#So like I know string theory is 'particles r like... closed loops of string'??? But not much more it's not cemented in my brain

[ralexsol]

I'm rly close to finishing Before the Big Bang by Laura Mersini-Houghton, so you know what that means. I need to go to Barnes & Noble to find random cosmology books that look interesting again

[nimblermortal]

Why Is the Universe 10-, 11-, or 26-dimensional, and Not Some Other Number?

This is in response to @kittydesade​‘s post questioning why those numbers, and the rough answer is: because it is referring, respectively, to either superstring, M-theory, or bosonic string theory. In any of these cases, the answer is “because that’s what makes the math work”, but I want a refresher on the distinctions. 

Cookies to me if I reach the end of this post without typing “Strong Theory” and failing to correct it.

Oh, and string theory is the idea that - well, you know how E=mc^2 which means that mass and energy are equivalent? Well, to link those, atoms are made up of subatomic particles like electrons, which are made up of sub-particles like quarks, which are made up of vibrating packets of energy in the form of strings, which may or may not form loops like hair ties. The mathematics of this is complex and it’s a generally questionable discipline of physics (or it was when I was a kid and first learned about it on PBS) but it’s Neat. It’s one way to understand gravity at a quantum level. Because despite its intuitiveness, gravity is really one of the most difficult forces to understand (which I am not actually sufficiently steeped in physics to understand, more’s the pity; I thought magnetism was at least as bad? but to some degree the more intuitive an idea is, the easier it is to ask why about, and therefore the less we end up understanding about it).

Bosonic String Theory is the oldest version, and so called because it includes bosons.

Bosons are not fermions; fermions are particles that, if identical, cannot occupy the same quantum state, whereas bosons can. Another way this is described is that fermions have half-integer spin (1/2, 3/2, etc) whereas bosons have integer spin (0, 1, 2). Exactly what this means is another discussion. This paragraph exists because I always forget the distinction, so there we go.

Bosonic string theory only covers bosons and does not cover supersymmetry, so in fact it is an outdated model and we can ditch the 26-dimensional option, which is nice because that’s a lot (as Caduceus Clay would put it)*. I want to go in more depth and reading about open/closed, orientable/non-orientable systems, and what a worldsheet is, and explain to you guys things about D25-branes (and branes in general), and work through the actual math in the Wikipedia article linked above, but this is supposed to be an overview of the differences between string theories, and I need to go grocery shopping. So I shan’t. For now.

Superstring Theory is the update to bosonic string theory. Apparently there have been multiple “string theory revolutions”, which is a fascinating statement to make and I want to know more. Superstring theory is nice because it incorporates supersymmetry, which we want to be true because it makes math elegant.

Elegance essentially means ‘small formulas, simple relations’. Historically speaking, the more elegant an equation is, the more likely it is to be at least broadly correct - think F=ma (Newtonian equation for force) or E=mc^2. Very short and nice! As opposed to our current Standard Model for particles, which requires more than one sheet of paper just to write down, and almost as John Mulaney* would put it, nobody likes that.

But we don’t have any proof of supersymmetry, we just want it really badly. If you are a (Christian-raised) atheist, you probably think of Christians as wanting there to be a God the way I am thinking of physicists wanting there to be supersymmetry. It makes everything easier, simpler, nicer to think that there is supersymmetry out there, and if we can just find proof then so many things will be more straightforward. (For the record, this is not how I think of God or theists.**)

Supersymmetry says that every boson has a partner, opposite fermion in mirror to it. In typically cute physicist fashion, they have decided that the partner for an electron is called a selectron. This solves all sorts of problems that I don’t have the space or time to go into here. The problem with this is, of course, the Fermi paradox: if these aliens* (particles) exist - where are they? And so far, we have not been able to answer this question, about either aliens or particles.

And the Fermi paradox is an ongoing problem with all of these theories. If there are these particles, where are they? If there are strings, where are they? If there are all these extra dimensions, where are they? The stupidest explanation I have heard for this last is that they are curled up very small inside each other. I have no idea how that works (fractals maybe?) but it seems much less intuitive than the simple answer “outside”. Referring back to A. Square’s perception of a sphere: Where is the sphere when A. Square does not perceive it as some form of a circle? Outside A. Square’s plane of perception. So that is what is intuitive to me, but maybe I am missing something.

M-theory unifies various superstring theories and apparently precipitated the second string theory revolution, which I think deserves capitals but sometimes we can’t have nice things. And apparently what happened was, we had about five major string theories, and then people started poking around and pointing out that if we used various dualities (situations in which two seemingly different things turn out to be the same in a nontrivial way) then several of these major string theories look to be the same thing.

One of the major dualities was S-duality, which says that strong couplings and weak couplings are the same thing, if you translate them into a different space (probably by something akin to Fourier analytics, which I am not going to explain here). And that’s... a lot, as Caduceus Clay* would say. First off, what is a coupling? Well, you know how if you are fitting data you can fit it to a polynomial? And if it’s random data, well, maybe a line (polynomial to the first power) doesn’t fit well, but a parabola (polynomial to the second power)is a little better, and if you keep raising that exponent you can get better and better fits, even as it becomes a meaningless fit? Well, some things are expressed better as polynomials of infinitely many terms. For example, Taylor series. And as long as the thing you are raising to a power is less than one, this gets more accurate without exploding (because, for example, 1/2^1 = 1/2, 1/2^2 = 1/4, 1/2^3 = 1/8, and so it keeps getting smaller and less significant). Those are weak couplings. Whereas if that term you are raising is greater than 1, it gets bigger and bigger (because, for example, 2^1 = 2, 2^2 = 4, 2^3 = 8...). Those are strong couplings, because naming convention is terrible.

But! There are ways and ways of looking at things. Because, say, I am looking at things in Cartesian coordinates, I plot something as being a units from the center on the x axis and b units from the center on the y axis, and it is at point (a, b). But if I switch to polar coordinates, I plot that as being c units from the center in straight radial distance and d degrees from the x axis if you rotate a line of length c from the x axis, and then that point is (c, d). And this is still the same point, but we are starting with different base assumptions for how we plot it. And if we want to move it, in Cartesian coordinates we might go a few units on the x axis, which is very simple, but in polar coordinates in order to do the same thing we would have to move some units on the radial axis and then some more rotations in order to get to the same point, and it is much more complex; or vice versa. And we can do that (not plotting, but changing how we look at things in order to make the problem we are working on simpler) in a lot of different ways, depending on our assumptions about how things work; the one I am most comfortable with is Fourier analysis. So if we do something akin to that to a strong coupling, it might start to look like a weak coupling - and then we have S-duality, and we start looking askance at just how different our major string theories are.

The other major duality was T-duality, which is an equivalence between quantum field theory and string theory. The simplest example has to do with strings propagating around a circle of radius R being equivalent to strings propagating around a circle of radius 1/R. Another example (or an example of this? my understanding is breaking down here) is that momentum is inherently particulate, a word I am using to mean ‘comes in discrete quantities’ because quantum is taken, and equivalent to the number of times a string wraps around one of these circles (which are the same circle). In terms of this being particulate, it makes sense in the same way that individual grains of sand become a beach, or Xeno’s Paradox - and again, it puts a floor on how finely one can divide something, which is something that humans a) keep trying to do and b) keep getting dissatisfied with and trying to break the smallest particle down again, most recently into strings. Anyhoo. In terms of these two circles being the same circle... well, Christians ought to be comfortable with that anyhow.

In general, T-duality relates two theories with different spacetime geometries - e.g. our two circles being the same circle. And now we really start looking askance at our different string theories, and asking questions, because both of these dualities apply to different options.

So one fellow, a Mr. (or more likely Dr.) Edward Witten sat down and looked at these two dualities, and he looked at something called eleven-dimensional supergravity, and he said, “What if there were in fact sufficient dualities that all of these string theories were the same?”

And physicists, presumably, went nuts trying to make this happen, because 1. it was elegant 2. it made things seem like they might make sense again 3. it was only 11 dimensions 4. he said that the M stood for “magic, mystery, or membrane” depending on what we eventually learn about what this theory actually is, and physicists love that sort of naming.

Personally, I like to think that M is the number of dimensions it takes (presumably 11), or theories (presumably 6), or physicists who die trying to make it happen (presumably infinite).

Every description of string theory I have ever seen has included a physicist who says something along the lines of, “String theory is the pot of gold at the end of the rainbow; it’s the Philosopher’s Stone. People keep devoting their lives to making it happen, but there’s just no evidence that it should. So it’s lovely to read about, but it’s such a waste of effort.”

But then - so is all art. Art is elegance, art is beauty, art is making a point about the nature of the world we live in; and so is string theory, whether it is true or not.

*See? I’m making this approachable by putting in popular culture references. Nyah.

**It is representative of how I think of (Christian-raised) atheists, because a lot of (Christian-raised) atheists are assholes about it.

Additional posts to make:

-What does spin mean, in a quantum sense? -What is an open, oriented system? What does it mean for a string to be closed and non-oriented? -What is a worldsheet? -What does the math for bosonic string theory mean, in terms of giving it a simple description? -What is a superstring theory revolution, how many were there, and what did they mean? -What are the following problems and how does supersymmetry solve them: the hierarchy problem, gauge-coupling unification, dark matter, “other technical motivations” -What is eleven-dimensional supergravity?