?

Log in

Previous Entry | Next Entry

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

Comments

( 12 comments — Leave a comment )
lars_larsen
Jul. 7th, 2005 03:03 pm (UTC)
I know you're not an experimentalist, but how would this be proven? Bigger particle accelerators? When would you guess something like this might be proven if it ever is?
spoonless
Jul. 8th, 2005 02:47 am (UTC)

I know you're not an experimentalist, but how would this be proven?

Well, you probably know this but I think it's worth being clear about it: no theory can be proven... you can prove that a theory is logically consistant, and you can observe that it's consistant with all known data up until current date... but any rules or principles you come up with which work well to explain and describe past experiments could always be broken by future experiments since you can never peform "all possible experiments" at once.

The theories which do well are those which have been well tested under a wide variety of conditions and stand up to all known experiments to date. But there's a little more to it than that, because any theory being investigated seriously by physicists is going to fall into this category... until it comes up against one (reproducible) experiment that doesn't fit, at which point it falls by the wayside and starts gathering dust. So another criteria used is to pick the simplest theory, via Occam's razor. Although sometimes there is trouble there because people disagree about which theory looks to be the simplest.

In the case of quantum gravity, we know that either quantum mechanics is wrong or general relativity is wrong. This must be true, because the two theories contradict each other (general relativity, if valid at very high energies would violate the Heisenberg Uncertainty Principle which is one of the basic tenets of quantum mechanics). But the energies at which they contradict each other are so far out of the range of anything acceleraters can get up to with our current technology that there's no easy way to tell *how* these theories are wrong or what should replace them. So even before String/M-theory is tested, it is superior to GR + QM because at least it's self consistant and matches everything seen up to date. One could argue that if science only exists to make cool technology, then it doesn't matter whether our theories are self contradictory until we can actually reach those energies... but some of us really want to know what's going on, which is why there is nevertheless still funding for it and interest in it.

That said, trying to get testable predictions out of string-based theories is a tough but active area of investigation. So to finally answer your question, there is no one experiment that we know of for sure that would be doable any time soon that could distinguish one theory of quantum gravity from another. But so far, there are no fully developed alternatives known. It could be that the real quantum gravity is something nobody has thought of yet. Or it could be that one of the scenarios laid out by loop quantum gravity people or other individuals working on their own ideas will eventually lead to a better alternative. Or there could be several different ways of understanding the real answer.

4000 character limit... see next post for continuation.
spoonless
Jul. 8th, 2005 02:50 am (UTC)
So what are some of the closest things to (short-term) predictions string theory (and spinoffs) have had? One is supersymmetry... it's definitely required by and therefore predicted by string theory, so if it's found at LHC that will be a good checkmark to say "yes, the theory matches a new observation we hadn't seen before"... unfortunately, it can't be used to refute it because we don't know what energy SUSY will be found at if it exists... it could be at accessible energies, or it could be way up too high where we can't see it. So part of the active research is coming up with ways to pin down better where SUSY should be found. Another possibility is cosmic strings... it's been proposed by some that giant cosmic sized strings should have been left over from
the big bang... so if we saw one of those in a telescope, it would be pretty difficult for anyone to deny that strings exist after that point. I don't know any of the details of that, though, and I think it's debated whether or not they should really be there which also needs to be sorted out. Other possible predictions involve properties of the dark matter. There are those who work on trying to figure out what the dark matter particles should look like, so that when we do measure them we can match it up with the various competing theories and get a clue as to which variant is right.

When would you guess something like this might be proven if it ever is?

Again, strike the word proven from it. The question is at what point (if ever) will it become widely accepted enough that it's considered the "standard" explanation and taught in textbooks. For comparison, look at quarks... nobody has ever "seen" a quark so you could not say they've been proven, but at some point they transitioned from being considered a "useful fiction" that only showed up in the mathematics, to being a basic building block of matter. I don't know when if ever this might happen for strings, but they do seem to be gaining more and more momentum and becoming more useful for doing high energy physics.
lars_larsen
Jul. 8th, 2005 03:50 am (UTC)
Its awesome that you went over the character limit. You're one badass motherfucker, motherfucker.

I read the book "supersymmetry", but was disappointed at the lack of math, it felt like one of those pansy-ass popular science books. It did talk a little bit about where SUSY would be found. And it seemed to say its at an energy just above what is currently testable. The problem is the book was written before the RHIC was built, and still no SUSY. I was sort of wondering if there was any new news on that front.

I'd say relativity is wrong, since quantum mechanics has been tested to about a zillion decimal places already. And experiments with relativity are fuzzy at best.

I guess I shouldn't say wrong, I should say incomplete.

And yeah, after I posted that I thought "damn, he's gonna ream me for saying proven". But I like that term. Maybe I should become a mathematician so I can use it properly :)
spoonless
Jul. 9th, 2005 07:22 am (UTC)

I read the book "supersymmetry", but was disappointed at the lack of math, it felt like one of those pansy-ass popular science books.

Hmmm... who was that by? I recently returned 3 books to the library, all with the title "Supersymmetry" or "Introduction to Supersymmetry" or similar variants. And all but one of them had a lot of math in them; the last one was a collection of essays by physicsists about its history and development.

But unless you've taken some very advanced math and physics courses (such as, what I finally got up to taking last quarter) you really don't want one of the ones with the math in!... unless you particularly enjoy staring at Greek letters. :)

The problem is the book was written before the RHIC was built, and still no SUSY. I was sort of wondering if there was any new news on that front.

Not really... everyone's placing bets, but there doesn't seem to be any consensus. Some theories predict high energy breaking, others low energy breaking. So whether we find it or not, it will toss away some theories and confirm others... however, it looks like there are too many variants for it to be a real test one way or another for something as general as string theory itself. There's also the difficulty of distinguishing supersymmetry from other possibilities such as technocolor or "large extra dimensions".

I'd say relativity is wrong, since quantum mechanics has been tested to about a zillion decimal places already. And experiments with relativity are fuzzy at best.

Then you'd be in good company with the string theorists. From what I've heard, one of the basic differences between loop quantum gravity (LQG) and string theory approaches to quantum gravity is that the LQG people mostly assume it's quantum itself that's wrong and the string people assume it's GR that's wrong. For this reason and others, I'd say that string theory is the more "conservative" approach, but that's probably because I share your opinion that GR hasn't been tested nearly as well as QM and therefore should be taken more with a grain of salt.
lars_larsen
Jul. 9th, 2005 05:24 pm (UTC)
Here is the sypersymmetry book. Its by Gordon Kane.

http://www.amazon.com/exec/obidos/tg/detail/-/0738202037/ref=pd_sxp_f/103-0794182-4433410?v=glance&s=books

So if people are just guessing at the energy level, what is the point of approaching this theoretically? If you have no idea what energy level it's at, then predicting things in advance is useless isn't it?

Has math ever predicted the energy level of things that were later discovered? Antimatter maybe? Or have we discovered something like that experimentally and then said "oh yeah, DUH! that makes perfect sense now".

If you could say we need X energy to see this particle, it would be a lot easier to get money out of congress. That would be nice. I don't think they like this "bigger is better, just trust us" approach.

I also read "3 roads to quantum gravity" by Lee Smolin. He seems to think one of the major problems is that everybody has holed themselves up in their respective little camps. Some just work on LQG, some just work on M-theory. They're invested in their own ideas, and just dismiss everything everyone else is doing. He advocates taking the most useful bits from each theory, and putting them all together into something more useful than any one of the existing theories.

With work this difficult, I dont blame you for focusing on what you do. It would be difficult for any one person to understand everything everyone else is doing.
spoonless
Jul. 9th, 2005 06:28 pm (UTC)

Here is the sypersymmetry book. Its by Gordon Kane.

Ah, I have seen that one in bookstores now that you mention it. Haven't read it though. The prof I took Quantum Field Theory III from last quarter had Gordon Kane for his advisor when he was in grad school. They've published a lot of papers together.

So if people are just guessing at the energy level, what is the point of approaching this theoretically? If you have no idea what energy level it's at, then predicting things in advance is useless isn't it?

Well, like I said... some specific theories that people have can predict certain ranges for those energies. But it depends on what particular set of assumptions they're using. So yes, in this case, there's not all that much point in guessing, we need to just wait and see.

Has math ever predicted the energy level of things that were later discovered? Antimatter maybe? Or have we discovered something like that experimentally and then said "oh yeah, DUH! that makes perfect sense now".

By energy here, we're really talking about the mass of the particles being discovered. Unfortunately, the masses of elementary particles are one of the very few things that quantum field theories cannot predict... they are the input parameters to the theory which is then used to predict everything else. There are 19 such input parameters to the standard model, most of them masses and a few other things.

But yes, in the case of antimater the exact mass was predicted... since they had to match the mass of their matter partners. In the case of the top quark, discovered in 1995, they had it narrowed down to a range just above where they had checked... but they didn't know the exact mass until they found it. The same is true for the Higgs. We know for sure that, if there is a Higgs at all, it must be just above where we're currently looking. If it's not, then the whole Higgs mechanism (which is a part of the Standard Model) will be discredited and thrown away. And it would need to be replaced with something else that explains the non-zero masses of the Standard Model... however, we have no other mechanism in mind yet so if it's not the Higgs then we'll just have to wait and see what it is.

If you could say we need X energy to see this particle, it would be a lot easier to get money out of congress. That would be nice. I don't think they like this "bigger is better, just trust us" approach.

Well, on the other side of the coin... if we knew exactly what the mass was then we'd probably know everything else about it too. Which means we wouldn't need to build an accelerator at all. So in some ways, the only reason it's worth spending the money is because we don't know.

I also read "3 roads to quantum gravity" by Lee Smolin. He seems to think one of the major problems is that everybody has holed themselves up in their respective little camps. Some just work on LQG, some just work on M-theory. They're invested in their own ideas, and just dismiss everything everyone else is doing. He advocates taking the most useful bits from each theory, and putting them all together into something more useful than any one of the existing theories.

Yeah, I recently read an interview with Smolin and he sounds very cool to me. I like his approach. Which is why I hope to read some LQG at some point in the next few years to at least get a feeling for what it is. But it wouldn't make much sense for me to work on it here because I wouldn't be able to have an advisor giving me hints and guiding me along. I guess if I did read more about it and think it was way better than string theory, then I could consider transferring. But for now, string theory makes a lot more sense for me to learn.
lars_larsen
Jul. 9th, 2005 06:54 pm (UTC)
I see. That makes sense.

In other news, I'm starting college for the first time ever this fall. Hoping to transfer to UVA and major in biomedical engineering. If I dont get accepted in that major (there are only 60 slots), I'm going into chemical engineering with a concentration in biotechnology and biochemistry.

Of course, thats if I get accepted for transfer in the first place. Which may be difficult because I'll probably have to transfer after one year, and they prefer students with 2 years and an AS degree already. I can't do that though because the second year of community college doesn't offer any classes that even approximate what the engineering students at UVA are taking in their second year. Whats the fucking point of a community college if it wont prepare you to transfer? Isn't that what its intended for in the first place?
spoonless
Jul. 11th, 2005 08:29 am (UTC)
Ah, cool. Well, I hope you get accepted... you seem very intelligent to me for someone who's never had any formal college training. I'd expect that once you get past the beaurocratic tape of getting in, you'll find it quite easy and fun.

Kind of sucks that half of getting ahead in life is convincing people dumber than you that you're worth taking seriously. But it's a chore that needs to be done :)
lars_larsen
Jul. 11th, 2005 04:11 pm (UTC)
I have to get a GED too!! How lame is that! I could complete all the classes for a degree but they wont give me the degree until I get a GED. What a huge pain in the ass. If I have to take an 8 hour test it better fucking be something useful like the MCAT.

Thanks for the encouragement.

The process so far has been very frustrating so far as I've already explained. Nobody seems to be of any help. They make me feel like I'm just another number.

Also, I've found that the process discriminates against adults, poor people, and unintelligent/uneducated people. That really pisses me off.
cola_fan
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?
spoonless
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.
( 12 comments — Leave a comment )

Profile

blueshirt
spoonless
domino plural

Latest Month

May 2017
S M T W T F S
 123456
78910111213
14151617181920
21222324252627
28293031   

Tags

Powered by LiveJournal.com
Designed by Lizzy Enger