# Edward Witen is not researching LQG-like quantum gravity

#### Fra

It's an amazing puzzle built deeply into the fabric of mathematical reality!
Hello John,

(Beeing triggered by "Mathematical reality")

I've enjoyed many of your great contributions and elaborations published on the internet, and I am curious if you might care to briefly comment on your view of the scientific method in general. Your choice of words indicates that your have some mathematical guidance, and if you are to characterize that, trying to think away your own brain so to speak, what if you were to "design a robot to work in your spirit". Along what principles, would it be designed to be successful?

What the connection/relation between this "mathematical reality", and "physical reality"?

So in short I am curious about your personal view of the scientific method, and philosophy of science in short. The reason why I am curious about you is of course that I've very much enjoyed alot of your elaborations in various places. And the reason why I am curious about this in general, have to do with my personal views of science about general learning models, in the context of the scientific method. And your views might be particularly interesting since I could relate them to the massive amount of work and education you've done through the internet.

I'm not expecting pages of philosophy, but if you'd care to comment shortly on the above I think it would be very interesting.

/Fredrik

#### Ratzinger

"exceptional" mathematical structures - things like the octonions, E8, the Mathieu groups, the Leech lattice, the Conway group and so on
So will exceptional mathematics be needed for understanding the final laws of nature?

If so could someone please write a book on exceptional math? Because that sounds exciting!

#### ccdantas

Hi John Baez,

Thank you for answering my first question -- it's quite clear now. It was the phrase "topological reasons" in his talk that seemed mysterious to me.

Concerning the Monster group, I've been organizing some links over at my blog about this issue (I welcome other more appropriate references, if that is the case, but in any case I´ll be linking to your TWF on Witten's paper there), and learn something about it over vacation. So if I understand you, you are saying that the main novelty in Witten's new paper would be the possibility that the Monster group could play some fundamental role -- although for the moment in an physically unrealistic 3D quantum gravity model, is that right? Then I would like to repeat my question (as given on my second post above, and also in addition to what you have written "this would be an incredible step forward") -- would you interpret then that Witten is implicitly arguing that the Monster group could also serve as a mathematical guidance to a more realistic 4D quantum gravity?

Thanks!

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#### john baez

Gold Member
Concerning the Monster group, I've been organizing some links over at my blog about this issue (I welcome other more appropriate references, if that is the case, but in any case I´ll be linking to your TWF on Witten's paper there), and learn something about it over vacation.
Great. The review articles by Terry Gannon cited in Witten's paper are very good, but http://arxiv.org/abs/math/0109067" [Broken] may be the most fun to start with.

So if I understand you, you are saying that the main novelty in Witten's new paper would be the possibility that the Monster group could play some fundamental role -- although for the moment in an physically unrealistic 3D quantum gravity model, is that right?
The Monster does play a fundamental role - in mathematics. It somehow determines the coefficients of the j function, which is the function you use to show that: the space of all ways to make a torus look locally like the complex plane is a sphere.

I hope you understand the italicized phrase: the grammar is a bit twisted! There are lots of ways to make a torus into a "Riemann surface", meaning a surface which is locally identified with the complex plane, so you can do complex analysis on it. And, there's a space of all such ways. And, the space of all these ways is the Riemann sphere!

So, if I hand you a torus made into a Riemann surface, you should be able to compute a point on the Riemann sphere - roughly speaking, a complex number that might be infinite. And, the recipe for doing this is called the j function.

You can expand this function as a power series in a variable q which is easy to compute given the complex structure on your torus (as explained in http://math.ucr.edu/home/baez/week66.html" [Broken]) and you get something strange:

$$j = q^{-1} + 744 + 196884 q + 21493706 q^2 + ...$$

And, it turns out that these funny coefficients are dimensions of representations of the Monster group: the largest finite simple group that doesn't fit into any family!

Now, the fact that such a simple issue could lead us quickly into the arms of a group with 808017424794512875886459904961710757005754368000000000 elements is really amazing. It's one of the deepest puzzles in mathematics. It's called Monstrous Moonshine. People have proved it's true, using ideas from string theory, and Borcherds won a Fields medals for his work on this. But his argument is complicated and indirect, so I don't think anyone feels they understand why Monstrous Moonshine is true. We want to know why!

If 3d quantum gravity is related to the Monster group as Witten argues, it would be an incredible step forwards to understanding this puzzle.

Then I would like to repeat my question (as given on my second post above, and also in addition to what you have written "this would be an incredible step forward") -- would you interpret then that Witten is implicitly arguing that the Monster group could also serve as a mathematical guidance to a more realistic 4D quantum gravity?
No, none of this work has much to do with 4d quantum gravity - at least, not that I can see. I don't think Witten is suggesting this, either.

I again urge you to stop seeking short-term physics applications for Witten's new work. That's not the right way to understand why it's so exciting! It's great because if it's true it means the world is far stranger yet simpler than anyone guessed.

If all I cared about was quantizing 4d gravity, I might not care about Witten's new paper very much. And, I'd be an unhappy and frustrated person, because I doubt anyone will succeed in quantizing 4d gravity in my lifetime. Luckily I care about many other things... so I can enjoy what Witten did.

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#### john baez

Gold Member
So will exceptional mathematics be needed for understanding the final laws of nature?
I'll tell you as soon as we discover those final laws. :tongue:

If so could someone please write a book on exceptional math? Because that sounds exciting!
I've considered writing such a book, just because this math is so beautiful and strange. So far I've only written some fragments, like http://math.ucr.edu/home/baez/dodecahedron/" [Broken]. Unfortunately these just skim the surface.

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#### john baez

Gold Member
So in short I am curious about your personal view of the scientific method, and philosophy of science in short.
I don't have anything quick and interesting to say about the philosophy of science. So, I'll just say some things about why I do science.

For a long time I've been fascinated by the http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html". I think it's one of the deepest mysteries of our universe. It's easy to understand in certain cases, at least if the universe needs to obey the laws of logic at all. But, why should comprehensible mathematical laws govern so much of physical reality? Do we really need to be made of particles that are representations of Lie groups, for example? I see no obvious reason why this needs to be true: it's just an empirical observation of a remarkable fact. One might go so far as to say it's the only real miracle in nature - apart from the fact that the universe exists at all, which is an even bigger miracle.

One possibility is that http://math.ucr.edu/home/baez/week146.html" [Broken]. This sound strange, but all the other possibilities seem even less plausible to me.

Anyway: I've been fascinated by these questions for a long time, but I quickly concluded I can't make much progress by a frontal assault. I think most of us should keep trying to understand mathematics and the laws of physics. We may never learn the "true laws of physics" - such things may not even exist - but if they do, and we learn them someday, maybe then we'll be in a better place to understand deeper mysteries: like why there are laws at all.

Here's another thing that seems worth saying:

When I first began learning mathematics, I was mainly interested in it for its applications to physics. After learning some more - maybe when I was around 40 or so - I realized that it has layers of depth and structure that are simply invisible when you're first getting started. Everything fits together in beautiful patterns that are in some sense simple, but these patterns fit together in larger patterns that are somehow simpler, and so on. I don't see that it stops anywhere. It keeps getting more interesting the more I learn - but unfortunately, it's very hard to explain why it's so interesting without getting into a lot of technicalities. I guess all I can say is that it keeps turning out to be much weirder and more beautiful than you'd ever expect... even if that's what you expect.

The possible relation between the Monster group and black holes in 3d quantum gravity is a great example. Another good one is the mysterious way in which the integers resemble a 3-dimensional space, with prime numbers being like knots, which can be linked inside this space. There are lots of others. I keep trying to explain these things in http://math.ucr.edu/home/baez/TWF.html" [Broken], but unfortunately there are too many.

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#### Ratzinger

I've considered writing such a book, just because this math is so beautiful and strange. So far I've only written some fragments, like Tales of the Dodecahedron, Platonic Solids in All Dimensions, and The Octonions. Unfortunately these just skim the surface.
Yes!!!

By the way, as much as we all love you John Baez, you need to write more books.

In the meantime, last week I bought Gannnon's book 'Moonshine beyond the Monster', despite the big price. Great read, recommended to everyone.

#### Fra

Thanks John, for your comments! That was just the kind of answer I was looking for and it makes sense. I'm not mathematician, but I certainly share some of your questions.

But, why should comprehensible mathematical laws govern so much of physical reality? Do we really need to be made of particles that are representations of Lie groups, for example? I see no obvious reason why this needs to be true: it's just an empirical observation of a remarkable fact.
How about that this is not a coicidence, and that the "physical reality" we perceive is the answers to the questions we ask? The formalism of our questions constrains the kind of answers we get. And when one asks quantitative questions mathematics is our formalism of choice, chose for it's effiency?

It seems to me different situations have different optimal languages. When we communicate human to human, english is not that bad at all, because the discussion here is mainly qualitative.

For a long time I've been fascinated by the unreasonable effectiveness of mathematics. I think it's one of the deepest mysteries of our universe.
I like to think that humans invented mathematics, discovered may be an alternative term, but I can't see the clear difference between invention and discovery. I'm fine with either.

And as such I'd assume that we have developed mathematics to be efficient. If it wasn't efficient it would probably never have grown. We would have found another more efficient language and perhaps even named that mathematics too?

I agree that the languages is kind of part of science, because the languages or formalisms seems to also evolve. In that way perhaps one can even attribute physical properties to the language as any lifeforms or system implementing some formalisms may get benefits in nature by making use of "more efficient reasoning".

/Fredrik

#### Mike2

john baez said:
I think it's one of the deepest mysteries of our universe. It's easy to understand in certain cases, at least if the universe needs to obey the laws of logic at all. But, why should comprehensible mathematical laws govern so much of physical reality? Do we really need to be made of particles that are representations of Lie groups, for example? I see no obvious reason why this needs to be true: it's just an empirical observation of a remarkable fact. One might go so far as to say it's the only real miracle in nature - apart from the fact that the universe exists at all, which is an even bigger miracle.
If it turns out that physics can be derived from logic, then I think it would not be such a mystery why a language developed from logic (mathematics) would be useful in describing reality. Both are derived from logic.

#### ccdantas

Great. The review articles by Terry Gannon cited in Witten's paper are very good, but http://arxiv.org/abs/math/0109067" [Broken] may be the most fun to start with.
Thanks! And thanks for your explanation. Quite interesting!

If 3d quantum gravity is related to the Monster group as Witten argues, it would be an incredible step forwards to understanding this puzzle.
Right. Now I see your point.

I again urge you to stop seeking short-term physics applications for Witten's new work.
A natural attitude for a non-mathematician, no? Being an astrophysicist in the first place drives me to think more in physical terms. In any case, I was not seeking short-terms applications of Witten's new work, but just trying to understand possible implications for future research.

Luckily I care about many other things... so I can enjoy what Witten did.
I think mathematics is, by far, less frustrating than physics. Actually I would love to turn myself into a mathematician, but the process would be too frustrating as well (!), so I limit myself to learn what it's possible to be learned in my lifetime.

BTW concerning the big mystery of why the physical world can be described by mathematics, that is certainly the most fundamental issue of all. And if physics is derived from logic (as Mike2 suggests, and would be great if he shows a proof that is so), one would have to end up with the same mystery: why the physical world can be described by logic. For me, the issue seems not to have progressed much and I would even say that it still revolves around a Kantian metaphysics on the basis of the human intellect. There is no obvious way to approach the question of the correspondence of the physical world (up to simplifying assumptions) to our mathematical internal formulations from a scientific point of view. There is still too much to be learned.

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#### ccdantas

the space of all ways to make a torus look locally like the complex plane is a sphere.
I'm reading the paper you have mentioned, it's quite interesting, though I needed some help from Nakahara to get some basic points. Thanks!

#### john baez

Gold Member
Some basic questions.
I've tried to give a general sketchy introduction to Witten's paper in http://math.ucr.edu/home/baez/week254.html" [Broken] - you might look at that.

Further, what is "holomorphic" factorization? (A pointer to the basic literature on this will suffice).
I don't really understand that term. It should be defined in http://arxiv.org/abs/hep-th/9205072" [Broken] --- this paper speaks of "meromorphic conformal field theories" instead of "conformal field theories with holomorphic factorization", but they must be the same thing. However, I'm having a bit of trouble finding the precise definition! I just know a bunch of properties of these theories.

First, the central charge c is an integer multiple of 24.

Second, as a consequence, the partition function is really a well-defined number, not just defined up to (24/c)th root of unity. In other words, it's "modular invariant".

These two are very important in Witten's paper.

Third, as another consequence, the http://en.wikipedia.org/wiki/Schwinger_function" [Broken] --- that is, holomorphic except for poles. This is not so important in Witten's paper, though.

Is it the only possible constraint?
Witten gives an argument that 3d quantum gravity has as its AdS/CFT dual a conformal field theory with c = 24k for some integer k = 1,2,3,... The main
nice thing is that - modulo a certain conjecture - Schellekens classified these conformal field theories for k = 1.

He argues that the (naive) partition function Z_0(q) differs from the "exact" Z(q) by terms of order O(q). Would this be correct for any k?
Yes, he argues this is true for any k. Then, around equation (3.13), he shows that this property, together with modular invariance of the exact partition function, completely determines the exact partition function! It's a certain explicit polynomial in the J function.

He finds that for k=1 the monster group is interpreted as the symmetry of 2+1-dimensional black holes. How sensitive is this result with respect to the value of k, and to respect to the other assuptions used in the derivation?
For k=1 he goes through Schelleken's list of 71 conformal field theories with c = 24 and picks the one that has the Monster group as its symmetries. He gives an argument for why this one is the right one, but it's not airtight.

He doesn't actually find the relevant conformal field theories with c = 24k for
higher values of k. He just figures out their supposed partition functions. Since the coefficients of their partition functions are - just as in the k = 1 case - dimensions of representations of the Monster group, it seems awfully plausible that these theories (if they really exist!) have the Monster group as symmetries.

However, this is something one would want to check. Nobody seems to know a c = 48 theory with Monster group symmetries, for example.

I will copy your questions and my answers to the http://golem.ph.utexas.edu/category/2007/07/this_weeks_finds_in_mathematic_15.html" [Broken], and hope some experts on conformal field theory (like Urs Schreiber and Jacques Distler) can help us out.

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#### ccdantas

Dear John Baez,

Thanks a lot. I'll go in more detail into what you have written and of course I'll read with great interest your new TWF and blog entry.

Over at http://egregium.wordpress.com/2007/06/29/witten-on-3d-quantum-gravity-and-the-monster-group/" [Broken], in special, I was thinking about the section "Holomorphic functions can be factored" of that article. Please let me know whether you think that is a right pointer or not. I'll add a link to the new TWF/n-Category Café entry over at my blog opportunely.

Thanks,
Christine

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