This week's question comes from the Bag O Questions that Andy delivered to our doorstep a number of weeks back. His question is:
Extra dimensional theories are claimed to work in 10 or 11 dimensions. Why these numbers and not, say, 42?There are actually two different questions in there, so before we answer them, we need to identify them. The extra dimensional theories that work with 10 or 11 dimensions are the various flavors of string theory, and he's quite right. 10, 11 and 26 are the numbers of the day for those theories. So, our first question is, "Why does string theory work with 10 or 11 dimensions and not, say, 42? The other question, which is more in line with the spirit of the original question, is "Why do extra dimensional theories in general only work with 10 or 11 dimensions?" The short answers are "They just do." and "It doesn't." OK, good talk. See you next week.
Kidding. Let's tackle the first question first. String theory is a theory of physics that describes the universe with strings, one dimensional extended objects, rather than particles. It is a mathematical theory at present, not having yet gotten to the point where it can be described as a physical theory. Our feeble instrumentation is unable to detect the results of predictions of string theory that differ from the predictions of other physical theories. In other words, the predictions of string theory that we can detect are easily explained by other theories. But I digress.
One of the interesting things about string theory is its ability to predict the number of dimensions in the universe. At a macroscopic level, we see the world in 4 dimensions, three spacial and one time. Earlier physical theories required the theorist to insert the number of dimensions, rather than have the theory drive the number of dimensions, as with string theory. I won't get into it too much, as I don't understand 99.9% of it, but it goes something like this. We know that in this universe, photons have no mass. In string theory, the mass of a photon depends on the energy of the string used to describe the photon. This string can have a number of quantum fluctuations that contribute to the energy of the string, and the number of fluctuations is determined by the number of dimensions in the universe. The more dimensions you have, the more fluctuations you can have. When the math is worked out for a photon to have no mass, which can be verified through experimentation, the universe has 10, 11 or 26 dimensions based on the flavor of string theory you use. To repeat my original answer, "they just do." Now it's possible that the extra dimensions are so small as to not really count, which can then reduce the number of dimensions to the 4 we're used to. Another idea is that we live in a 4 dimensional subset of the universe, which sounds vaguely like something Stoner Bob comes up with after a few rotations.
Now, for the second question, which asks why extradimensional math only works with 10 or 11 dimensions, we revisit my original answer for that question, which was "it doesn't." In other words, there are various extradimensional mathematical spaces and theories out there that allow for a range of dimensions. One such space is the Hilbert Space which allows for either a finite number of dimensions or an infinite number of dimensions, based on what you're doing. This is of particular importance because Hilbert spaces are used in quantum mechanics to contain the wavefunctions, which then describe all of the possible states of a system. In that particular application, a Hilbert space with infinite dimensions is infinitely more useful, if you'll excuse the pun.
This isn't a situation where we're adding dimensions to the 4 we already know of. Hilbert spaces contain coordinates, each of which represent a probable or possible amplitude for a quantum state which may apply to the position of a particle, or speed, or spin, so on and so on. Heisenberg's Uncertainty principal tells us that not all of these attributes can be observed simultaneously, for example, you can not measue the position of a particle at the same time as the momentum. Now, because they can't all be observed at the same time, only an infinite number of dimensions can contain all of the possible number of quantum states.
For the purpose of explaining this further, let's take a bastardized version of the Schroedinger's Cat thought experiment. Let's say you have a cat in a box and you don't know if it's alive or dead. Until such time as you open the box and see, there are two possibilities, alive and dead. Simple enough. Now, let's say you don't know the color of the cat. Until such time as you open the box and see, there are an infinite number of possibilities, when one considers color, spots, stripes, patches, etc. Any mathematical model that you used to express the possible colors for the unseen cat would need to be able to express the fact that there are an infinite number of possibilities. So it is with quantum mechanics, and so it is with Hilden spaces. Note, that the Hilden space can't tell us which of the infinite number of choices will exist in reality, just as our cat math wouldn't be able to tell us what the color of the cat will be, simply what it could be.
Quantum mechanics is one of the most interesting and beautiful branches of science that I've come across in my studies. There is a quiet elegance to the notion that there is a universe of possibilities that exist outside our frame of reference, all waiting to be observed so that they can be born into our reality. Now, there is a huge discrepancy between what happens on a quantum scale and what happens on a macroscopic scale, in fact some of the math derived from Hilden spaces shows that if you could take a physical object apart into its component particles, apply some sort of spin or rotation to the particles and then put the particles back together, you'd now have two copies of the original object. Despite this discrepancy, I can't help but think that the universe of possibilities that exist on a quantum level can make it to us and that rather than be dictated by our surroundings and pushed to our final destinations, we need to just realize the sea of potential around us and act. As Jake once said, "Go then. There are other worlds than these."
OK. Enough philosophizing as clearly I'm quite shitty at it. Hilden spaces aren't the only models that use dimensions other than 11, 10 or 26. There's something called the Hausdorff dimension that corresponds to the number of balls of radius r needed to cover item X completely. As you can well imagine, that dimension can easily grow towards infinity if either the object X increases in size or the radius r decreases in value. There's also the Krull dimension of commutative rings which describes the number of ways that early 80's fantasy movies seemed really cool when you were a kid, but manage to suck totally once you watch it as an adult.
So there you have it. You need not limit the number of dimensions to 10, 11, 26, 17 or eleventy billion. You can have whatever number of dimensions you want, provided you're willing to learn the math to back it up. Now if you'll excuse me, I have one very pissed off cat to let out.
Wikipedia - String Theory
Wikipedia - Hilbert Space
Wikipedia - Dimension
The QED Project - "A Semi-Pop Non Mathematical Tutorial on Hilbert Space in Quantum Mechanics" - Jack Sarfatti
Metamath Home Page - Hilbert Space Explorer Home Page