I've spent part of the weekend thinking over some applications of an idea I worked on several years ago. This idea involves the concept of large extra dimensions, which

I mentioned briefly once before. I promised then I'd have more to say on the topic, so here goes the first installment.

The

hierarchy problem in particle physics is the problem of reconciling two wildly disparate mass scales; the weak scale (10

^{2} GeV) and the Planck scale (10

^{19} GeV). This hierarchy is technically unnatural in particle physics, since, in general, the effect of quantum mechanics (here known as

renormalization) is to make the observable values of such scales much closer in size.

One approach to this problem is to introduce a mechanism that cancels many of the quantum corrections, allowing the scales to remain widely separated even after quantum mechanics is taken into account. An example of such a mechanism (and the most popular one, for sure) is supersymmetry (SUSY), which I may discuss another time.

However, another perspective is to view the hierarchy problem no longer as a disparity between mass scales, but rather as an issue of length scales, or volumes. The general hypothesis is that the universe as a whole is 3+1+d dimensional (so that there are d extra, spatial dimensions), with gravity propagating in all dimensions, but the standard model fields confined to a 3+1 dimensional submanifold that comprises our observable universe. This submanifold is called the brane (as in

membrane).

This is really a superstring-inspired modification of the Kaluza-Klein idea that the universe may have more spatial dimensions than the three that we observe. As in traditional Kaluza-Klein theories, it is necessary that all dimensions other than those we observe be compactified (wrapped up nice and small), so that their existence does not conflict with experimental data. The difference in the new scenarios is that, since standard model fields do not propagate in the extra dimensions, it is only necessary to evade constraints on higher-dimensional gravity, and not, for example, on higher-dimensional electromagnetism. This is important, since electromagnetism is tested to great precision down to extremely small scales, whereas microscopic tests of gravity are far less precise (although remarkable advances have been made in recent years, prompted in part by these theoretical ideas).

Since constraints on the new scenarios are less stringent than those on ordinary Kaluza-Klein theories, the corresponding extra dimensions can be significantly larger, which translates into a much larger allowed volume for the extra dimensions. This extra volume is a big deal!

You can imagine the strength of gravity as being a bit like the force due to a steady stream of water emerging from the nozzle of a hosepipe. Suppose that the water is confined, by some fancy nozzle, say, to emerge in a stream that is essentially one dimensional - a very fine stream. If you've ever fitted a tight nozzle to a hosepipe, then you'll know that such a stream is very powerful, and the force it exerts on you, if pointed your way, is very high.

Now imagine that, instead, the water is spread out to emerge in a plane (OK, that would require one fancy nozzle, but I'm sure you can

imagine it). In this case, if your body is in the way of the water in some direction (the same distance from the nozzle as in the first case), it will still exert a force on you, but less than when you were being hit by the one-dimensional stream. This is because you're not being hit by all the water, but instead by a portion of it - there are other directions available for the rest of the water to go.

If we now fit a nozzle that allows the water to spread out in a spherically-symmetric three-dimensional pattern, then the force on your body will be yet weaker because there are still further directions for the water to spread.

The analogy I'm drawing here is with lines of gravitational flux, the density of which, in Newtonian gravity, describes the strength of gravity. The more directions (think dimensions) available for the water (think gravitational flux) to spread, the weaker is the force experienced.

Thus, in the

large extra dimensions picture, it is the spreading of gravitational flux into the large volume of the extra dimensions that allows gravity measured on our brane to be so weak, parameterized by the Planck mass M

_{P}, while the fundamental scale of physics M

_{*} is parameterized by the weak scale, M

_{W}, say.

Given this, the problem of understanding the hierarchy between the Planck and weak scales now becomes that of understanding why extra dimensions are stabilized at a linear size (~0.1 mm, for example) that is large with respect to the fundamental length scale (1/M

_{*}). This is the rephrasing of the hierarchy problem in these

large extra dimension models.

Nemanja Kaloper, John March-Russell and

Glenn Starkman and I

proposed a modification to the above picture, in which we argued that there exist attractive alternate choices of compactification (the way in which the extra dimensions are wrapped up). These compactifications employ a topologically non-trivial internal space - a so-called d-dimensional compact hyperbolic manifold, and throw into a new light the problem of explaining the large Planck/Weak hierarchy. It'll be fun to blog about these, but since I'd like to keep that discussion distinct from the primer above, I think I'll leave it here for now.