Matt Strassler [15 Jan 2012]
In this article and the next, we will learn why extra dimensions lead to “Kaluza-Klein (KK) partner” particles (described in the previous article in this series, which you should read before this one.) If a known type of particle of mass m can travel in a dimension of which we are unaware — an “extra” dimension — then we will eventually discover many other types of particles, similar to the known one but heavier, with masses M>m.
In this “Step 1” article I’m going to begin to explain to you why this is true, but I will give you only half the argument. This half will tell you why there are KK partners, and why they all have masses M > m. But this part of the argument will incorrectly suggest that there is a type of KK partner particle for every mass M that is larger than m. Only in Step 2 (the next article), when we include a little bit of quantum mechanics, will we get the right answer: that these particles come with very specific masses, a discrete set, with the first new one somewhat heavier (and perhaps much heavier) than the known one. See Figure 1.
Let’s start from a naive question about the strip — the ship canal in our previous examples, which you should definitely read about if you have not already. An observer that, like the freighter in our examples, knows nothing about the short dimension thinks the ship canal is just a line, not a strip. But if this observer is scientifically minded, he or she still knows a thing or two. First, the observer knows about motion forward and backward along the strip; velocity and momentum along the strip make sense to this observer. Second, the observer also knows about energy, and knows that energy is related to mass and motion. Specifically, a particle’s energy E is related to its mass m and its motion (in particular, its momentum p along the strip) by Einstein’s famous formula
- E2 = m2 c4 + p2 c2
which says that a particle’s energy is given by a combination of mass-energy and motion-energy. For a particle that is motionless and has therefore no momentum (p=0), this formula reduces, as it must, to E2 = m2 c4, or in other words, E = m c2 .
So if there is a small boat motionless in the canal, an observer who knows about both dimensions and observes the boat will say: Its momentum is zero, and its energy is all mass-energy: E = m c2. And an observer who knows only about the dimension along the strip will say exactly the same thing. See Figure 2, upper panel, where the viewpoint of the observer who knows about two dimensions is shown in the upper part of the panel, while the view of the observer who knows about only the long dimension, and thinks the ship canal is just a line, is shown in the lower part.
If instead the small boat is moving along the canal, an observer who knows about both dimensions and observes the boat will say: Its momentum along the canal, palong, is non-zero, and its energy-squared is
- E2 = m2 c4 + (palong)2 c2 .
And an observer who knows only about the dimension along the strip will again say exactly the same thing. See Figure 2, middle panel.
But what does an observer think he or she is looking at when studying a particle that is not moving along the strip but is instead moving across the strip? See Figure 2, lower panel.
An observer who knows about both dimensions and observes the boat will say: Its momentum along the canal, palong, is zero, but its momentum across the canal, pacross, is not zero, so its energy-squared is E2 = m2 c4 + (pacross)2 c2. Notice this means, necessarily, that E > m c2, because the boat has motion-energy as well as mass-energy.
However, an observer who knows only about the dimension along the strip cannot possibly say the same thing, because this observer knows nothing about the possibility of pacross. What the observer will think, looking at this particle, is that it is not moving. After all, it is not moving along the strip, and that is the only type of motion the observer can detect. And so, according to the observer, all of the particle’s energy, whatever it is, must be due to its mass.
So when looking at a particle that has palong=0 and non-zero pacross , the one-dimensional observer makes a mistake, but a very interesting one. This observer says: Hmmm. The momentum of this object is zero, and so its energy should be equal to its mass times c2, as was the case for the boat in the upper panel. But its energy E is larger than m c2, so this can’t be the same boat that we saw in the upper panel. Apparently nature has another type of small boat that we didn’t originally know about, similar to the first one, but with a different and larger mass: M = E/c2.
In other words, if a boat of mass m is moving with momentum pacross across the canal, the naive observer unaware of the extra dimension will incorrectly infer that what is being observed is a boat of mass M > m, with
M2 = m2 + (pacross /c)2 > m2 .
From this line of reasoning, we learn a correct piece of information: the signature of an extra dimension is the presence of particles that are similar in character to known particles (which are the particles that move only along the strip) but which appear heavier (due in fact to their unobserved motion across the strip.) And this immediately generalizes from the strip (with its one long dimension and one short dimension) to our own universe (with three long dimensions and perhaps one or more short dimensions); when known particles move in extra (i.e. unknown) dimensions, they appear to us to be heavier versions of themselves.
But the intuition we’ve obtained here is also wrong, because naively it would seem that a particle moving across the strip could have any momentum that it wants, and therefore an observer should see particles with every possible mass M that is bigger than m, as shown at the bottom of Figure 1. And that’s not correct. Instead, only very specific values of M are allowed, as shown at the top of Figure 1 — a discrete set, not the continuous set of masses that our argument would have implied. And that’s because of quantum mechanics. It’s because “particles” are actually quantized waves, or “quanta,” which was not true of the little boat used in the examples above. In the next article we’ll see why this makes such a difference.