# What mathematics background must one have to be conversant in M Theory?

The Discovery Science Channel ran an absorbing program, “Parallel Universes” which I recorded and watched with some interest. However, that is not to say I understood it. So, I wonder what kind of math do I need to truly comprehend all this heady material?

Anyway, here’s what I could delve from the program

Initially, String Theorists believed that all matter is composed of teeny tiny strings, and that their research would explain the Big Bang and everything in the Universe - i.e., the Theory of Everything.

But they were thwarted by the fact that they kept finding more and more different strings to explain the same thing. Thus, instead of a TOE they were getting a TON: a Theory of Nothing. All their strings, they said, consisted of 10 dimensions — 9 were Spatial, 1 was Time.

Meanwhile a particularly brilliant guy, Michael Duff (perhaps a Brit, judging from his accent), was working on Super Gravity, and he insisted that the string was composed not of 10, but 11 dimensions. Sadly he was pooh poohed by the know-it-all String savants and for maybe 10 years poor Duff was in the scientific equivalent of limbo.

Eventually, however, things took a change for the better when the String theorists finally realized he was right. Eleven dimensions was right on the money, and with that, they found there were not 5, but just one string after all. Those other pain in the ass “different” strings were simply manifestations of the same string and in fact were quite the same. What’s more this one string was infinitely long and just 10 to the minus 20 mm wide.

But, somewhere along the line, this 1 string morphed into a membrane, and it gets even more incomprehensible after that. In a nutshell, here’s what I think I gleaned from the program:

Our universe is a Membrane — aka Brane. There are an infinity of other Branes, other Universes, out there all sailing along in (for lack of a better word) Space. The Branes appear in variety of shapes - Donuts, Balls, Sheets, etc.

When two Branes collide some very violent stuff happens. And this, say the M-Theorists, is precisely how this Universe of ours was formed.

Imagine an outline of two ripple-shaped Branes, looking more or less like colorless, cruller-shaped clouds. If these two Branes collide, when one Brane’s cruller outcroppings touch the other’s outcroppings, a super gigantic explosion results — a Big Bang.

And since these Branes are surging endlessly through Space, such collisions probably happen regularly, resulting in ever more Big Bangs, ever more Branes, ever more Universes.

What I like about this theory, is it seems to negate the damned Singularity. I just cannot accept the fact that all the matter in this universe was compacted in a tiny mass, maybe as large as an atomic nucleus, maybe as big as a thumbnail (Carl Sagan?), maybe the size of a clenched fist. Whatever. It is inconceivable (to me) that billions upon billions of stars in untold billions of galaxies and star clusters could be contained in such a small mass.

Another pleasant aspect is that the theory looks to what happened before our Big Bang, something that some scientists (including some Dopers) have insisted is pointless.

The upshot of M-Theory is that we don’t live in a Universe, but a Multiverse. And I don’t understand any of it.

So that’s my question:

What math background do the String and M-Theorists have? And even if by some improbable show of perseverance, I should acquire such knowledge, would I have any chance of getting a hold on the subject?

Somehow, I doubt it.

I can’t answer your question, E=mc², but I just wanted to thank you for your simple and concise definition of string theory. I didn’t understand it too well myself, and whatever I tried to read on the subject seems to have been written by scientists, for scientists.

If you ask me, you seem to have a pretty good hold on it. I’d converse with you on the subject.

It’s been a while but I believe that the objects which theorists say compose the universe should not be understood literally as “strings,” but that that is a metaphor for the mathematical construct. Michi Kaku’s book Hyperspace is an interesting read on this and related subjects.

Excuse me that’s Michio Kaku. Wish there was an edit function.

I’m reading Brian Greene’s book “The Elegant Universe” right now - they made a NOVA documentary about it. So far, it seems quite accessible. Ask again when I’ve finished in a few weeks.

Please don’t worry about typos, iwakura43. I think most of us here read so quickly we don’t notice small errors like that. Besides, what you said is far more important than how you spelled it.

Anyway, Kaku was a major commentator on this “Parallel Universe” video, mentioned in the OP. And if I’m not mistakemn, I think he’s the the same guy, who on TV years ago, pushed for some genius he proclaimed to be the smartest in the world, to lead the way in String Theory research.

Michio sure knows how to put pressure on someone.

And, by the way, notice how I deftly corrected the typo in ‘mistakemn.’

And I just bought Kaku’s Hyperspace. Thank you for the recommendation. It gets good reviews at amazon, whereas Greene’s book is severely panned by some. (My apologies, Smeghead.

I’m not completely familiar with the formalisms of M theory, but at the very least I’d guess you’d need a pretty strong background in topology and differential geometry. Those are subjects that undergraduate math majors might be introduced to, but at the very least we’re talking introductory to intermediate graduate classes.

I don’t seem to own Hyperspace, though I have Kaku’s earlier work, Beyond Einstein, which I heartily recommend. Also, Greene has out a book called The Fabric of the Cosmos which has a lot of background about the development of new ideas in physics and an introduction on unified theories at the end.

So far, nobody’s given a hint of the math requirements, but I did happen upon a a book by Barton Zwiebach,s A First Course in String Theory at amazon.

In a review of this tome, a gentleman named Douglas Mckenzie admitted he couldn’t understand “…the math of String Theory so instead I spent two years preparing by learning Quantum Field Theory and Differential Geometry…”

You’ll also be happy to know that Mckenzie’s interest “…now extends into other areas for example Maxwell fields on D-branes.”

The chuckling you hear is me, laughing at myself for even thinking about learning the requisite math. :smack:

To be coctail-party conversant, Brian Greene’s books on top of what you’ve got now.

To be truly conversant… well, truth be told I’m not really there myself. First you’ll want a solid background in differential geometry for the GR basics. You’ll also want a full understanding of Lagrangian mechanics (essentially more differential geometry), and Hamiltonian mechanics (symplectic geometry this time), as well as their adaptations to classical field theory.

Now you’ll need a whole lot of algebra to grok the operator theory (part of functional analysis) that comes into quantization, which even the physicists don’t really get in a rigorous mathematical way. Also the representation theory to really get down the quantum field theory limit, and even more differential geometry for the gauge fields. Don’t forget a lot of Lie algebra stuff.

Oh, and the people who work in the field are running away with a whole lot of algebraic geometry techniques, like moduli spaces and Donaldson theory (which has been reborn as Seiberg-Witten theory).

Actually, there’s probably a fair bit more than that, but what I’ve laid out should take you through the first five or ten years. When you’re done with that let me know.

Hmph.

Don’t worry. In all honesty, even Ed Witten hasn’t even given a hint of the math requirements

Ooh, I forgot a metric assload of n-category theory, which you might have to wait on since it hasn’t really been worked out yet. Things like path-integrals (and worse) don’t seem to be workable without it.

I sincerely apologize, ultrafilter.

Don’t sweat it.

If someone has formulated some concept of the medium - especially its size, if that has any meaning whatsoever - in which the branes exist, I don’t recall seeing it anywhere. And I do follow that stuff, as much as I can comprehend of it anyway. Since this universe is at least 13.7 GY old, and hasn’t been … um … restarted, it seems implausible to me that brane collisions are terribly frequent.

And how many people do you think actually do understand it??

Remember, even a great many physicists disclaim to understand quantum physics. They can do the math okay; they just find the concepts frustrating (even as we, the semi-ignorant masses). I don’t have the quote handy, but Einstein had big problems with quantum stuff. And if I’ve got even that much correct, you gotta have a fair grasp on quantum mechanics in order to master strings.

My best guess is that you’d need 30-40 credits’ worth of grad level math (as Mathochist says). You know, getting the tools (basic stuff), to master the tools (higher level basics), to master the tools (lots of geometry & topology, at least, plus quantum & other physics).

Here’s my suggestion: Get yourself (or ask for it as a present from someone) a subscription to Scientific American (or check each month’s issue as it comes out). They have at least two or three articles a year on cosmology, and in recent years they’ve not only stopped putting equations in the physics articles; they’ve also … uh … dumbed them down to the level of interested layperson. :great relief: That doesn’t mean they leave out difficult concepts, they just put more effort into explaining them.

They also have periodic special issues (only sold at stores, or ordered direct, which is much more expensive because of postage, unless you download it). There was one about cosmology last year (I think; I tried to find my copy, but it wasn’t where I usually put them when I’ve finished reading them). Here’s a link that’ll let you take a look at the contents of one I think you’ll want: The Edge of Physics . It includes an article by Duff. AAMOF, that’s the one I was trying to find. I remember reading some of the articles.

If possible, I recommend going to the nearest library with a subscription and checking the contents pages. I think you’ll find there are many articles which will give you various points of view on this subject. There’s an article in the August 2003 issue I think you’d find particularly fascinating (even though it doesn’t talk about branes), titled “Are you a hologram?” It’s about black holes and quantum physics, mostly, but it talks about information density. The author blurb says he “has contributed to the foundation of black hole thermodynamics … and gravitation.” His work is clearly based in superstring (Duff’s) theory.

He’s the guy!!

I am absolutley, positively, unequivocally, dead solid certain Witten is the one who Kaku was publicly exhorting to go into string theory.

Or maybe it was Yogi Berra.

I will follow your advice - all of it. But as far as expenditures, I’ll have to wait a couple of weeks. My wife has already dropped father’s day hints to our son -
so we can assume I’ll be getting Brain Greene’s The Elegant Universe & The Fabric of the Cosmos (thanks to Mathochist’s recommendation), plus other books on order such as Kaku’s Hyperpspace

Now, if someone would please invite me to cocktail party in the year 2007…

Oh. I think Einstein’s Quantum-related quote you were looking for was something like, “God doesn’t play dice with the Universe.”

And if someone wants a video cassette of Parallel Universe, I’ll make one and send it to you. (I have my copy on DVR) Maybe you could pass it along to others who are similarly interested. Last one on the list gets to keep it.

Email me or a mod your snail-mail addresses and we’ll get this thing going.

Of course, if this is against SDMB rules, the offer is rescinded.