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What is Matrix Theory?

< BEGIN overdramatized theatrics >

I knew there was a reason I bought spoonless.net in 2000 and made it my personal hostname, and subsequently made all my account usernames spoonless since. There was a destiny awaiting me, a question I had to answer, and I could feel it but I just hadn't seen it yet. But then I remember that I don't really believe in all this fate crap, now, do I? Maybe I just need to eat a cookie, and by the time I'm done... I'll feel right as rain.

This summer I've been trying to answer a question. Well, not answer it... but help those who've been unplugged a lot longer get a step closer to the answer. It's the question that drives us. You know the question, we all know it: What is the Matrix? Or, more precisely: What is Matrix Theory? (*thunder crash!*)

< /END overdramatized theatrics >

So, I'm trying to decide how to explain this so that a decent fraction of my friends-list will have a clue what I'm talking about. But there's just no good way. So I'm gonna just wing it, and see how it comes out. Here's a very condensed version of a long story, one that I'm still learning myself (but I have a very good teacher):

There are 4 known forces of nature. 3 of them are described by the Standard Model of particle physics: the electromagnetic force, the weak force, and the strong force. These each have a quantum field theory describing them, based on the gauge groups U(1), SU(2), and SU(3) respectively. Gravity is the 4th known force of nature, and although almost everyone expects it to be quantized somehow, so that it can be unified with the other 3, nobody has succeeded in coming up with a fully consistant, fully developed, fully understood, and fully tested quantum theory of gravity. The closest anyone has is 5 different "superstring" theories which are consistant, as far as they've been checked, but haven't yet been tested to see which of them (if any) corresponds to the world we're living in. In 1995, Edward Witten (recently described by CNN as "physics' sharpest mind since Einstein") figured out a way in which all 5 of these might be limits of the same overarching theory which he dubbed "M-Theory". At first, M was assumed to stand for Magic, Mystery, Membrane, or Mother. But years later there is growing suspicion that it may actually stand for Matrix. This is an odd way of saying it, since it should stand for whatever Ed wanted it to stand for, right? Well, yes and no. He left it intentionally ambiguous since M-Theory is not really a known theory but an unknown theory; at the time he proposed it, it was nothing more than a conjecture (theory is the strongest word you can use to describe something in science... weaker equivalents which are used to describe things which are not-quite-theories are: conjecture, hypothesis, scenario, or model). Matrix Theory in its current form was proposed a year or two later (by Banks, Fischler, Susskind, and Shenker... Tom Banks being the one whom I'm doing this summer project for). Unlike M-"Theory", Matrix Theory is a lot closer to deserving the title of Theory. Although it's in some ways even further from being tested, since it's not even known yet whether it reproduces the right properties we expect out of M-Theory. This is where little-old-me comes in. Matrix Theory is a framework from which you could in principle (with enough computing power and talented human resources) calculate anything you could possibly want to know. But the answers you get may or may not correspond to our reality... this is a question that experiments will ultimately have to answer. The first step is showing that they correspond to what's known as "11-dimensional supergravity" (local supersymmetry in 11 spacetime dimensions, which gives rise to: General Relativity which has been well tested, plus a few other things like gravitino fields which may exist but haven't been seen yet)... the assumed low energy limit of M-Theory. This has been done to a limited extent in certain rough approximations, but I'm trying to show more rigorously and precisely how they correspond. This would be a step forward in answering the question "What is Matrix Theory" or equivalently, the question: "Does M stand for Matrix?"... because what we want to know here is whether Matrix Theory is really M-Theory or if it's something different. If it's something else, then it's probably not terribly interesting and we can just forget about it. If it is M-Theory, then that opens up a lot of different possibilities, and puts M-Theory into a much firmer framework. Of course, it would still need to be tested experimentally, which would be the final step... but due to current tightly constrained technological limitations, that step could be very far off. (Or it could be right around the corner, we just won't know until we know more about what the theory says.)

There's a heck of a lot more I could try and summarize, but I'll leave most of it for later. But I'll try and answer one question which might be the first that comes up: why is it called "Matrix Theory"? Well, if you know any math beyond balancing a checkbook, you probably know that matrices are grids of numbers that do magical things when multiplied together. One of the main differences between matrices and regular numbers is that they do not "commute", that is A*B is not equal to B*A. This makes them very useful in quantum mechanics where different measurements of observables do not commute (the order in which you look at things matters). But what Matrix Theory (or any kind of "Matrix Models" in particle physics) does is apply the same thing to spacetime geometry itself. Instead of having spacetime coordinates which are numbers, you have coordinates given by non-commuting matrices. So the direction in which you go matters... west and then north is not the same as north and then west! This makes the structure of spacetime look very weird at small scales where quantum gravity comes into play. It's called a "non-commuting" geometry. This is a pretty difficult subject in its own right, but fortunately I don't really have to know anything about it yet in order to do what I'm doing.

Well, that's it for now. Next I want to post a bit on light-cone coordinates and the infinite momentum frame soon because I think they're pretty neat, and I've been starting to use them a lot lately. But that will have to wait.

"What is a Matrix?" - Werner Heisenberg
"What is the Matrix?" - Neo
"What is Matrix Theory?" - spoonless


Jul. 7th, 2005 07:36 pm (UTC)
So your plan is to further specify an intriguing theory that is only yet postulated? Sounds very cool.

Your explanation of why use matrices based on symmetry was interesting.

very basic questions: (was never sharp at physics)
1. quantization means different things, in audio it means things are chopped up into rectangular blocks. In physics it implies particles, right? So we have strong reason to believe that gravity is a particle?
2. what does 11 dimensions explain? (why do we think they are there)

So what are the desirables shown by M-theory that you want to show are possible with Matrix theory?
Jul. 8th, 2005 03:45 am (UTC)

1. quantization means different things, in audio it means things are chopped up into rectangular blocks. In physics it implies particles, right? So we have strong reason to believe that gravity is a particle?

In physics, it also means chopped up into blocks... where the blocks are called quanta, or sometimes "particles". But the word particle in that context means something a lot more general than most people would think of when they picture a particle. It could mean any sort of quantized chunks of energy, really--matter being one kind of energy.

2. what does 11 dimensions explain? (why do we think they are there)

11-dimensions doesn't explain anything directly, it's just something that's necessary in order for the theory to be self-consistant. It is somewhat surprising that you could have a theory that works only in a particular number of dimensions, but in this case that's what happens. Try it in any other number and it breaks down and makes no sense... well, with the caveat I'll add below(*). One of the first questions people ask here is usually "why do we only see 4 dimensions (3 space and 1 time) if there are really 11?" and the answer is that... based on our observations, we can only conclude that there are 4 large dimensions. To assume we know exactly how many there are would be an unjustified leap. The others could be very small so that they loop around after a really short distance, and we wouldn't have seen them yet. So this is what is assumed in string theories & M theory.

(*) there's a few ways in which what I'm saying here is a simplification. There are other ways in which even more dimensions could be hidden... so the number could be higher than 11 if it's done properly. And the 5 string theories which are limits of M-Theory only have 10 effective dimensions where 1 is too small to matter. But keeping things as simple as possible, we could assume 11 for now. The other thing is, there is a sense in which the number of dimensions isn't even completely fixed or specified. It could just depend on how you look at things or how far you zoom in. But this gets into the fringes of what I know, so I'm probably not going to be of much help there.


domino plural

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