Of Particular Significance

How to Look for Signs of Extra Dimensions

Matt Strassler  [12 Jan 2012]

I’ve given you some examples suggesting how extra dimensions — dimensions of space of which we are unaware — might be present in nature. But I haven’t explained to you yet how scientists could figure out that they are there.

There are several basic strategies that one can employ, and over time I’ll describe a few of them.  But for the moment I’ll just focus my attention on one basic consequence of extra dimensions that is very general, and leads to a particle physics strategy that is relevant in many contexts, including research at the Large Hadron Collider.

I’m going to explain it in two steps. In the first step, using some rather simple physics, I’ll give you intuition that is rather simple, but imperfect (because it leaves out a basic feature of quantum mechanics), and gives an answer which is partly wrong. In the second step, I’ll fix the wrong part, which will require one more step in sophistication, and then you’ll see the full answer.

But before I explain it, let me tell you the answer first, so you know what it is that I have to explain to you.  Here it is, in a few different versions to help you get the point.

Any type of particle that moves in extra dimensions, as well as in the dimensions we know about, will appear, to naive observers such as ourselves who are unaware of the extra dimensions, as multiple types of particles that move only in the known dimensions and which differ little from each other except in their masses.

Said another way: if a type of particle can move in all the dimensions, it will appear to the uninformed observer as though nature has not only this particle (moving only in the known dimensions) but a set of partner particles, called “KK partners,” also moving only in the known directions, that differ very little from the original one, except that they are heavier.  “KK” stands for Kaluza and Klein, for reasons to be explained later.

To be more concrete: suppose that we live in four spatial dimensions, where three of the dimensions are large (the ones we know about) but the fourth is rather short in extent (as in the short dimension across a strip, the case I used extensively in previous examples.) [Short may be very short indeed, much smaller across than a proton.] Let’s call the distance across that dimension L.

Now suppose there is a type of particle that is very tiny, much smaller than L, and can move around freely within all four spatial dimensions. And suppose that there are clever observers who know this: they will say, “here is a type of particle that can move around in four dimensions, and it has a mass m”. Now consider naive observers such as ourselves, who are unaware of the small spatial dimension and think we live in a three-dimensional world. What we will say, after doing some experiments, is shown in Figure 1: “here is one type of particle that can move around in three dimensions, and it has a mass m; and lo, here is another type of particle that also can move around in three dimensions, and it is something like the first except that it has a mass M, much larger than m; and wow, here is yet another type of particle that moves in three dimensions, is something like the first, but it has a mass M’ larger than M; and now another type, of mass M”; and now another, and another …

Fig. 1: A type of particle that can travel in both known and unknown (i.e. extra) dimensions will appear to human scientists as a set of types of particles (the original particle of mass m and a family of heavy KK partners of mass greater than m) that can travel in the known dimensions. The details of the masses of the KK partners can be used to help determine the number, size and shape of the extra dimensions. The smaller the extra dimensions, the heavier the KK partners will be.

What sets the masses M, M’, M” and so forth is a combination of the fundamental mass m and the geometry of the extra spatial dimensions — in particular, MM’M” and so on are all inversely proportional to L. The smaller is L, the larger are M, M’, etc., and the more difficult it is to discover the heavy KK partners.  Moreover, the pattern of masses exhibited by the KK partners gives direct insight into the number and size and shape of the extra dimensions. [For those musically inclined, this fact is related to the observation that the precise harmonics that a musical instrument emits can give insight into its internal shape and size.]

Explicitly: if photons (particles of light) could move in one or more extra dimensions, like the small boat on a ship canal, then an observer who knows about the extra dimensions would describe them as massless particles (m=0) moving in all the dimensions. But what human scientists (who at present only know about a massless photon that travels in the three known dimensions) will discover, eventually, is a set of heavy photon-like particles. The smaller the size of the extra dimension, the larger the masses of the KK photons, and the harder they would be to discover — more precisely, the heavier they are, the more energetic a particle accelerator would have to be in order to have a chance of producing them.

Now it may very well be that several types of particles can move in the extra dimension(s), and in this case human scientists will discover heavy KK partners for each of these types of particles (see Figure 2). The discovery of a small number of heavy particles that resemble some known lightweight particles, and which exhibit a similar pattern of masses, as in Figure 2, would strongly suggest these new particles are KK partners, and the presence of one or more extra dimensions.

Fig. 2: If several particles can travel in the known and unknown (i.e. extra) dimensions, then each one will appear to human scientists as exhibiting its own set of KK partners (see Figure 1). If the various particles are really traveling in the same extra-dimensional space, the patterns of their KK partner masses will resemble each other, especially at higher masses.

So — that’s the basic answer: a type of particle that travels in extra dimensions as well as the known ones would reveal itself to us through the discovery of  its heavy KK partners. Later I’ll talk more explicitly about how one might try experimentally to produce and discover the KK partners. But first, why is this answer correct?

Click here for the partial explanation (relatively simple), and then here for the rest of the story [in preparation] (definitely more challenging, but also really, really interesting!)

53 Responses

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  4. To Matt Strassler: Q1) you said “The smaller is L, the larger are M, M’, etc., ..” ->why their masses depend on L?
    Q2)why KK photons look heavier? instead of looking bluer and staying massless?

    1. Hmm… how much physics do you already know? [Do you understand standing waves? If so, you should spend some time thinking about them, and then you should be able to guess the answers to these questions. If not, I have to answer differently.]

      1. 1)As L gets smaller, the wavelength that can fit in L must be smaller(shorter), which means it vibrates faster and comes with more (vibrational) energy ->more massive? is this correct? 2)Photon energy is given by h times its frequency but why is this true? is this still applicable to KK photon? Do KK photons all have the same color(frequency) in our observable dimensions? or do they get bluer as they get more massive? First of all, how can photons become massive? they cannot, or?

  5. hi matt just found this link on your site ref extra dimensions. Why is it assumed or preferred to think that the next dimension ( framework ) considered should have anything to do with matter or mass? in fact if it had any matter or mass could preclude it from being one? Is that not an assumption? Referring to your End of the World comments i.e. it’s gone past its sell by date being 12/12/12.

    Bible patrons are ‘still in date’ for their prediction: ‘ Mathew’ – sorry can’t remember which ‘verse 25’ I think from childhood RE. “The end of the world will be caused by a hot flaming ball from the East! No date attached so this prediction is somewhat open ended!

    1. When physicists say “extra dimensions”, they mean extra dimensions of space. That means that those dimensions, just like the three we know about, are places where things happen; objects have position, velocity, momentum in those directions, and motion in those directions contributes to motion-energy. Other types of dimensions that you might imagine are not the ones that are referred to by this term. Different examples of spatial and non-spatial dimensions are given here: http://profmattstrassler.com/articles-and-posts/some-speculative-theoretical-ideas-for-the-lhc/extra-dimensions/extra-dimensions-how-to-think-about-them/worlds-of-1-spatial-dimension/

  6. I hope you will be available to enlighten us a little bit about this phenomena [entangled particles] or at least suggest a book/paper to read it. There are many non-academic articles having and creating confusion about this topic.
    Thank you.

    1. In a comment elsewhere on this site, I recommended The Quantum Challenge by Greenstein and Zajonc. Their section 6.4, “Does Quantum Nonlocality Violate the Principle of Relativity?”, builds upon discussions of the Einstein–Podolsky–Rosen argument and Bell’s theorems earlier in chapters 5 and 6. I think the book has many good explanations, but it’s not easy going — it’s an actual textbook, which I used in an introductory “Modern Physics” class many years ago. (Disclosure: that class was taught by Zajonc.) Unfortunately, it has a typical textbook price, so I suggest you look for it in a library if you are interested.

      1. What I think is really needed for the public is a careful laying out of the logical possibilities — causality, locality, probability, and so forth — without the math. And then a discussion of what we do and don’t encounter in daily life, and how experiment contradicts what we’d expect from normal experience. That’s very tough to do well. Moreover, at some point it risks being so controversial that people won’t even agree about it… and it is very hard to explain things to the layperson that experts still can’t agree on.

    2. I’ll keep my eye out for a good layperson’s article. In two or three years I expect to have a lot more time to study this subject carefully enough to perhaps do it myself. Right now I couldn’t do a good enough job. The best articles are probably being written by experts in atomic physics (people working on laser-cooled atoms or ions, or Bose-Einstein condensates, …)

    1. Re: point #2 — there’s no physics in the choice of units — natural units are just a slightly unusual way of measuring things. You can measure a distance L with a ruler (meters), as usual. Or you can measure the distance L with a clock (seconds), by asking: “how much time would it take for light to travel a distance L?” Since the speed of light c is a constant in empty space, as far as we know, this question makes sense. (The answer, of course, is that the time T is L/c.) This does not mean that spatial dimensions are the same as time dimensions. (And the whole thing has *nothing* to do with how much energy is around.) It simply means that we may use the constancy of the speed of light to measure distances in space using a clock instead of a ruler.

      What makes time dimensions physically different from space is cause and effect: two events that are at the same time but at different points in space cannot affect each other, while two events that are at the same point in space but at different times can indeed affect one another — more precisely, the earlier one can affect the later one. This is the fundamental physical distinction between time and space in our universe. Choosing to use natural units — a choice that you are free to make or not to make — clearly doesn’t affect the physics.

      Several very big confusions are in your point 4.

      1) The time dimension in our universe is enormous — at least the age of the known universe (billions of years) in duration (which, if you use natural units and convert the time to a length, using the fact that in a certain amount of time light can travel a certain distance, is billions of light-years, an immense distance.) It is comparable in size to the 3 spatial dimensions that we are aware of, which also stretch no less than the distance across the visible universe. Again, I am not sure how you got the idea that the time dimension is small — that doesn’t make any sense, since even your own lifetime tells you it is not small.

      2) Particles such as the electron or photon may or may not have volume; no finite volume has ever been measured. The radius of an electron is known to be smaller than about 0.000,000,000,000,000,001 meters, and its volume smaller than (roughly) the cube of that radius. The proton has a volume which is roughly the cube of 0.000,000,000,000,001 meters.

      3) It does not entirely make sense to ask whether a particle occupies a certain amount of time, but there is a sense in which a particle doesn’t really exist if it doesn’t exist for a time which is of the order of Planck’s constant h-bar divided by the energy of the particle

      minimum time ~ h-bar/ E

      (where E cannot be smaller than m c-squared, m the mass of the particle, but it could be larger.) For any particle you ever come across in nature, that time will be far, far shorter than the age of the universe.

      1. How about entangled particles? Can’t they interact when they are at different points in the space at the same time?

        1. They cannot. Entangled means their properties are correlated, but it does not mean that they can interact with each other. It’s an extremely subtle point; I’m not going to try to do a decent job on it right now…

      2. To Matt Strassler: You said “3) It does not entirely make sense to ask whether a particle occupies a certain amount of time, but there is a sense in which a particle doesn’t really exist if it doesn’t exist for a time which is of the order of Planck’s constant h-bar divided by the energy of the particle

        minimum time ~ h-bar/ E”

        Q) Why is “minimum time ~ h-bar/ E” true? why is the minimum time (a particle can exist?) is related to its energy? where does this come from?

        1. This comes from quantum mechanics. A particle (which more accurately is a “quantum”) is a ripple in a field (specifically the ripple of smallest intensity). The vibration time for that ripple is hbar/E. If you try to say that a particle exists for a shorter time, it hasn’t even vibrated once… in which case, it isn’t really sensible to talk about it. It would be like asking about the tone made by the vibration of a guitar string when you stop the sound before the string vibrates even once.

          1. A stable particle, or wave (quantum) must satisfy that condition but what about non-coherent chaotic waves or (non-quantized?) disturbances? Can they exist shorter time than minimum time ~ h-bar/ E?

  7. Hummm!

    I might have learned something that gives me wrong answers.

    1. We are aware of 3 space dimensions and one time dimension.

    2. Einstein said that space and time is not distinguishable above certain energies,( Can I say “smallness”?)
    At certain high energy, time can be a space dimension.

    3. We also have virtual particles and vacuum fluctuations that come from somewhere other than the 3 dimensions that we are aware of and that have limited duration, (time component), before they go back to where they came from.

    4. A particle exist in 3 dimensions simultaneously and has a time component to the concept of “exist’

    What did I learn that was wrong that will prevent me from accept the truth/possibility of your next article?


    1. 1. Correct

      2. Not correct. I’m not sure where you learned this. Einstein certainly did not say that time can be a space dimension at high energy. What he said was that time and the three dimensions of space have to be thought of, in the presence of gravity, as a four-dimensional system that can curve together. That is, it is not enough to imagine the universe as a curved 3-dimensional space along with an independent time, one must instead imagine it as a curved 4-dimensional space-time. But still at each point in this 4-dimensional space there are three dimensions of space and one of time. (Since scientists do not understand time very well there are many speculations about it; perhaps you heard someone say this as part of a speculation?)

      3. Virtual particles/fluctuations appear spontaneously within the usual 3 space dimensions, and are not “coming from” anywhere else. To fluctuate randomly is simply an intrinsic property of quantum fields. This property is completely independent of anything having to do with extra dimensions.

      4. I don’t understand the English in (4); can you try to restate it, please?

  8. How can a KK particle be told apart from a winding mode particle?
    Are they looking for these too at the LHC, or is it completely impossible to see them (if they exist…) ?

  9. I’m not a physicist, but couldn’t something like the Muon be an electron in another dimension (or the other way around)? Even Rabi commented “Who ordered that?”

  10. KK particles…..KK photons…wow Matt.


    Trying to get a grip on this subject has been extremely difficult, but as Bee of Backreaction has shown with her efforts or with Lisa Randall’s Book on Warped Passages, it has still been hard to synopsize.

    Was Einstein faced with the same dilemma?

    “Einstein’s special relativity was developed along Kant’s line of thinking: things depend on the frame from which you make observations. However, there is one big difference. Instead of the absolute frame, Einstein introduced an extra dimension. Let us illustrate this using a CocaCola can. It appears like a circle if you look at it from the top, while it appears as a rectangle from the side. The real thing is a three-dimensional circular cylinder. While Kant was obsessed with the absoluteness of the real thing, Einstein was able to observe the importance of the extra dimension”

    It has not been without understanding some conceptual breakthroughs to see that our resulting framework of three plus one is reduce from higher dimensional thinking? Relativity as a result.


    a) Compactifying a 3-D universe with two space dimensions and one time dimension. This is a simplification of the 5-D space­time considered by Theodor Kaluza and Oskar Klein. (b) The Lorentz symmetry of the large dimension is broken by the compactification and all that remains is 2-D space plus the U(1) symmetry represented by the arrow. (c) On large scales we see only a 2-D universe (one space plus one time dimension) with the “internal” U(1) symmetry of electromagnetism. Link on physics web pic.

    Again trying to understand this with the coordinated arrows of direction with a turn at each direction was a standard expression I had come across yet it really still did not make sense.


    The “musical example” does make sense. I was looking for example by Calle.

    “To find extra dimensions of the type studied by the CERN group, experimenters are on the alert for what they call Kaluza-Klein towers, which are associated with carriers of the nongravitational forces, such as the photon of electromagnetism and the Z boson of the weak force. Excitations of energy within the extra dimensions would turn each of these carriers into a family of increasingly massive clones of the original particle—analogous to the harmonics of a musical note.” Outdated link sorry.

    I would ask then if it is appropriate to say metaphorically that while we might see “as an example” water in a glass, is to say that space exists even within the confines of that water. We are able to say that solubles could be added without raising the water line and the volume of the water?


    1. http://3.bp.blogspot.com/_cldxKGOzgeM/RuKu4_rrIXI/AAAAAAAAAio/HRnTDwx5ARc/s320/cosmic.jpg

      “Most people think of “seeing” and “observing” directly with their senses. But for physicists, these words refer to much more indirect measurements involving a train of theoretical logic by which we can interpret what is “seen.”

      “Remarkably, we can potentially “see” or “observe” evidence of extra dimensions.” Lisa Randall

      So we have to theoretically see extra-dimensions as possible? You are then trying to develop the framework and model for this evidence( while calorimeter signs of particles are present from collisions..at the same time the presence of these KK particles are present….then show the tracks of KK particles in extra-dimensions?). So truly then it has to become more then theoretically possible?

      When you look at array of ICECUBE measure field…..Cherenkov is then seeing advanced detail of collision process with cosmic particle collisions? You know what I mean?:)


      1. The word “see” is shorthand in this context, and I try to avoid its use. “Observe” means “detect with scientific apparatus, which serves as an extension of our senses.” The chain of logic is not theoretical — it is simply an extension of the same chain of logic that allows the detection of photons by our eyes to be translated within our brains into an image of the world.

  11. you’ve got a lot so i might have missed it,
    but would spin (and isospin) be explained by the extra dimensions?

    1. Spin is certainly a separate issue; it is an intrinsic property of particles in three spatial dimensions. Weak-nuclear-force isospin is something very different (the names are similar because there is some analogy in the mathematics, but the physics of spin and isospin are profoundly different.) The notion that isospin might arise from extra dimensions (isospin symmetry might actually be a rotational symmetry of the geometry of some extra dimensions) has certainly been considered. But it’s not necessary that it arise that way, and there’s no evidence one way or the other as of yet.

  12. Great article

    But I was wondering how this idea of a dimension having a ‘size’ and wondered how this would apply to our own known spatial dimensions? So do our known dimensions: x, y, z have a set size that we know of, or are they infinite?

    1. The dimensions that we can see can only be said to be as large as the distance that we can see across the universe. That’s about 10 billion light years or so. What happens beyond there we don’t know. For all practical purposes (in terms of anything we can measure on earth or observe in the sky) there’s no way to tell if they are infinite or not. And therefore, for practical purposes, we may treat them as though they are infinite (even though this might be a simplifying approximation) when describing all the physical processes we observe in daily life and in our laboratories.

  13. Thank you for your excellent articles, Matt. Especially this latest series on extra dimensions. A few questions about the scenario above:
    1) You say several times that the larger the mass of the “heavy partners”, the harder those partners are to find. Why is this?
    2) In the photon example (and maybe all the others) is the mass associated with the hidden dimensions the result of hidden momentum in those dimensions? That is, the result of the fact that the photon’s “real” momentum vector has components that we can’t see?

    1. Heavy particles (larger mass M) must be made in collisions of higher energy (E=Mc2) and collisions of higher energy require larger and more expensive particle accelerators. Simple as that.

      Your answer to (2) is correct. I will explain that in more detail in the next articles.

  14. (Total layman) Could the three generations of fermions potentially be an example of this? I’m thinking particularly of the charged leptons, since as far as I know mass is the only difference among the electron, muon, and tau. You also mention the particles would have to be small to travel across any hidden dimensions – could the charged leptons potentially do this since they are point-like, or would their charge radii prevent this sort of movement?

    1. Charge radii are not fundamental in quantum mechanics and they don’t affect the true sizes of particles. The electron’s charge radius is comparable to the size of an atomic nucleus, and electrons are known to be at least 1000 times smaller than that.

      Regarding the three generations of fermions: it is a good question, but the answer is no, for reasons that are not instantly obvious. I’ll address that in a later article.

  15. Speaking of Kaluza and Klein, are you aware of any English-language translations of their seminal papers on Kaluza–Klein theory? I’ve been unable to track any down.

    Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) 1921: 966–972.
    Klein, Oskar (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie”. Zeitschrift für Physik A 37 (12): 895–906. Bibcode 1926ZPhy…37..895K. doi:10.1007/BF01397481.

    1. English translations of the two original Kaluza and Klein papers can be found in
      Modern Kaluza-Klein Theories, Applequist, Chodos, and Freund,
      (Addison-Wesley, 1987)

  16. Multiple particles that are very similar except for mass? I am curious why the three generations of most particles don’t suggest extra dimensions (unless they do). OK, taus and muons aren’t *identical* to electrons except for mass, but they are pretty similar.

    1. Actually taus and muons are pretty darn similar to electrons, so your question is a good one. But in fact, for reasons that are not yet obvious given what I have told you, the answer turns out to be no. I’ll explain that later.

      1. Oh, I knew they didn’t suggest extra generations — if they did you’d’ve been saying why they were evidence for extra dimensions, if not actually writing your Nobel acceptance speech (or analyzing someone else’s), not explaining what this hypothetical evidence might look like. But I am looking forward to hearing what those subtleties are.

        (And in general, I have been gobbling up your essays with considerable pleasure.)

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