly assumed. It is assumed in Loop Quantum Gravity, and a lot of people don’t think this makes sense.

“Why could there not be smaller distances and times, just as there are larger masses?”

The question isn’t so much whether there

*are*distances shorter than (this question might not even be meaningful), but whether you can ever

*measure*anything shorter than. And it’s this resolution of short distances that seems to be spoiled by quantum gravitational effects.

There is very relevant distinction you must draw here, which is that in most cases physicists speak of the Planck length as a “minimum length scale”, not just a “minimum length.” The reason is that not any quantity with a dimension length is a length of something, and not any quantity with a dimension of energy is an energy of something. It is not the distance itself that matters but the space-time curvature, which has the dimension of an inverse distance squared. So if a physicist says “quantum gravity becomes important at the Planck length” what they actually mean is “quantum gravity becomes important when the curvature is of the order of an inverse Planck length squared.” Or “energy densities in the center-of-mass frame are of the order Planck mass to the forth power”. Or at least that’s what they should mean...

And so, that the Planck length acts as a minimal length scale means that particle collisions that cram increasingly more energy into a smaller volumes will not reveal any new substructures once past Planckian curvature. Instead they may produce larger extended objects such as (higher-dimensional) black holes or stringballs and other exotic things. But note that in all these cases it is perfectly possible to exceed Planckian energies; this is referred to as “superplanckian scattering.”

Some models, such as deformed special relativity, do have an actual maximal energy. In this case the maximum energy is that of an elementary particle, which seems okay since we have never seen an elementary particle with an energy exceeding the Planck energy. In these models it is assumed that the upper bound for energies does not apply to composite particles, that you probably had in mind. The Planck energy is a huge energy to have for elementary particles, but its mass equivalent of 10

^{-5}gram is tiny on every-day scales. Just exactly how it should come about that in these models the maximum energy bound does not apply for composite objects is not well understood and some people, such as I, believe that it isn’t mathematically consistent.

Thanks for an interesting question!

## 24 comments:

Hi,

two quick comments: It was one of the excitements about D-Branes in the second half of the 1990s that these objects might allow you to probe distance scales shorter than the string length (which is likely the Planck length under these circumetances), as their length scale differs by a factor of the (dimensionless) string coupling g_s which could be small. The pioneering paper at the time was by Douglas, Kabat, Pouliot and Shenker http://arxiv.org/abs/hep-th/9608024

comment number two is that almost exactly 10 years ago, I tried to explain a similar point in my blog by comparing it to atomic physics where by dimensional arguments you can guess the Bohr radius: http://atdotde.blogspot.de/2005/09/natural-scales.html

Best

Robert

Hi Bee,

good comment on your question

Kay

Hi Bee,

I saw a small video by Brian Greene. He said that the quantization of gravity is so difficult, since Gravity is a smooth theory and quantum mechanics is so chaotic. Do you agree?

Take care Kay

Well said.

Hi Robert,

Thanks for your comment. Yes, you are right. I didn't want to go into the difference between the string length and the Planck length. I discussed the paper you mention in section 3.2.3 of my review :o) Best,

Sabne

Hi Kay,

Strikes me as odd. I'd have thought it's exactly the opposite, it's quantum mechanics that is linear and gravity isn't. Not sure that this is why quantizing it is so difficult, but it certainly is a difference between the theories. Do you have a link to the video? Best,

B.

Thanks for the fantastic explanation. That was the clearest and most concise statement of the fundamental issues in quantum gravity I've come across. A couple (perhaps naive) questions came to me while reading that I would love to get clarification on from you since you seem to have a gift for exposition (or understanding).

You said that 'quantum gravity implies that space and time themselves are subject to quantum uncertainty and fluctuations'. If everything at sufficiently small space and time scales is (increasingly) seen to exhibit quantum effects, why isn't it believed that quantum mechanics itself implies this?

Also, you said (paraphrasing) wavelength is inversely proportional to energy. Is there, or why is there, a limit to the range of wavelengths? Does an increasingly short wavelength eventually lead to something we would consider 'massive' and no longer described by quantum mechanics? In other words, can we make sense of the notion of a conversion factor between wavelength and mass? ...And on the other end of the spectrum, does increasingly long wavelength become what we refer to as space? (Which, in a way, brings us back to the earlier question of why space itself is not considered quantum mechanical).

Thanks again for the above blog.

-JT

Jt,

The question is not so much whether or not space-time should be considered quantum-mechanically, but how to do it. For this you need a theory of quantum gravity, since gravity is the theory that tells you how space and time behave. It *is* believed that space and time also do have quantum properties, but just saying 'quantum properties' isn't a proper theory. Best,

B.

Thanks for your reply. When you say 'it is believed that space and time also have quantum properties' it makes me believe you didn't understand my question, likely because I didn't express it well enough.

If we see gravitational effects (time dilation, space time curvature) become more prominent at increasingly massive scales, and quantum mechanical properties (entanglement, uncertainty, etc) become more prominent at increasingly massless scales, then why don't we consider quantum mechanical properties (entanglement, uncertainty, etc), properties of masslessness (space) instead of properties of a class of things (particles) that have little mass?

In other words, we don't say that massive things have gravitational properties and now we have to understand why black holes (the greatest expression of mass) do as well; we understand them under the same logical structure of gravity. Likewise, instead of saying that light and other low energy (compared to massive objects) particles exhibit quantum mechanical properties, but now we have to understand in what way space does, why don't we take the perspective (as we do for gravity) that quantum mechanical properties (entanglement, uncertainty, etc) are (the properties of) space, which become increasingly less apparent when mass and the resulting gravitational effects are at play? Just as black holes are the most extreme expression of gravitational properties, shouldn't space be considered the full expression of what we refer to as quantum mechanical properties?

I'm saying: aren't black holes to gravity as 'space' is to quantum mechanics. I put space in quotations because I mean it in the sense of whatever it's properties would be in the complete absence of mass (which, given the perspective of being the opposite of black holes, would be expanding space).

Jt,

You are getting several things wrong there. First, it is wrong to say that gravitational effects become 'more prominent at massive scales'. The relevant scale is curvature or energy density, not mass. Second, it is wrong to say that quantum mechanical properties become more prominent at 'increasingly massless scales'. Quantum properties are extremely important for example in high energy scattering, and in many cases the quantum effects do get stronger with increasing energy (depends on the running of the coupling constant). Third, black holes are not 'greatest expressions of mass', black holes can exist with any mass. Fourth, the 'properties of space' do not become 'less apparent when gravitational effects are at play', I'd say the opposite is the case. Most importantly though, it's all words and physics doesn't work this way. You have to come up with a theory - that is a mathematical framework - that reproduces both quantum field theory and gravity in the appropriate limits and also achieves to quantize gravity. Best,

B.

Dear Sabine!

You have wrote "but whether you can ever measure anything shorter than". But I suppose that we cannot measure not only scales at 10(-35) but at length 10^(-30) either. So as Your sentence may be an explanation, it has not any other deeper meaning...

I especially like the derivation of Planck length by equating the Compton wavelength with Schwarzschild radius.

But anyway we must be cautious with any bottom-up derivations since in general QG≠GR+QM

fiksacie,

I think you misunderstood what I meant. We cannot, right now, in practice measure any structure below about 10^-20 meter or so, which is what the LHC currently probes. But there is nothing *in principle* prohibiting us from doing better than that. Until you get to the Planck scale, where, according to the arguments I list, resolution doesn't improve any more. Best,

Sabine

Giotis,

Yes, equating the Schwarzschild radius with the Compton wavelength is eventually always what it comes down to, but I don't like to phrase it this way. The reason you can put it this way is purely dimensional, as Planck himself already observed. The Planck scale is just the only scale you get when you combine hbar, c, and G, so any argument that only involves these scales will eventually get you there. I don't like the argument with the Schwarzschild radius because it makes many people believe that if you accelerate a particle it will eventually become a black hole, which is nonsense (I elaborated on this here). What you really need to make this argument work is to either use more and more massive particles (then you have to eventually deal with particles of Planck mass which is awkward) or you have to instead talk about com energy in a collision (which is what I normally do), but then you have to do some handwaving to still use the Schwarzschild solution (so instead you make a general scaling argument). Best,

B.

What mathematical properties can be present as a description of the Planck length regime, and help us extend geometry into higher dimensions? If strings are not point particles, then what is a point particle being described as string amplitudes. Did changing the mathematical property become necessary when seeking to grasp an understanding of quantum gravity?

Markus,

I think this is basically correct. The problem with Penrose's idea is not the conformal invariance at the beginning of the universe, but at the 'end' of the universe. For this you somehow have to get rid of all particle masses again. Best,

B.

Anyway as I said these are bottom-up heuristic arguments.

The thing we theoretically know for sure from known physics and we can trust is that at Planck scale new physics necessarily appears.

This is because in perturbative QG if we scatter gravitons at energies higher than Planck energy we violate the unitarity bound for the scattering amplitude.

I like to summarize this schematically as follows:

QG=QFT+GR for E≤Ep since known physics (perturbative QG) is theoretically consistent

QG≠QFT+GR for E>Ep since known physics becomes theoretically inconsistent

I'm well aware that a theory needs to be mathematical and predictive! I do think you're wrong that 'it's all words and physics doesn't work that way' because I am simply trying to get to better understanding, which you have helped establish with words.

I should have used energy density instead of mass. To your second point, how would you describe the conditions in which qm effects are more prominent? Just small in time and length? And to your fourth point, I know that is what you would say, as it is what you stated previously. I was simply saying that if we can understand why space exhibits the properties (qm) it does at the 'other end of the spectrum' in regards to energy density in the same way (starting with a principle) we understand it to exhibit strong gravitational effects in black holes, then we would understand qm to be a necessary feature of space at a low energy density. Put differently, shouldn't we say that different 'properties of space' are more and less prominently featured at different energy density scales, rather than saying that gravity alone exposes these properties?

Again, I'm not trying to put forth a theory, just asking questions.

Jt,

"shouldn't we say that different 'properties of space' are more and less prominently featured at different energy density scales, rather than saying that gravity alone exposes these properties?"

What I am telling you is that this isn't the case (in any interpretation that I can give to these words) in current theories, and if you insist that this is what we "should say" then you do propose a new theory, which I kindly ask you to please do elsewhere. Thanks,

Sabine

I hear you and appreciate your straightforward and honest response. I was simply asking questions but recognize it went in the direction of suggestion. Thanks for the original response and I do look forward to future posts.

There is a simple and cool way to guess the Planck lenght from a principle. I learned this reading to Jack Ng... Take the Compton wavelenght (quantum mechanical!) for a particle of mass M, say λ=ℏ/Mc. Take now the Schwarzschild radius of a stationary black hole, say R=2GM/c². The Schwarzschild radius becomes comparable to the Compton wavelength when Rλ=L² (or =2L to give up the O(1) numerical factor). Then, simple arithmetic provides L²=(ℏ/Mc)(2GM/c²)=Gℏ/c³=(Planck Length)² :) So, the Planck length "naturally appears" when we combine gravity with Quantum Mechanics. Of course, what happens beyond Planck length is a mystery due to a lack of a quantum theory of gravity at those scales. Wheele's spacetime foam is the "broad" idea (not yet probed) that at distances around Planck length there is no spacetime continuum but a foamy version. It is quite likely that any notion of "usual geometry" can not be defined with (real) numbers beyond this length. Some radical ideas involve ultrametric and non-archimedean metric at that scale but they are an oddity yet. Any notion of fixed geometry at that scale is meaningless because we know that spacetime will fluctuate to that scale, even in topology, so what is left? Obviously, if classical spacetime fluctuates it fluctuates from a vacuum and from some constituents, what are they? What is the quantum gravity vacuum?

J,

You could have learned this on my blog.

Indeed, it could be I learned that from Ng AFTER I read your post years ago. Aging is definetively affecting my memories, even the gravitational. Anyway, thank you for your polite remark! I did not remember your post...

I wish I could reignite my blog soon, I am eager to post about stuff I have learned in the last months and transmit them. Also, I am aiming to search for invited posts when I reach #200. Are you interested? By the way, I am Juan, from http://thespectrumofriemannium.com, (we also met in Sweeden in the workshop for Science Writers) and I am trying to think about how to communicate better between the ridge we have between layman and professionals. I think I can do something else w.r.t. it. I will think about it next weeks/months. A bridge between 2 worlds: 2017 is going to be the year...

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