Matt Strassler [20 Jan 2012]
In the previous article in this series, I explained why a type of particle that can travel in an extra dimension will seem, to naive observers who do not know about the extra dimension, to have Kaluza-Klein (KK) partners — heavy versions of the original type of particle. What I pointed out to you was that when the original particle (with mass m) moves in the direction of an extra dimension, it appears to a naive observer to be stationary and heavier than it should be, as though it is another type of particle. This class of apparently new particles, similar to the original one but heavier, are called KK partners.
In the case of a strip, if a particle of mass m is moving across the strip with momentum pacross, an observer who believes the strip is a line will think the particle is a KK partner with zero momentum and mass M, where
M2 = m2 + (pacross /c)2 > m2 .
Though largely correct, this argument seemed to imply that there should be one KK partner for each and every mass M that is larger than m. But that’s not the case… because our world is a quantum world. (See Figure 1 of the previous article.) Let’s now learn why quantum mechanics changes this picture.
The key feature of quantum physics that we need is this: for a quantum “particle” traveling in a dimension of finite length, not all possible values of pacross are allowed. Stated more generally: quantum mechanics tells us that a “particle” that is moving in a dimension that is finite in length can only have certain particular values of momentum in that direction.
This is one of the most important and odd consequences of quantum mechanics! At first glance it is very counterintuitive, because what could stop you from giving a particle that has momentum p a tiny little push, so that it has just a tiny little bit more momentum than p ?
What is a “quantum”?
You may have already noted that I have started putting quotation marks around the word particle, because in the current context I need to distinguish the term“particle,” as used to describe electrons, photons, muons, quarks, gluons, and all the rest of the known elementary “particles”, from the intuitive notion of particle that we inherit from our experience of dust, sand, salt, and grit. Electrons and photons are not like little specks of dust, or grains of sand. In fact, it is better to call a “particle” such as an electron, photon, quark, etc. a “quantum”, which is a more subtle object. These “quanta” [plural of quantum, as memoranda is plural of memorandum] are ripples in fields, more like waves than particles in many respects. In fact, the best way to understand a quantum is to first think about waves. [Another crucial linguistic point: when I say, ``wave'', I don't mean something like a single ocean wave that breaks once on the beach --- I really mean a ``wave train'', with many crests and many troughs.]
An example of such waves would be electromagnetic waves, among which are the light waves that our eyes can detect. Now imagine taking such a wave — say, the light from a laser — and dimming the light, further and further. How dim can you go? It turns out that in our quantum world, there is a dimmest possible flash of light, which we call a “quantum” of light, or a “photon”. A photon is the light wave whose wave-height, and whose intensity, is the smallest possible. [These concepts and names are due to Einstein, who --- despite his reputation for having been unhappy with the conceptual implications of quantum mechanics --- was one of the founders of the theory. Caution: like most words, quantum can have multiple meanings, though you won't see any of the others in this article.]
There’s nothing intuitive — at least nothing that I personally find intuitive — about the fact that light waves are made from quanta, because no process of which we are consciously aware exhibits this effect. But our bodies, through processes of which we are not aware, use this fact all the time. In fact, though light from a light bulb looks continuous to our brains, our eyes are actually absorbing photons one at a time. Moreover, I’ve seen, with my own eyes, clear evidence that light is made from quanta; it’s not something I know just from books. I’ll explain that some other time.
We physicists often call this quantum of light a “particle of light” because in many respects it does behave like a particle. Any given photon that is traveling in a straight line on its own has a definite energy and momentum; all photons have the same mass (in particular, zero); a photon cannot be divided into smaller pieces; and a photon can be emitted or absorbed only as a whole. These properties roughly (though imperfectly) correspond to what our intuition would expect of particles, such as grains of sand, marbles, dust specks, etc.
But the word “quantum” is in many ways a better word than “particle”, because some of the properties of a quantum are similar to those of particles and some are similar to those of waves. A well-known example of wave-like behavior is that a quantum can pass through two doors simultaneously, and can interfere (in the same sense that waves can interfere, where a wave crest and wave trough can cancel each other out) with itself. And we’re about to see another example.
You should keep in mind that what is true for the photon is true for all the known “particles”. Really, each one is a type of quantum — a wave of smallest possible height in a corresponding field. The electron is a quantum in the electron field. An up quark is a quantum in the up quark field. The Z particle is a quantum of the Z field… and so on.
Kaluza-Klein Partner Quanta
Now it’s time to learn what I’ve been promising to explain to you: why the wavelike nature of quanta implies that the masses of Kaluza-Klein partners take certain specific values, rather than all possible values larger than the mass m of the original “particle.” The famous physicist Louis de Broglie (following initial insights of Einstein) was the first to state clearly that the relation between waves, particles and quanta implies that for a quantum there is a relation between
- its momentum (a particle-like property) and
- its wavelength (a wave-like property) [where again you must read the word ``wave'' as ``wave-train'', not as a single wave at the beach --- and the wavelength is the distance between crests of the wave-train]
and this relation is simply: momentum = h / wavelength .
Here h is Max Planck’s famous constant, as fundamental a constant of nature as is the speed of light. Planck introduced this constant in 1900 while trying to resolve a puzzling physical phenomenon that I’ll explain elsewhere; it was the first step toward our quantum picture of the world. Wherever you try to describe a phenomenon in which quantum mechanics plays an important role, h will appear. [In many formulas you see the quantity ℏ, which is just h divided by 2π , because often that quantity is more convenient for keeping formulas simple.]
Now in some ways it turns out to be a little simpler to explain what happens for a quantum traveling on a tube, rather than on the strip that we’ve used in all the previous examples. [I discussed the strip and the tube together in the context of Worlds of 2 Spatial Dimensions.] Almost everything I’ll say for the tube turns out to be true on the strip too. For this reason I’ll explain both of them together.
The Lightest KK Partner
Let’s think about a quantum traveling on the strip (of width W) or tube (of circumference S). First, imagine a quantum traveling in the long dimension [long meaning infinite, or so long that it might as well be infinite as far as any of us can tell.] A wave traveling along the strip or tube can move along the long dimension in either direction, and can have any wavelength (the distance between its wave crests). See Figure 2. Such a quantum can therefore have any momentum along the strip or tube, according to de Broglie: the momentum can be zero, very small, small, large, in either direction, etc. In principle you can make the momentum a little bigger (or a little smaller) by pushing on the quantum in the direction (or opposite the direction) of its motion.
But now consider a quantum (i.e. “particle”) traveling across the strip or tube. First of all, it clearly cannot have a wavelength that is longer than the distance across the strip or around the tube! This is easy to see on the tube: there has to be at least one crest and at least one trough somewhere around the tube, moving around the tube, as shown in Figure 3. If the wavelength is larger than S, the wave will not connect smoothly back on itself, as shown in Figure 4 (left). The largest wavelength the wave can have is exactly S; the one and only trough in the wave must be on exactly the opposite side of the tube from the one and only crest.
The crest and trough of the wave in Figure 3 move around and around the tube, somewhat analogous (see Figure 5) to an ordinary non-quantum particle (here I really do mean something like a grain of sand, not a quantum or “particle”) running around the tube, but crucially different: while the ordinary intuitive particle could without a problem go slightly slower or slightly faster, giving it a slightly larger or smaller momentum, the quantum corresponding to the wave cannot have a slightly larger or slightly smaller momentum, because that would correspond to an unacceptable wavelength, such as in Figure 4.
On the strip it is a little trickier, but as shown in Figure 6 it turns out that there has to again be one crest at one wall and one trough at the other, which change over time as shown in the figure: the crest doesn’t move, but it decreases in size and then turns into a trough, while the trough turns into a crest. [Note that unlike Figure 3, where the crest and trough stay the same size but move around the tube, the crest of this wave doesn't move as it shrinks. This is therefore called a ``standing wave''. Think of a wave on a guitar string or a violin string for a similar (but not identical) example. ] Intuitively this standing wave corresponds to an ordinary non-quantum particle going back and forth across the strip. (Less intuitively, but more accurately, it corresponds to an ordinary particle doing both at the same time. But that rather strange and cool quantum fact is not essential right now.) This is indicated in Figure 7.
In both cases there is a largest possible wavelength (S for the tube, 2 W for the strip). And that means there is a smallest possible momentum (h/S and h/ 2 W for the tube and strip). And finally, that means there is a lightest-possible Kaluza-Klein particle! which has mass M, where
- M2 = m2 + (pacross / c)2 = m2 + (h / c S)2 (tube)
- M2 = m2 + (pacross / c)2 = m2 + (h / 2 c W)2 (strip)
Notice for a massless particle (m=0) the formulas simplify to
- M = h / c S (tube)
- M = h / 2 c W (strip)
and these last formulas are approximately right if S or W is also very small, as will often be the case in plausible speculations.
So now we have learned that because “particles” are really quanta, with some wave-like properties,
- the lightest KK partner has a mass M which is quite a bit larger than m, and
- since the formulas above for M have 1/W and 1/S in them, the smaller is the extra dimension, the heavier is the lightest KK partner.
- In fact once S and W are so small that M is much larger than m (or if m is zero to start with), then M is approximately proportional to 1/S or 1/W.
Ok!! That’s the main point, so please make sure you understand it before proceeding. There are a few more things to explain though:
- why there are many KK partners, with different masses M, M’, M”, etc. (where by definition M < M’ < M”, etc. )
- why the masses are separated from one another
- why the masses all grow as the extra dimension(s) become smaller
- why the KK partners of different types of particles that all may travel in the same extra dimension(s) will have similar masses, especially for heavier KK partners
- why the KK partner masses tell us directly about the shape, size and number of extra dimensions
The answers follow very quickly from what I’ve already told you.
Beyond the Lightest KK Partner
Next: why are there many KK partners? That is simply because the quantum waves on the strip or tube can have many different wavelengths. In Figures 8, 9 and 10 you can see waves that have 1/2 or 1/3 of the maximum wavelength, corresponding (according to Einstein and to de Broglie) to quanta of double and triple the minimum momentum.
More generally, any wavelength is allowed for which there are n crests and n troughs, where n is any positive integer (1,2,3,4,…), so that the wavelength of the wave is S divided by n (or 2 W divided by n), and the wave fits neatly inside the circle of circumference S or the line of length W. Any other wavelength (see Figure 4) is not allowed. Correspondingly, given the de Broglie relation momentum = h / wavelength, any momentum of the form n h / S (or n h / 2 W) is allowed, and for each value of n we have a KK partner of mass
- M2 = m2 + (pacross /c)2 = m2 + (n h / c S)2 (tube)
- M2 = m2 + (pacross /c)2 = m2 + (n h / 2 c W)2 (strip)
This now answers most of the questions above, at least for the tube and strip:
- there are many KK partners (one for each integer n>0),
- their masses are well separated from one another (since when you change n by 1, the mass M changes by a lot),
- their masses all grow as the extra dimensions shrink (because the last terms in the formulas become larger as W or S become smaller),
- heavy KK partners for different particles with different masses m have similar masses M (because for large enough n, the second terms in the formulas are large compared to m2, making the KK partner masses approximately M = n h / c S for the tube and n h / 2 c W for the strip, almost independent of m.)
Now we need to answer the last question: why does the number, size and shape of the extra dimensions determine the masses of the KK partners — and thus, conversely, why does measuring the masses of many KK partners permit a determination of the properties of the extra dimensions, much the way listening to the sound of a musical instrument can allow one (in principle) to determine its shape, size, and materials?
Take the simplest (and overly simple!) example first. Let’s consider two extra dimensions and re-use our classic ship canal, including (as we did at the end of a recent article giving examples of extra dimensions) the fact that the canal has depth too, so we can think about waves moving around inside it. (The sound you hear in a hallway, or any underwater sound in the ship canal, would involve waves of this type.) The cross-section of the canal (if we slice it at some point along the longest dimension) is just a rectangle, of width W and depth D. Now just as an ordinary non-quantum particle, even if stationary from the point of view of the long dimension, could move along either or both of the extra dimensions (and thus have momentum either in the width or the depth dimension), so a wave will have a wavelength in both extra dimensions. [This simple separation of the wave into what it is doing in the width dimension and what it is doing in the depth dimension is quite special to a rectangular canal, and will not generally occur in other examples.] For instance, as shown in Figure 11, at the top, one allowed wave would have three troughs in the width direction and one in the depth dimension. We can specify how many troughs and crests there are in the width direction by an integer n1 and similarly in the depth dimension by a number n2, and for each n1 and n2 (where one or both is greater than zero) we have KK partners. For a massless (m=0) or nearly massless quantum, its KK partners have masses
- M2 = m2 + (pwidth/c)2 + (pdepth/c)2 = m2 + (n1 h / 2 c W)2 + (n2 h / 2 c D)2
Thus you see that the pattern of masses is different from the case of one extra dimension, and that the pattern can tell us both W and D.
If instead the cross-section of the canal is a different shape — say, a triangle, or a half-disc, as shown in Figure 12 at the bottom, then we will get a different pattern of masses that reflects the precise shape of the triangle or half disc. And we can stop thinking about practical ship canals, and imagine a three-dimensional space whose cross-section is any other finite two-dimensional shape seen in Figure 1 of my article on Worlds of Two Spatial Dimensions: a full disc, or even a sphere or torus. Each of these various shapes will give us a different pattern of KK partner masses. And if the number of extra dimensions is three, or four, or five… vastly more patterns are possible.
Examples of a few patterns, for massless particles, and with the size of the extra dimensions chosen so that the first KK partner mass is the same in each case, are shown in Figure 12. Clearly quite a few KK partner masses must be measured if one is to infer the shape and size of the extra dimensions (or even confirm that any newly discovered heavy particles actually are KK partners) so understanding the nature of any extra dimensions would take time.
Could the Known Heavier Matter Particles be KK Partners of the Lighter Ones?
Now an obvious additional question arises. We know that among the known elementary particles, there is the electron, and there are heavier versions of the electron: the muon and the tau. There is the up quark, and heavier versions of it, the charm and top quarks. There is the down quark, and heavier versions, the strange and the bottom quarks. Are the heavier known particles KK partners of the lighter ones?
It’s tempting at first glance, but the answer is a resounding no. Sorry. It’s not a dumb question by any means. It’s just that it has a smart, negative answer.
The muon and tau, the charm and the top, the strange and the bottom all get their masses from the Higgs field, not from momentum in extra dimensions. This is abundantly clear from detailed experiments. I’ll write more about this later, but as a first hint, read the article on what the world would be like if the Higgs field were zero. Notice that were the Higgs field zero on average, the electron, muon and tau would all be massless [and in fact would be broken up into two types of particles each.] That’s not consistent with the muon and tau being KK partners of the electron.
And there are many other reasons. Perhaps the most serious is that it turns out that since the electron is electrically charged and is surrounded by an electric field, the photon must travel in any dimension that the electron travels in (though the reverse is not true). So if the electron has KK partners, the photon should have KK partners too. But from our formulas (and similar ones that are more general), since the photon is massless and the electron mass (0.0005 GeV/c2) is light compared to the mass of the muon (about 0.1 GeV/c2), if the muon is a KK partner, the photon should have a KK partner of a similar mass. But such a particle, if it existed, would have been discovered decades ago. In fact there are no KK partners for the photon observed in experiments up to masses considerably larger than the Z particle mass… many hundreds of GeV/c2. (The Z particle itself can’t be a photon KK partner; it isn’t similar enough to the photon.) That means that any KK partners of the electron have to be at least that heavy too.
Where do we go next? Experiment.
We’ve gone from theory (the possibility of extra dimensions, and associated math and geometry) to predictions (KK partners.) Next step: what do we know about extra dimensions from experiments? We haven’t seen any KK partners in experiments yet, but we should still ask what we learn from their absence? Quite a lot, actually, as I’ll explain soon… along with a description of how efforts to find signs of extra dimensions continue at the Large Hadron Collider and elsewhere.