Edward Witen is not researching LQG-like quantum gravity

[josh1]

The conference http://gesalerico.ft.uam.es/strings07/040_scientific07_contents/041_speakers.htm is about string theory, not other approaches to quantum gravity.

It should be quite clear to anyone who understands LQG-like approaches to quantum gravity that there is not one single word in Witten's presentation that has anything whatsoever to do with LQG-like approaches to quantum gravity.

What he is doing is returning to a subject for which he's made a number of important contributions to in the past which is (2+1) dimensional gravity. As he says, his motivation for doing so is to obtain a quantum theory of BTZ black holes which he believes is possible for negative cosomological constant since it is only for these values of the cosmological constant that black hole solutions exist in (2+1) dimensional gravity.

He invokes a number of ideas, all of which are related to string theory and none of which are related to LQG. Most prominent are conformal field theory and the AdS/CFT correspondence. He also brings up the subject of gauge theory descriptions of (2+1) dimensional gravity, but only to demonstrate the serious problems associated with these sorts of gauge theory descriptions in this context.

On the other hand, people in lqg continue their search for legitimacy by trying to find ways to connect their ideas to string theory.

 

[setAI]

seems like more sour grapes- I guess you don't want to argue in the actual thread because there is far too much evidence about Witten's recent change in thinking- it is well known that Witten has been distancing himself from strings over the last year or two- it has come straight from his own mouth- denial and wishful thinking can't rescue a sinking ship josh

it's just tough love kid- nothing personal

 

[josh1]

I guess you don't want to argue in the actual thread because there is far too much evidence about Witten's recent change in thinking- it is well known that Witten has been distancing himself from strings over the last year or two- it has come straight from his own mouth

The title of the thread in question is "Edward Witen is not researching LQG-like quantum gravity". But not working on string theory does not mean he is working on LQG-like versions of quantum gravity. What he has been working on is pure mathematics, much of it being string theory-inspired or -related.

seems like more sour grapes-- denial and wishful thinking can't rescue a sinking ship setAI

it's just tough love kid- nothing personal

The core of ensabah6's thread is his use of Wittens lecture to bolster his claim that Witten is doing LQG-like quantum gravity. Given how obvious it is that this lecture shows nothing of the sort, it must be a reflection of your own desperation to try to convince yourself that ensabah is right.

Since it`s always more satisfying to vanquish an opposing viewpoint by using the facts, I can only assume that your choice to personally insult me instead is an indication that you do not have anything to back up the claim by ensabah6's you`re trying to defend.

For my part, I was specific about the physics that was discussed by Witten in his lecture. I don`t see what more I can do.

 

[garrett]

Hi josh, I can believe everything you're saying except the last sentence, which is blatantly false.

I am writing from the Loops '07 conference currently underway in Morelia. I can assure you that, at least from what I've experience in the past few days, no researchers here are attempting to establish credibility for their LQG approach by connecting with string theory -- even though there are almost as many different approaches being discussed as there are conference attendees.

When I got here I was expecting to find some antagonism towards string theory, but I was surprised to find a much more mature viewpoint overall. I'm getting the strong impression that people here accept that string theory may or may not be a valid approach to describing nature, but they would rather work on LQG or some related direction they consider promising. After all, string theory has plenty of people working on it already.

A commenter (not a speaker, but I don't remember who) did mention yesterday that Witten was at the strings conference talking about (2+1) gravity, and not directly about strings. But this was just a humorous interjection, for the small irony.

But, this morning, Moshe Rozali presented a plenary talk entirely on string theory and the role of the metric therein. He fielded many questions during his talk, but these were for clarification and not at all antagonistic. After his talk, there was a good round of applause. Really, people here accept strings as another approach among many.

That is what I've experienced. Now, my personal opinion is less mature. I think string theory involves too many insufficiently justified constructions and assumptions, piled one upon the other. In short, I think it's far-fetched, probably not how nature works, and largely a waste of time. LQG, I think, encompasses the more conservative approaches, with many paths showing progress.

I would have greatly enjoyed spending time here talking trash about string theory. But I got the strong impression that such derision was NOT welcome, so I held my tongue -- well, mostly. ;)

 

[josh1]

Hi Garret,

Firstly, thankyou for your very thoughtful response. Now...

I am writing from the Loops '07 conference currently underway in Morelia. When I got here I was expecting to find some antagonism towards string theory, but I was surprised to find a much more mature viewpoint overall. I'm getting the strong impression that people here accept that string theory may or may not be a valid approach to describing nature, but they would rather work on LQG or some related direction they consider promising. After all, string theory has plenty of people working on it already.

Where did your expectation that at the loops conference there would be more acrimony towards string theory come from?

 

[garrett]

Where did your expectation that at the loops conference there would be more acrimony towards string theory come from?

Mostly it came from the projection of my own feelings. I have been working on my own for a decade now on foundational physics that is not stringy. This is a tough situation to be in, since string theory has been more popular and received the lion's share of funding. Basically, I was envious of the (relatively) stable careers, funding, and other opportunities open to people studying strings, but I didn't want these things bad enough to put aside my misgivings about the theory and spend my time on it instead of spending it on the research I wanted to do. This was a hard practical decision to make, as a life choice -- it is very hard to be poor, especially when you were used to being ranked at or near the best student in your class. It made me a little bitter about the situation -- but my life is very good, so not extremely bitter.

I expected many in LQG to have been through similar struggles -- working away at their ideas while the choice academic positions in theory go to string theorists -- and be bitter. But it is not so. I'm not sure why, but I can speculate -- it may be the sense of community is so strong that it keeps people happy, or it may be the rise and success of P.I. as a home base for good but non-mainstream theory. Probably both.

In any case, life is going so well for me right now that I believe I will lose this bitterness with time. Don't expect me not to hold a grudge though. ;)

But I will never tell people not to work on what they want to -- be it string theory or anything. In principle, it should be a choice made without ulterior motives. In practice, we're human.

 

[josh1]

Mostly it came from the projection of my own feelings. I have been working on my own for a decade now on foundational physics that is not stringy. This is a tough situation to be in, since string theory has been more popular and received the lion's share of funding. Basically, I was envious of the (relatively) stable careers, funding, and other opportunities open to people studying strings, but I didn't want these things bad enough to put aside my misgivings about the theory and spend my time on it instead of spending it on the research I wanted to do. This was a hard practical decision to make, as a life choice -- it is very hard to be poor, especially when you were used to being ranked at or near the best student in your class. It made me a little bitter about the situation -- but my life is very good, so not extremely bitter.

I expected many in LQG to have been through similar struggles -- working away at their ideas while the choice academic positions in theory go to string theorists -- and be bitter. But it is not so. I'm not sure why, but I can speculate -- it may be the sense of community is so strong that it keeps people happy, or it may be the rise and success of P.I. as a home base for good but non-mainstream theory. Probably both.

In any case, life is going so well for me right now that I believe I will lose this bitterness with time. Don't expect me not to hold a grudge though. ;)

But I will never tell people not to work on what they want to -- be it string theory or anything. In principle, it should be a choice made without ulterior motives. In practice, we're human.

I understand. But can you help me understand the bitterness of setAI and other PF members towards string theory?

 

[garrett]

I understand. But can you help me understand the bitterness of setAI and other PF members towards string theory?

Sure, string theory gets too much attention and funding, for very little payoff. It's mostly held together by lots of old failed promises to get the standard model from strings. And, unless you get something resembling a prediction right quick, it's inertia will run out.

 

[Demystifier]

In my opinion, it is not really relevant whether Witten is doing string theory, LQG-like theory, or something else. A more important question to which I would like to see the answer is the following:
Do LQG people find his results useful for LQG physics?

 

[garrett]

Well, I've enjoyed some of his papers. And I pulled the explicit E8 structure constants from his "Superstring Theory" book. But I'm too dumb to understand most of his papers now -- I'm not sure, but he may have spawned a paper writing style I hate: "words, words, words, math result." I hate that. When I read a paper, I want to be able to work through the calculations in it, so I really know what's going on. I can do that with his earlier papers, but not his later ones.

 

[Demystifier]

But I'm too dumb to understand most of his papers now -- I'm not sure, but he may have spawned a paper writing style I hate: "words, words, words, math result." I hate that. When I read a paper, I want to be able to work through the calculations in it, so I really know what's going on. I can do that with his earlier papers, but not his later ones.

I hate this stile too. The worst thing is that it is not only the style of Witten, but of all most famous string theorists. To me, it seems to be a sign that they do not yet completely understand things that they attempt to understand.

 

[john baez]

In my opinion, it is not really relevant whether Witten is doing string theory, LQG-like theory, or something else. A more important question to which I would like to see the answer is the following:
Do LQG people find his results useful for LQG physics?

Back when I did loop quantum gravity, I found his results on 3d quantum gravity and Chern-Simons theory to be incredibly important. And it wasn't just me: if you look, you'll see that almost everyone who has worked seriously on loop quantum gravity and spin foams has checked their work by seeing what it reduces to in 3d spacetime, and comparing the results with Witten's ideas.

His new work is an outgrowth of his old work, and if it holds water it's again likely to be influential in loop quantum gravity. Why? Because it tries to address the question of degenerate metrics!

Witten's old work, and loop quantum gravity, allows the metric to become degenerate. The main problem with the Barrett-Crane spin foam model seems to be that the path integral in this model is dominated by http://arxiv.org/abs/gr-qc/0208010" . While Witten's new work only applies to 3d quantum gravity, and it seems quite speculative, any progress in this direction would be a wonderful thing.

One doesn't need to sympathize with string theory or loop quantum gravity to enjoy Witten's work. If his http://arxiv.org/abs/0706.3359" ![/I]

People who are fighting about whether his new work belongs to string theory or not should look up from their petty squabbles and think a bit about what this means! It's absolutely mind-boggling. It's as if the sky just sprouted pink polka dots.

Even if this new paper is wrong, Witten's the most brilliant mathematical physicist of our era. He works this kind of magic routinely; the rest of us just gape in awe.

 

[Haelfix]

Wittens paper is extremely important. Not the least because it shows something incredibably weird going on in 2+1 gravity. Various descriptions that were previously thought identical, are now giving different answers. It seems to me to be quite a revolution in the field, not the least b/c a tremendous amount of research has gone into this area and entire research programs based on the 'guesses' that one gets in this regime.

The resolution of this problem either leads to something extremely beautiful with the monster group of all things, or the fact that there may indeed be *no*solution for pure gravity for 2+1, which would falsify a lot of things.

 

[ensabah6]

Back when I did loop quantum gravity, I found his results on 3d quantum gravity and Chern-Simons theory to be incredibly important. And it wasn't just me: if you look, you'll see that almost everyone who has worked seriously on loop quantum gravity and spin foams has checked their work by seeing what it reduces to in 3d spacetime, and comparing the results with Witten's ideas.

His new work is an outgrowth of his old work, and if it holds water it's again likely to be influential in loop quantum gravity. Why? Because it tries to address the question of degenerate metrics!

Witten's old work, and loop quantum gravity, allows the metric to become degenerate. The main problem with the Barrett-Crane spin foam model seems to be that the path integral in this model is dominated by http://arxiv.org/abs/gr-qc/0208010" . While Witten's new work only applies to 3d quantum gravity, and it seems quite speculative, any progress in this direction would be a wonderful thing.

One doesn't need to sympathize with string theory or loop quantum gravity to enjoy Witten's work. If his http://arxiv.org/abs/0706.3359" ![/I]

People who are fighting about whether his new work belongs to string theory or not should look up from their petty squabbles and think a bit about what this means! It's absolutely mind-boggling. It's as if the sky just sprouted pink polka dots.

Even if this new paper is wrong, Witten's the most brilliant mathematical physicist of our era. He works this kind of magic routinely; the rest of us just gape in awe.

Hi, Are you now working on another QG or given up entirely? A lot of papers have been done in LQG on the Kodama state (i.e Eyo Ita, Randanomo) that could use peer review :rofl:

 

[Chronos]

. . . The resolution of this problem either leads to something extremely beautiful with the monster group of all things, or the fact that there may indeed be *no*solution for pure gravity for 2+1, which would falsify a lot of things.

I'm leaning towards the 'no solution' hypothesis. . . in the 'not even wrong' sense. The more important thing to do now is wait for better observations. Most researchers are crawling all over one another to reach the top of the wrong mountain - IMO.

 

[ccdantas]

he's claiming 3d quantum gravity is intimately connected to the Monster group!

Some basic questions. Why is k "an integer for topological reasons"? (k is a parameter that appears in a second term -- a multiple of the Chern- Simons invariant of the spin connection -- added to the action).

Further, what is "holomorphic" factorization? (A pointer to the basic literature on this will suffice). Is it the only possible constraint?

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?

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?

Thanks,
Christine

 

[josh1]

His new work is an outgrowth of his old work…

This is precisely what I said!

People who are fighting about whether his new work belongs to string theory or not…

Witten’s work may or may not impact “LQG-like quantum gravity, but it’s just as likely to impact other fields as well. Therefore to say that Witten is now working on LQG-like quantum gravity is simply wrong.

Now, squabbling about whether or not Witten's work belongs to string theory is not something one would find happening very much within the physics community since everyone knows what John say's is correct. But this forum is not the physics community, and given the penchant that members here have of purposefully misleading people who don't know any better to believe that strings is on it's way out, it was worthwhile to respond to the thread entitled "Witten is researching LQG-like quantum gravity".

 

[Thomas Larsson]

When I read Witten's paper I came to think about something that Rovelli said in his first talk at http://www.fuw.edu.pl/~kostecki/school.html : "Don't write a paper on 3D quantum gravity" (7:50). Personally I find the wrong sign of the cc more disturbing than the restriction to 3D, though. Assuming AdS when the cc is known to be positive seems dangerously close to denial of experimental facts.

If you take the positive cc seriously, there is AFAIU only one possible conclusion: QG has local observables. Since de Sitter space has no global observables (no observable observables anyway), and there should better be some observables, there must be local observables. If this is true, and everybody is looking for a theory of QG without local observables, the lack of progress is not surprising.

 

[jal]

http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.3359v1.pdf
Three-Dimensional Gravity Reconsidered
Edward Witten
22 June 2007
Can I assume that his approach has enough symmetry to accommodate ALL approaches to try to build a working model.
References
p.6 .. 1.3. A Non-Classical Restriction
p.7 …1.4. Plan Of This Paper
-------------
jal

 

[marcus]

When I read Witten's paper I came to think about something that Rovelli said in his first talk at http://www.fuw.edu.pl/~kostecki/school.html : "Don't write a paper on 3D quantum gravity" (7:50). Personally I find the wrong sign of the cc more disturbing than the restriction to 3D, though. Assuming AdS when the cc is known to be positive seems dangerously close to denial of experimental facts.
...

I remember that quote. He was advising the grad students and other young researchers at the Zakopane school how to make their work relevant to solving the QG problem. Not to make it merely interesting mathematics.

In the same spirit he might well have added "Don't write on PURE gravity, include matter in the picture," and also what you said about not making such an unrealistic cc assumption, but I don't remember him actually saying that. Given the overall message it probably wasn't necessary.

He did qualify what he said about 3D IIRC something like: do 3D only if you are developing tools and methods you think will help you with the real problem in 4D----don't rely on the very SPECIAL nature of 3D. I don't recall the exact words but there was some escape clause like that. And it applied neatly to Laurent Freidel's work on 3D-quantum-gravity-with-matter where he makes the point that he is intentionally avoiding using tools that work only in lower dimension. A good summary of that work, slides and audio, is at the ILQGS site---a 15 May 2007 seminar talk Freidel gave. The inclusion of matter and the use of tools generalizable to 4D is crucial, and made explicit in the talk. You get to hear reactions and questions from Rovelli, Ashtekar, and others in the seminar. SOME 3D gravity-and-matter is obviously interesting (needless to say Freidel wasn't assuming negative cosmological constant, TL)

http://relativity.phys.lsu.edu/ilqgs/
Tuesday, May 15th
Laurent Freidel, Perimeter Institute
Matter coupling to 3d quantum gravity and effective field theory I

 

[ccdantas]

Assuming AdS when the cc is known to be positive seems dangerously close to denial of experimental facts.
(...)
If you take the positive cc seriously, there is AFAIU only one possible conclusion: QG has local observables.

I think Witten is clear enough about his first motivations (although the conclusions he arrives at do not seem clear enough to me at this point).

He aims at arriving at an exact description of the quantum theory of black holes in the trivial, pure gravity in 2+1 D and (I presume :uhh:) draw some conclusions from it. With Lambda < 0, it seems he can make use of techniques from dual conformal field theory and other assumptions (as well as draw some correlations) and calculate things. He says he cannot do that with Lambda > 0, as in this case it "doesn’t really make sense as an exact theory in its own right but (like an unstable particle) must be studied as part of a larger system" (from his talk slides).

I don't understand what were his conclusions doing this exercise. If he gets to an exact 3D quantum gravity theory for lambda <0, that's an achievement, but as far as I can understand this, it means he cannot use the same methods for other cases (Lambda >=0) and 4D gravity. What lessons can be learned from this exercise then? If there is no way to extend it to a more realistic scenario, then I presume it is a purely academic exercise, no matter how interesting it is...

On the other hand, could this possible and mysterious connection of 2+1D gravity with the Monster group be a way out to infer about 4D models with Lambda>0?

 

[marcus]

... What lessons can be learned from this exercise then? If there is no way to extend it to a more realistic scenario, then I presume it is a purely academic exercise, no matter how interesting it is...
...

several of us seem worried by how many unrealistic assumptions

some might be bothered by it being 3D (like Rovelli especially if it led to using tools that were special to the situation)

some, like Thomas, are bothered by the cosmological constant assumed negative

to me, at this stage of the game, excluding matter and focussing on PURE gravity seems like taking a step backwards

I do not see on what grounds our universe can be thought to be like an "unstable particle" that can only be studied in the context of a "larger system". That sort of drastic assumption would seem more appropriate if ongoing approaches had proven dead-ends. So I must say it surprised me when I read it.

 

[jal]

...unrealistic assumptions

I do not have a problem with Edward Witten’s paper.

“We aim to solve three-dimensional gravity with Lambda < 0, at some distinguished values of ℓ/G at which it makes sense.”

Lambda < 0 is assumed in order to be able to look behind the event horizon.
This is then equated to the central charge c and also equated to be a negative constant.

“The values that emerge – with the help of a small sleight of hand in the choice of the gauge group for the Chern-Simons theory –are interesting. They are the values at which cL and cR are integer multiples of 24, and complete holomorphic factorization of the dual CFT is conceivable.
More generally, the same is true for the left- and right-moving central charges cL and cR.
So the central charges of the dual CFT cannot depend on a continuously variable parameter ℓ/G. It must be [18] that the theory only makes sense for specific values of ℓ/G.

According to our interpretation of three-dimensional gravity, it is described by a conformal field theory with c = 24k and no primary field other than the identity of dimension less than k + 1”.

c = 24k
?meaning?
In 2d the smallest value that can be assumed is a value that can make an event horizon and it would have a negative energy of 24. An outside observer would not be able to observe anything else on the “brick wall” of the event horizon.
How does that conflict with what I have been saying, “The Quantum Minimum Length Structure would be a sphere of 24 units?”

Are there objections to having c = 24k?

Jal
----------------------

 

[john baez]

quantization of the cosmological constant

Some basic questions.

I'm about to jump on a plane, so I'll only answer one now. I'll read Witten's paper more carefully on the plane. I plan to write an issue of This Week's Finds about this work of his. If that leaves some of your questions unanswered (and it almost surely will), it would be great if you'd repost them over at the n-Category Cafe as comments to that issue.

Why is k "an integer for topological reasons"? (k is a parameter that appears in a second term -- a multiple of the Chern- Simons invariant of the spin connection -- added to the action).

The Chern-Simons action S is invariant under small gauge transformations (those connected to the identity by a continuous path), but changes by multiples of a certain constant c under large gauge transformations. What shows up in path integrals is the exponentiated action exp(ikS) where k is some coupling constant. The consequence is clear: exp(ikS) remains unchanged under large gauge transformations if and only if exp(ikc) = 1, meaning that k has to be an integer multiple of 2 pi / c.

If you set up all your normalization conventions nicely, c = 2 pi, so k has to be an integer.

This stuff is explained a bit more in my book Gauge Fields, Knots and Gravity, in section II.4, Chern-Simons Theory. Also see the end of section II.5.

In 3d quantum gravity, the consequence is that the cosmological constant can only take certain discrete values!

 

[john baez]

I don't understand what were his conclusions doing this exercise. If he gets to an exact 3D quantum gravity theory for lambda <0, that's an achievement, but as far as I can understand this, it means he cannot use the same methods for other cases (Lambda >=0) and 4D gravity. What lessons can be learned from this exercise then? If there is no way to extend it to a more realistic scenario, then I presume it is a purely academic exercise, no matter how interesting it is...

You shouldn't evaluate this work of Witten based on its short-term implications for physics.

Yes, the theory he's studying is completely unrealistic as a physical theory. Conclusions drawn from it do not apply to the physical world, since spacetime has constant curvature away from matter in 3d gravity, unlike in 4d gravity. Massive particles don't attract each other in 3d gravity, and there are no gravitational waves either.

All this is well-known by everyone who works on 3d gravity.

The importance of Witten's work lies elsewhere. To understand it, you need to think about Monstrous Moonshine. It's an amazing puzzle built deeply into the fabric of mathematical reality! Briefly: one of the most basic functions in complex analysis, the j-function, turns out to know a lot about the representations of the Monster group: the largest sporadic finite simple group, an insanely large group with 808017424794512875886459904961710757005754368000000000 elements.
The Monster group is the king of all the "exceptional" mathematical structures - things like the octonions, E8, the Mathieu groups, the Leech lattice, the Conway group and so on. The fact that it shows up as you soon as you seriously study complex analysis is completely bizarre and mysterious.

The Monster group is already known to be related to string theory - but only in a rather sneaky and complicated way. If it turned out to play a role in a (seemingly) simple theory like 3d quantum gravity, this would be an incredible step forward.

 

[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 a lot 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!

 

[jal]

jal
How does that conflict with what I have been saying, “The Quantum Minimum Length Structure would be a sphere of 24 units?”

Are there objections to having c = 24k?

I guess the only problem is that it shows up toooooo much.
http://golem.ph.utexas.edu/category/2007/06/more_mysteries_of_the_number_2.html
More Mysteries of the Number 24
Posted by John Baez
------------
jal

 

[john baez]

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" 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" ) 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.

 

[john baez]

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/" . Unfortunately these just skim the surface.

 

[john baez]

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" . 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" , but unfortunately there are too many.

 

[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

 

[josh1]

...we all love you John Baez...

No we don't.