This Week's Finds in Mathematical Physics (Week 148)

John Baez

Last week I talked about some millennium-related books. This week, some millennial math problems! In 1900, at the second International Congress of Mathematicians, Hilbert posed a famous list of 23 problems. No one individual seems to have the guts to repeat that sort of challenge now. But the newly-founded Clay Mathematics Institute, based in Cambridge Massachusetts and run by Arthur Jaffe, has just laid out a nice list of 7 problems:

1) Clay Mathematics Institute, Millennium Prize Problems, http://www.claymath.org/prizeproblems/index.htm

There is a 1 million dollar prize for each one! Unlike most of Hilbert's problems, these weren't cooked up specially for the occasion: they have already proved their merit by resisting attack for some time.

Here they are:

1. P = NP?

This is the newest problem on the list and the easiest to explain. An algorithm is "polynomial-time" if the time it takes to run is bounded by some polynomial in the length of the input data. This is a crude but easily understood condition to decide whether an algorithm is fast enough to be worth bothering with. A "nondeterministic polynomial-time" algorithm is one that can check a purported solution to a problem in an amount of time bounded by some polynomial in the input data. All algorithms in P are in NP, but how about the converse? Is P = NP? Stephen Cook posed this problem in 1971 and it's still open. It seems unlikely to be true - a good candidate for a counterexample is the problem of factoring integers - but nobody has proved that it's false. This is the most practical question of the lot, because if the answer were "yes", there's a chance that one could use this result to quickly crack all the current best encryption schemes.

2. The Poincaré conjecture

Spheres are among the most fundamental topological spaces, but spheres hold many mysteries. For example: is every 3-dimensional manifold with the same homotopy type as a 3-sphere actually homeomorphic to a 3-sphere? Or for short: are homotopy 3-spheres really 3-spheres? Poincaré posed this puzzle in 1904 shortly after he knocked down an easier conjecture of his by finding 3-manifolds with the same homology groups as 3-spheres that weren't really 3-spheres. The higher-dimensional analogues of Poincaré's question have all been settled in the affirmative - Smale, Stallings and Wallace solved it in dimensions 5 and higher, and Freedman later solved the subtler 4-dimensional case - but the 3-dimensional case is still unsolved. This is an excellent illustration of a fact that may seem surprising at first: many problems in topology are toughest in fairly low dimensions! The reason is that there's less "maneuvering room". The last couple decades have seen a burst of new ideas in low-dimensional topology - this has been a theme of This Week's Finds ever since it started - but the Poincaré conjecture remains uncracked.

3. The Birch-Swinnerton-Dyer conjecture

This is a conjecture about elliptic curves, and indirectly, number theory. For a precise definition of an elliptic curve I'll refer you to "week13" and "week125", but basically, it's a torus-shaped surface described by an algebraic equation like this:

y2 = x3 + ax + b

Any elliptic curve is naturally an abelian group, and the points on it with rational coordinates form a finitely generated subgroup. When are there infinitely many such rational points? In 1965, Birch and Swinnerton-Dyer conjectured a criterion involving something called the "L-function" of the elliptic curve. The L-function L(s) is an elegant encoding of how many solutions there are to the above equation modulo p, where p is any prime. The Birch-Swinnerton-Dyer conjecture says that L(1) = 0 if and only if the elliptic curve has infinitely many rational points. More generally, it says that the order of the zero of L(s) at s = 1 equals the rank of the group of rational points on the elliptic curve (that is, the rank of the free abelian summand of this group.) A solution to this conjecture would shed a lot of light on Diophantine equations, one of which goes back to at least the 10th century - namely, the problem of finding which integers appear as the areas of right triangles all of whose sides have lengths equal to rational numbers.

4. The Hodge conjecture

This question is about algebraic geometry and topology. A "projective nonsingular complex algebraic variety" is basically a compact smooth manifold described by a bunch of homogeneous complex polynomial equations. Such a variety always has even dimension, say 2n. We can take the DeRham cohomology of such a variety and break it up into parts Hp,q labelled by pairs (p,q) of integers between 0 and n, using the fact that every function is a sum of a holomorphic and an antiholomorphic part. Sitting inside the DeRham cohomology is the rational cohomology, The rational guys inside Hp,p are called "Hodge forms". By Poincaré duality any closed analytic subspace of our variety defines a Hodge form - this sort of Hodge form is called an algebraic cycle. The Hodge conjecture, posed in 1950 states: every Hodge form is a rational linear linear combination of algebraic cycles. It's saying that we can concretely realize a bunch of cohomology classes using closed analytic subspaces sitting inside our variety.

5. Existence and mass gap for Yang-Mills theory

One of the great open problems of modern mathematical physics is whether the Standard Model of particle physics is mathematically consistent. It's not even known whether "pure" Yang-Mills theory - uncoupled to fermions or the Higgs - is a well-defined quantum field theory with reasonable properties. To make this question precise, people have formulated various axioms for a quantum field theory, like the so-called "Haag-Kastler axioms". The job of constructive quantum field theory is to mathematically study questions like whether we can construct Yang-Mills theory in such a way that it satisfies these axioms. But one really wants to know more: at the very least, existence of Yang-Mills theory coupled to fermions, together with a "mass gap" - i.e., a nonzero minimum mass for the particles formed as bound states of the theory (like protons are bound states of quarks).

6. Existence and smoothness for the Navier-Stokes equations

The Navier-Stokes equations are a set of partial differential equations describing the flow of a viscous incompressible fluid. If you start out with a nice smooth vector field describing the flow of some fluid, it will often get complicated and twisty as turbulence develops. Nobody knows whether the solution exists for all time, or whether it develops singularities and becomes undefined after a while! In fact, numerical evidence hints at the contrary. So one would like to know whether solutions exist for all time and remain smooth - or at least find conditions under which this is the case. Of course, the Navier-Stokes equations are only an approximation to the actual behavior of fluids, since it idealizes them as a continuum when they are actually made of molecules. But it's important to understand whether and how the continuum approximation breaks down as turbulence develops.

7. The Riemann hypothesis

For Re(s) > 1 the Riemann zeta function is defined by

ζ(s) = 1/1s + 1/2s + 1/3s + ....

But we can extend it by analytic continuation to most of the complex plane - it has a pole at s = 1. The zeta function has a bunch of zeros in the "critical strip" where Re(s) is between 0 and 1. In 1859, Riemann conjectured that all such zeros have real part equal to 1/2. This conjecture has lots of interesting ramifications for things like the distribution of prime numbers. By now, more than a billion zeros in the critical strip have been found to have real part 1/2; it has also been shown that "most" such zeros have this property, but the Riemann hypothesis remains open.

If you solve one of these conjectures and win a million dollars because you read about it here on This Week's Finds, please put me in your will.

Okay, now on to some other stuff.

This week was good for me in two ways. First of all, Ashtekar, Krasnov and I finally finished a paper on black hole entropy that we've been struggling away on for over 3 years. I can't resist talking about this paper at length, since it's such a relief to be done with it. Second, the guru of n-category theory, Ross Street, visited Riverside and explained a bunch of cool stuff to James Dolan and me. I may talk about this next time.

2) Abhay Ashtekar, John Baez and Kirill Krasnov, Quantum geometry of isolated horizons and black hole entropy, preprint available at gr-qc/0005126 or at http://math.ucr.edu/home/baez/black2.ps

I explained an earlier version of this paper in "week112", but now I want to give a more technical explanation. So:

The goal of this paper is to understand the geometry of black holes in a way that takes quantum effects into account, using the techniques of loop quantum gravity. We do not consider the region near the singularity, which is poorly understood. Instead, we focus on the geometry of the event horizon, since we wish to compute the entropy of a black hole by counting the microstates of its horizon.

Perhaps I should say a word about why we want to do this. As explained in "week111", Bekenstein and Hawking found a formula relating the entropy S of a black hole to the area A of its event horizon. It is very simple:

S = A/4

in units where the speed of light, Newton's constant, Boltzmann's constant and Planck's constant equal 1. Now, in quantum statistical mechanics, the entropy of a system in thermal equilibrium is roughly the logarithm of the number N of microstates it can occupy:

S = ln N.

This is exactly right when all the microstates have the same energy. Thus we expect that a black hole of area A has about

N = exp(A/4)

microstates. For a solar-mass black hole, that's about exp(10^76) microstates! Any good theory of quantum gravity must explain what these microstates are. Since their number is related to the event horizon's area, it is natural to guess that they're related to the geometry of the event horizon. But how?

It's clear that everything will work perfectly if each little patch of the event horizon with area 4 ln(2) has exactly 2 states. I think Wheeler was the first to take this seriously enough to propose a toy model where each such patch stores one bit of information, making the black hole into something sort of like an enormous hard drive:

3) John Wheeler, It from bit, in Sakharov Memorial Lecture on Physics, Volume 2, eds. L. Keldysh and V. Feinberg, Nova Science, New York, 1992.

Of course, this idea sounds a bit nutty. However, the quantum state of a spinor contains exactly one bit of information, and loop quantum gravity is based on the theory of spinors, so it's not as crazy as it might seem.... Still, there are some, ahem, details to be worked out!

So let's work them out.

The first step is to understand the classical mechanics of a black hole in a way that allows us to apply the techniques of loop quantum gravity. In other words, we want to describe a classical phase space for our black hole. This step was done in a companion paper:

4) Abhay Ashtekar, Alejandro Corichi and Kirill Krasnov, Isolated horizons: the classical phase space, Advances in Theoretical and Mathematical Physics 3 (2000), 418-471. Preprint available at gr-qc/9905089.

The idea is to consider the region of spacetime outside the black hole and assume that its boundary is a cylinder of the form R x S2. We demand that this boundary is an "isolated horizon" - crudely speaking, a surface that light cannot escape from, with no matter or gravitational radiation falling in for the stretch of time under consideration. To make this concept precise we need to impose some boundary conditions on the metric and other fields at the horizon. These are most elegantly described using Penrose's spinor formalism for general relativity, as discussed in "week109". With the help of these boundary conditions, we can start with the usual Lagrangian for general relativity, turn the crank, and work out a description of the phase space for an isolated black hole.

If we temporarily ignore the presence of matter, a point in this phase space describes the metric and extrinsic curvature of space outside the black hole at a given moment of time. Technically, we do this using an SU(2) connection A together with an su(2)-valued 2-form E. You can think of these as analogous to the vector potential and electric field in electromagnetism. As usual, they need to satisfy some constraints coming from Einstein's equations for general relativity. They also need to satisfy boundary conditions coming from the definition of an isolated horizon.

Since the black hole is shaped like a ball, the boundary conditions hold on a 2-sphere that I'll call the "horizon 2-sphere". One thing the boundary conditions say is that on the horizon 2-sphere, the SU(2) connection A is completely determined by a U(1) connection, say W. This U(1) connection is really important, because it describes the intrinsic geometry of the horizon 2-sphere. Here's a good way to think about it: first you restrict the spacetime metric to the horizon 2-sphere, and then you work out the Levi-Civita connection of this metric on the 2-sphere. Finally, since loop quantum gravity is based on the parallel transport of spinors, you work out the corresponding connection for spinors on the 2-sphere, which is a U(1) connection. That's W!

The boundary conditions also say that on the horizon 2-sphere, the E field is proportional to the curvature of W. So on the horizon 2-sphere, all the fields are determined by W. This is even true when we take the presence of matter into account. When we quantize, it'll be the microstates of this field W that give rise to the black hole entropy. Since W is just a technical way of describing the shape of the horizon 2-sphere, this means that the black hole entropy arises from the many slightly different possible shapes that the horizon can have.

But I'm getting ahead of myself here! We haven't quantized yet; we're just talking about the classical phase space for an isolated black hole.

The most unusual feature of this phase space is that its symplectic structure is a sum of two terms. First, there is the usual integral over space at a given time, which makes the E field canonically conjugate to the A field away from the horizon 2-sphere. But then there is a boundary term: an integral over the horizon 2-sphere. This gives the geometry of the horizon a life of its own, which ultimately accounts for the black hole entropy. Not surprisingly, this boundary term involves the U(1) connection W. In fact, this boundary term is just the symplectic structure for U(1) Chern-Simons theory on the 2-sphere! It's the simplest thing you can write down:

ω(δ W, δ W') = (k/2π) ∫ δ W ^ δ W'

Here ω is the U(1) Chern-Simons symplectic structure; we're evaluating it on two tangent vectors to the space of U(1) connections on the 2-sphere, which we call δ W and δ W'. These are the same as 1-forms, so we can wedge them and integrate the result over the 2-sphere. The number k is some constant depending on the area of the black hole... but more about that later!

I guess this Chern-Simons stuff needs some background to fully appreciate. I have been talking about it for a long time here on This Week's Finds. The quantum version of Chern-Simons theory is a 3-dimensional quantum field theory that burst into prominence thanks to Witten's work relating it to the Jones polynomial, which is an invariant of knots. At least heuristically, you can calculate the Jones polynomial by doing a path integral in SU(2) Chern-Simons theory. It also turns out that Chern-Simons theory is deeply related to quantum gravity in 3d spacetime. For quite a while, various people have hoped that Chern-Simons theory was important for quantum gravity in 4d spacetime, too - see for example "week56" and "week57". However, there have been serious technical problems in most attempts to relate Chern-Simons theory to physically realistic problems in 4d quantum gravity. I think we may finally be straightening out some of these problems! But the ironic twist is that we're using U(1) Chern-Simons theory, which is really very simple compared to the sexier SU(2) version. For example, U(1) Chern-Simons theory also gives a knot invariant, but it's basically just the self-linking number. And the math of U(1) Chern-Simons theory goes back to the 1800s - it's really just the mathematics of "theta functions".

As a historical note, I should add that the really nice derivation of the Chern-Simons boundary term in the symplectic structure for isolated black holes was found in a paper written after the one I mentioned above:

5) Abhay Ashtekar, Chris Beetle and Steve Fairhurst, Mechanics of isolated horizons, Class. and Quant. Gravity 17 (2000), 253-298. Available at gr-qc/9907068.

Originally, everyone thought that to make the action differentiable as a function of the fields, you had to add a boundary term to the usual action for general relativity, and that this boundary term was responsible for the boundary term in the symplectic structure. This seemed a bit ad hoc. Of course, you need to differentiate the action to get the field equations, so it's perfectly sensible to add an extra term if that's what you need to make the action differentiable, but still you wonder: where did the extra term COME FROM?

Luckily, Ashtekar and company eventually realized that while you can add an extra term to the action, you don't really need to. By cleverly using the "isolated horizon" boundary conditions, you can show that the usual action for general relativity is already differentiable without any extra term, and that it yields the Chern-Simons boundary term in the symplectic structure.

Okay: we've got a phase space for an isolated black hole, and we've got the symplectic structure on this phase space. Now what?

Well, now we should quantize this phase space! It's a bit complicated, but thanks to the two-part form of the symplectic structure, it basically breaks up into two separate problems: quantizing the A field and its canonical conjugate E outside the horizon 2-sphere, and quantizing the W field on this 2-sphere. The first problem is basically just the usual problem of loop quantum gravity - people know a lot about that. The second problem is basically just quantizing U(1) Chern-Simons theory - people know even more about that! But then you have to go back and put the two pieces together. For that, it's crucial that on the horizon, the E field is proportional to the curvature of the connection W.

So: what do quantum states in the resulting theory look like? I'll describe a basis of states for you....

Outside the black hole, they are described by spin networks. I've discussed these in "week110" and elsewhere, but let me just recall the basics. A spin network is a graph whose edges are labelled by irreducible representations of SU(2), or in other words spins j = 0, 1/2, 1, and so on. Their vertices are labelled as well, but that doesn't concern us much here. What matters more is that the spin network edges can puncture the horizon 2-sphere. And it turns out that each puncture must be labelled with a number m chosen from the set

{-j, -j+1, .... j-1, j}

These numbers m determine the state of the geometry of the horizon 2-sphere.

What do these numbers j and m really MEAN? Well, they should be vaguely familiar if you've studied the quantum mechanics of angular momentum. The same math is at work here, but with a rather different interpretation. Spin network edges represent quantized flux lines of the gravitational E field. When a spin network edge punctures the horizon 2-sphere, it contributes area to the 2-sphere: a spin-j edge contributes an area equal to

8 π γ sqrt(j(j+1))

for some constant γ.

But due to the boundary conditions relating the E field to the curvature of the connection W, each spin network edge also contributes curvature to the horizon 2-sphere. In fact, this 2-sphere is flat except where a spin network edge punctures it; at the punctures it has cone-shaped singularities. You can form a cone by cutting out a wedge-shaped slice from a piece of paper and reattaching the two new edges, and the shape of this cone is described by the "deficit angle" - the angle of the wedge you removed. In this black hole business, a puncture labelled by the number m gives a conical singularity with a deficit angle equal to

4 π m / k

where k is the same constant appearing in the formula for the Chern- Simons symplectic structure.

I guess now it's time to explain these mysterious constants! First of all, γ is an undetermined dimensionless constant, usually called the "Immirzi parameter" because it was first discovered by Fernando Barbero. This parameter sets the scale at which area is quantized! Of course, the formula for the area contributed by a spin-j edge:

8 π γ sqrt(j(j+1))

also has a factor of the Planck area lurking in it, which you can't see because I've set c, G, and ħ to 1. That's not surprising. What's surprising is the appearance of the Barbero-Immirzi parameter. So far, loop quantum gravity cannot predict the value of this parameter from first principles.

Secondly, the number k, called the "level" in Chern-Simons theory, is given by

k = A / 4 π γ

Okay, that's all for my quick description of what we get when we quantize the phase space for an isolated black hole. I didn't explain how the quantization procedure actually works - it involves all sorts of fun stuff about theta functions and so on. I just told the final result.

Now for the entropy calculation. Here we ask the following question: "given a black hole whose area is within ε of A, what is the logarithm of the number of microstates compatible with this area?" This should be the entropy of the black hole - and it won't depend much on the number ε, so long as its on the Planck scale.

To calculate the entropy, first we work out all the ways to label punctures by spins j so that the total area comes within ε of A. For any way to do this, we then count the allowed ways to pick numbers m describing the intrinsic curvature of the black hole surface. Then we sum these up and take the logarithm.

What's the answer? Well, I'll do the calculation for you now in a really sloppy way, just to sketch how it goes. To get as many ways to pick the numbers m as possible, we should concentrate on states where most of the spins j labelling punctures equal 1/2. If all these spins equal 1/2, each puncture contributes an area

8 π γ sqrt(j(j+1)) = 4 π γ √3

to the horizon 2-sphere. Since the total area is close to A, this means that there are about A/(4 π γ √3) punctures. Then for each puncture we can pick m = -1/2 or m = 1/2. This gives

N = 2 A/4 π γ √3

ways to choose the m values. If this were exactly right, the entropy of the black hole would be

S = ln N = (ln 2 /4 π γ √3) A

Believe it or not, this crude estimate asymptotically approaches the correct answer as A approaches infinity. In other words, when the black hole is in its maximum-entropy state, the vast majority of the spin network edges poking through the horizon are spin-1/2 edges.

So, what have we seen? Well, we've seen that the black hole entropy is (asymptotically!) proportional to the area, just like Bekenstein and Hawking said. That's good. But we don't get the Bekenstein-Hawking formula

S = A/4

because there is an undetermined parameter in our formula - the Barbero-Immirzi parameter. That's bad. However, our answer will match the Bekenstein-Hawking formula if we take

γ = ln 2 / π √3

If we do this, we no longer have that annoying undetermined constant floating around in loop quantum gravity. In fact, we can say that we've determined the "quantum of area" - the smallest possible unit of area. That's good. And then it's almost true that in our model, each little patch of the black hole horizon with area 4 ln(2) contains a single bit of information - since a spin-1/2 puncture has area 4 ln(2), and the angle deficit at a puncture labelled with spin 1/2 can take only 2 values, corresponding to m = -1/2 and m = 1/2. Of course, there are also punctures labelled by higher values of j, but the j = 1/2 punctures dominate the count of the microstates.

Of course, one might object to this procedure on the following grounds: "You've been ignoring matter thus far. What if you include, say, electromagnetic fields in the game? This will change the calculation, and now you'll probably need a different value of γ to match the Bekenstein-Hawking result!"

However, this is not true: we can redo the calculation including electromagnetism, and the same γ works. That's sort of nice.

There are a lot of interesting comparisons between our way of computing black hole entropy and the ways its done in string theory, and a lot of other things to say, too but for that, you'll have to read the paper... I'm worn out now!

Addenda: I thank Herman Rubin and Lieven Marchand for some corrections of errors I made while describing the Riemann hypothesis and P = NP conjecture. I also thank J. Maurice Rojas for pointing out that Steve Smale was an individual who did have the guts to pose a list of math problems for the 21st century, back in 1998. This appears in:

6) Stephen Smale, Mathematical problems for the next century, Mathematical Intelligencer, 20 (1998), 7-15. Also available in Postscript and PDF as item 104 on Smale's webpage, http://www.cityu.edu.hk/ma/staff/smale/bibliography.html

I believe this also appears in the book edited by Arnold mentioned at the beginning of "week147".

... for beginners engaging in research, a most difficult feature to grasp is that of quality - that is, the depth of a problem. Sometimes authors work courageously and at length to arrive at results which they believe to be significant and which experts believe to be shallow. This can be explained by the analogy of playing chess. A master player can dispose of a beginner with ease no matter how hard the latter tries. The reason is that, even though the beginner may have planned a good number of moves ahead, by playing often the master has met many similar and deeper problems; he has read standard works on various aspects of the game so that he can recall many deeply analyzed positions. This is the same in mathematical research. We have to play often with the masters (that is, try to improve on the results of famous mathematicians); we must learn the standard works of the game (that is, the "well-known" results). If we continue like this our progress becomes inevitable. - Hua Loo-Keng, Introduction to Number Theory