Mathematical Mystery Believed to Have Been Solved
The Scotsman ^ | Mon 6 Sep 2004 | John von Radowitz
http://www.freerepublic.com/focus/f-news/1208744/posts
http://news.scotsman.com/latest.cfm?id=3461424

One of the seven great unsolved mysteries of mathematics may have been cracked by a reclusive Russian who is not remotely interested in the £560,000 prize his solution could win him, it emerged today.

The Poincare Conjecture involves the study of shapes, spaces and surfaces and makes predictions about the topology of multi-dimensional objects.

Basically, it says that a three-dimensional sphere can be used in an analogous way to describe higher-dimensional objects that are impossible to visualise.

Since Henri Poincare suggested the theorem in 1904, some of the greatest mathematicians of the 20th century have struggled to prove it either right or wrong.

All have failed. But now the world of maths is buzzing with the news that an answer might at long last have been found.

Dr Grigori Perelman, from the Steklove Institute of Mathematics at the Russian Academy of Sciences in St Petersburg, has published two papers offering a solution to a larger-scale problem called the Geometrization Conjecture.

This is also concerned with geometry, and experts say that contained within it is proof that the Poincare Conjecture works.

If Perelman can satisfy his peers that this is the case, he stands to win a one million dollar cash prize from the Clay Mathematics Institute in the United States.

The Institute is offering million dollar prizes for solutions to each of the mathematical conundrums it calls the Seven Millennium Problems.

But there is a more fundamental problem the general community of mathematicians needs to solve first. Perelman does not seem to be interested.

Dr Keith Devlin, a leading mathematician from Stanford University in California, explained: “He´s very reclusive, and won´t talk to anyone. He´s shown no indication of publishing this as a paper, and he´s shown no interest in the prize whatsoever.

“Has it been proved? We don´t know, but there´s good reason to think it has been. My guess is that in about 12 months people will start to say okay, this is right, but there´s not going to be a golden moment.”

Dr Perelman published his two papers in November, 2002 and March last year.

A third is yet to be published.

By all accounts, Poincare will come out of the first two papers, said Dr Devlin.

If the conjecture was proved it would have profound ramifications, he told the British Association Festival of Science at the University of Exeter.

Scientists working on the frontiers of cosmology and physics frequently dealt with hyperdimensions. A solution to the Poincare Conjecture would greatly increase their understanding of the shape of the universe.

Dr Devlin compared proving Poincare with setting off an avalanche. If you are on top of a mountain, and it is spring, and you jump up and down, a little bit of snow moves. But at the bottom a whole lot of snow comes down.

“It can´t fail to have enormous implications; it will just be huge.”

He said solving mathematical problems such as the Poincare Conjecture was more like writing a story than doing a sum, which was why it took so long.

“It´s just so damn complicated, he said. It really can take two or three years to certify the thing.”

Proving the Poincare Conjecture would be the first great mathematical breakthrough since Andrew Wiles solved Fermats Last Theorem in 1994.

This year, Professor Louis de Branges de Bourcia, from Purdue University in the United States, claimed to have proven another of the Millennium Problems called the Riemann Hypothesis.

The hypothesis is a 150-year-old theory about Prime Numbers – numbers that divide only by one and themselves and are considered the atoms of arithmetic.

De Branges claimed to have confirmed a conjecture made by the German mathematician Bernhard Riemann in 1859 about the way prime numbers were distributed.

But, unlike in the case of Poincare Conjecture, the worlds mathematicians are becoming increasingly convinced that he has got it wrong.

Marcus du Sautoy, Professor of Mathematics at Oxford University, said: “The mathematical community is sceptical whether the methods of Louis de Branges are capable of proving the Riemann Hypothesis.”

If de Branges turned out to be right, it would have a dramatic impact on both global business and national security.

Encrypted codes are based on the randomness of prime numbers. If a system could be found that made them predictable, no secret would be safe.

“What mathematics has been missing is a sort of maths prime spectrometer, like the machine chemists use to tell them what things are made of,” said Prof du Sautoy. “If we had something like that it would bring the world of e-commerce to its knees overnight.”

It is indeed not true that only supergravity plays a role in these calculations. In particular anomalous scaling dimensions of operators on the SYM side is matched to at least first non-sugra order on the string side by a whole small industry. Search the arXive for papers by Tseytlin and by Zarembo (2004), for instance.

I know that the area operator constructed in LQG has discrete spectrum (BTW, which spectrum precisely? Last time I checked there were at least two different ‘proposals’ for this spectrum.)

But the area operator is not gauge invariant, but has single spin networks as eigenstates. But any state that actually solves all of the constraints, including the Hamiltonian constraint (if it can ever be defined) must be a superposition of spin network states (first of all it must be a knot state (solving the spatial diffeo constraints), but surely even a superposition of knot states) and most probably (nobody knows!) a continuous superposition.

This means that it is not clear that on physical states the area operator has any eigenstates at all.

Often LQG properties are argued in terms of spin network states. But these are just a particular choice of basis – of the kinematical Hilbert space. Statement like ‘the universe is a huge spin network’ or things like that make an unjustified identification of a choice of basis with a physical observable.

BTW, does anyone know about the status of the ‘Phoenix Project/Master Constraint Programme‘ to actually construct the Hamiltonian constraint? As long as this hasn’t been done it seems pretty vain to discuss any properties of LQG.

BTW, as I discuss here the 1+1D example once again helps to understand the situation: When you decide to restrict to the space of spatially rep invariant functions in 1+1 D gravity and then try to apply the Hamiltonian constraint as an operator on that, you know you cannot succeed. That’s because this amounts to imposing the constraints

L_n – \bar L_{-n} for all n

together with

L_0 + \bar L_0 .

The first infinite set of constraints can be imposed all right, and even with the methods used in LQG, because these generators have no anomaly among themselves. But then standard results in 1+1d gravity tell you that the Hamiltonian constraint L_0 + \bar L_0 cannot annihilate any of the resulting states.

I am speculating that this is in fact the reason why Thomas Thiemann in his LQG-string paper decided not to precisely follow the standard LQG procedure (which consists of applying the quantized Hamiltonian constraint as an operator on the space of spatially rep-invariant states) but decided to treat both L_n and \bar L_n by ‘relaxed canonical quantization’.

let’s try to avoid bickering and general accusations and just focus on the discussion of some technical facts and problems. I’d be very interested to see Lee Smolin reply here (maybe mediated by Peter Woit).

You claimed that something is a problem for string theory and I asked you to please demonstrate it. Your reply

As all string theorists, you consistently require much higher standards from the competition than from yourself.

leaves me in puzzlement. From past discussions I can guess that you are thinking that somehow results about anomaly cancellation for chiral fermions have to do with anomalies in the canonical quantization of gravity. I frankly admit that I don’t see what you have in mind. But if you can make it precise you should probably do so and show it to somebody who knows more about it than I do.

It is precisely this problem with discussing full 1+3d nonperturbative quantum gravity that makes me rather wish to discuss a toy example like 1+1d gravity. All open questions of LQG are also non-trivial and unsolved in this context.

I do indeed agree with your claim that for an understanding of a non-perturbative quantization of gravity it should be very helpful to have an idea of the existing quantum reps of the symmetry generated by the classical constraints (though I would not call that alone a quantization of gravity, as you sometimes do).

I have never claimed (I hope) to have quantized gravity. I do claim, however, that I have succeeded in quantizing the symmetry principle underlying gravity, and I believe that this will prove to be an essential ingredient in quantum gravity itself.

But I don’t claim that I know how to similarly ‘fix’ the LQG approach in higher dimensions. If you do and if you think that you can make a nonperturbative quantization of the action of closed string field theory and if you think you can deduce some kind of problem for string theory from that please do so and let us know about your result.

As all string theorists, you consistently require much higher standards from the competition than from yourself. I create (not alone, but anyway), the mathematics that makes it possible to discuss diffeomorphisms on the same footing as conformal transformation, and you say this may be interesting if I also succeed in using this to quantize gravity. The last guy who succeeded in creating both mathematical infrastructure and physics was called Newton, and I think that my results may be valuable even though I am a lesser soul than him, especially since everybody else has failed to make any progress towards on quantum gravity. Not even Einstein created his own math; Ricci and Levi-Civita created tensor calculus for him to use.

Meanwhile, you seem completely undisturbed by the fact that string theory makes no testable predictions whatsoever and that recent landscape ideas show that it never will, a situation which Witten describes as “[String theory] is also more predictive than conventional quantum field theory.” I don’t believe in LQG, but as long as Smolin does not claim that LQG is more predictive than field theory, he deserves good marks for honesty. Misrepresentation is Fraud’s cousin.

BTW, did you see our comments on your 2-form gauge theory over here? (Possibly you were on vacation when these were posted.) Since you are only using a 2-form B you have to face the problem that the surface holonomies that you compute are not independent of the ‘parameterization’ of your surface, i.e. of the order in which you multiply the group elements on the plaquettes.

The amplitudes depend on the triangulation (or quadrangulation, rather); even the number of indices depends on that. This is why it’s only the lattice model, were there is a canonical triangulation, that is well defined. However, there is no order dependence in the sense of non-associativity. Contraction of all index pairs associated with internal links is order independent; finite sums can be performed in any order.

there is non-perturbative quantization of gravity, where we try to solve the canonical constraints exactly, and there is perturbative quantization, where we compute quantum corrections to classical solutions. The discussion here is about how to do the non-perturbative quantization.

I do indeed agree with your claim that for an understanding of a non-perturbative quantization of gravity it should be very helpful to have an idea of the existing quantum reps of the symmetry generated by the classical constraints (though I would not call that alone a quantization of gravity, as you sometimes do).

This is precisely what I claim is missing in the LQG approach, and in the special case of 1+1 dimensions one indeed finds that the idea you are proposing gives us the correct constraint algebra (namely the Virasoro algebra), while LQG misses it.

But I don’t claim that I know how to similarly ‘fix’ the LQG approach in higher dimensions. If you do and if you think that you can make a nonperturbative quantization of the action of closed string field theory and if you think you can deduce some kind of problem for string theory from that please do so and let us know about your result.

BTW, did you see our comments on your 2-form gauge theory over here? (Possibly you were on vacation when these were posted.) Since you are only using a 2-form B you have to face the problem that the surface holonomies that you compute are not independent of the ‘parameterization’ of your surface, i.e. of the order in which you multiply the group elements on the plaquettes. To ensure well-defined surface holonomy you’d need to add a 1-form A and the condition B+F=0 and then you’d make contact with Girelli&Pfeiffer’s work.

BTW, one maybe interesting piece of trivia in the triangle of topics constituted by LQG, strings and non-abelian 2-form field theories is the following:

If we restrict as in LQG to spatial rep-invariant states we obtain, in 1+1 dimensional gravity, the so-called boundary states. A particularly natural such state is the loop space function which assigns the holonomy of some connection A to a loop. Incidentally, this function is precisely the boundary state which describes a non-abelian 2-form B = -F_A .

See this entry and hep-th/0408161 for the details.

The following statements are almost identical:

1. LQG is only quantization in a weaker, nonstandard way. In particular, if the quantization of 3+1D gravity in any way resembles quantization of 1+1D gravity on a world sheet, then the diffeomorphism constraint should acquire an anomaly which cancels against ghosts.

2. All symmetries, including gauge symmetries like diffeomorphisms, need representations of lowest-energy type.

Thus, apparently to your great embarrassment, we are saying roughly the same thing (except that ghosts are problematic, because normal ordering doesn’t just ruin nilpotency but makes the BRST operator ill defined). However, spacetime diffeomorphisms do not use lowest-energy representations in string theory neither, although it does treat worldsheet gravity in this correct way.

So exactly why do you expect LQG people to take your argument seriously, if you don’t take the logical implications seriously yourself?