There was a previous thread about some scientists believing there are 10 dimensions. I remember there was also a group that thought there were 36. Im sure this has something to do with topology. I know it’s a hard answer to explain to a neophyte, but go for it. How can the manipulation of numbers come up with stuff like this?
[deep breath] OK, here goes.
First off, it’s not math, it’s physics (or, perhaps, mathematical physics) that leads to this conclusion.
The starting point is quantum mechanics. The most accurate theories of physics are all based on quantum mechanics (except for gravity, i.e. General Relativity (GR)). So let’s assume QM is correct.
Step two: strings. The “standard model” of particle physics is based on the idea that particles are point-like (zero-dimensional) objects. In the 70’s some theorists started playing around with the idea that the fundamental objects are string-like (one-dimensional) instead. They came to the surprising conclusion that such a theory would automatically include GR as a subset of the theory.
Step 3: strings + QM. Now try to make a quantum-mechanical string theory. It turns out that you can only do it in 26 (not 36) dimensions. This is disappointing, because the observed number of spacetime dimensions is 4 (3 space plus time). But it’s also exciting because no one had ever come up with any quantum theory that included GR before. Another problem, tho, is that this theory has a tachyon - a particle that moves faster than light.
Step 4: strings + QM + supersymmetry. Now add in the clever but wrong idea that there is a symmetry between fermions (particles that obey the Fermi exclusion principle) and bosons (the other ones). I say this idea is wrong because, for example, we know there is an electron (a fermion), but we’ve never seen a particle with the same properties as an electron but which is a boson, as required by supersymmetry. Never mind, we can work around that slight problem. Anyway, the result of strings + QM + supersymmetry is a theory that only works in 10 dimensions, and doesn’t have any tachyons.
Step 5: Reducing the dimension. Well, 10 is closer than 26 but still wrong. However, the additional 6 dimensions can be hidden in an “internal space” that affects that particle’s properties, but aren’t observable as spacetime dimensions. Do any of these theories explain the observed properties of elementary particles (actually strings)? Well, it’s hard to say, because string theories are terrifically difficult to predict anything with. The math is just too hard.
Which is why some physicists think the whole topic is “advanced mathematical theology”, not real physics.
The ‘simple’ answer for why 10 (I thought it was 11 but I could be wrong) and 26 dimensions is that the math doesn’t work at any other level. Why it doesn’t work is a question for someone FAR better versed in math than I am.
Scientists are looking for a Quantum Theory of Gravity. Again for reasons I don’t really understand gravity does not wrap up nicely with the other three forces when you try to develop a mathematical model that describes everything (Grand Unified Theory or GUT). Using String Theory the extra dimensions allows ‘room’ in which the math can be performed and seems to allow a way to pull all four fundamental forces together under one umbrella. Of course, as FriendRob pointed out, String Theory also has some aspects pop out that scientists aren’t really comfortable with.
When thinking of ‘room’ to do math consider Newton’s equations and Einstein’s equations. Newton’s equations can be considered a subset of Einstein’s equations. F=MA is a basic piece of math but it isn’t very accurate. For 99.999% of the things you are likely to do here on earth it works just fine. However, get into the realm of relativistic effects and it falls apart. Einstein’s equations allow a greater degree of precision in describing what will happen at the extremes. If you felt like it you could use Einstein’s equations to do the math but when you simplify his equations for basic day-to-day stuff Newton’s equation pops back out. Since F=MA is much easier to deal with than Einstein’s equations most people stick with F=MA…the greater precision of what Einstein developed just isn’t necessary most of the time. When you move up into String Theory you get the ‘room’ to do even more math such that Einstein’s equations become a subset of String Theory in a similar manner as Newton’s is a subset of Einstein.
I heard somewhere that String Theory is an oddity in that the theory was developed before the math. Usually some new mathematical techniques are developed and then clever people work them into new theories. String Theory was the other way around. The theory developed and now people are searching for the math to round out the theory. Last I heard they expected about 20 years or so for the math to catch-up with the theory in this case. That’s a vague memory though so take it FWIW.
The interesting thing about many string theories is that they have been shown to be equivalent to other string theories.
from what I know, the 11-D model is the 4-D General Relativity
expanded to 11 dimensions to describe Electromagnetism, Weak,
and Strong Nuclear forces besides Gravity.
26-D model is a Special Relativity/Quantum Mechanical theory
which describes things as charge and spin of the various particles.
Part of the problem is GR dosen’t describe charge, and QM dosen’t
describe mass.
Pretty much the only way we can have a quantum theory of
gravity is to observe quantum effects…andd for that we must have access to a quantum singularity.
No problem. I’m pretty sure a quantum singularity is what powers Romulan Warbirds. If we can only get the Romulan Empire to play nice we should be in business 
.
As I recall, the reason that the theory demands 10 or 26 dimensions is that all the infinities are multiplied by n - 10 or n - 26, where n is the number of dimensions. Put in 10 or 26 for n, and the infinities go away.
Theoretically, it’s not the most sound method, but by and large it seems to work.
Sort of. It’s more to do with a general feature of quantum field theories called anomalies. If you start with a particular classical theory and try to turn it into a quantum one, then often you’ll find that some symmetry that’s present in the classical theory appears to be violated in your quantum theory. And usually you want to arrange things so that this isn’t the case. Now, since you’ve got divergences to worry about anyway, the violation usually does involve an infinity, but that’s not the fundamental aspect of an anomaly. It’s more like an inconsistency in the theory: you’re getting different answers depending how you look at things and that shouldn’t be case in a correct theory.
In the simplest string theories, you get anomalies like a violation of energy-momentum conservation. But quantise in 26 dimensions and you don’t get them (because the formulae expressing the violations turn out to have n-26 in them). Hence you quantise in 26 dimensions.
Anomalies are actually an absolutely standard feature of even bog-standard quantum field theories, including the Standard Model.