Also available at http://math.ucr.edu/home/baez/week148.html
June 5, 2000
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:
A) 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. This is the most practical question
of the lot, because P *were* equal to NP, there's a chance that one could
use this result to quickly crack all the current best encryption
schemes.
B) The Poincare 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? Poincare 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 Poincare'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 Poincare conjecture remains uncracked.
C) 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:
y^2 = x^3 + 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.
D) 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 H^{p,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
H^{p,p} are called "Hodge forms". By Poincare 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.
E) 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).
F) 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.
G) The Riemann hypothesis. For Re(s) > 1 the Riemann zeta function
is defined by
zeta(s) = 1/1^s + 1/2^s + 1/3^s + ....
But we can extend it by analytic continuation to most of the complex
plane - it has a pole at 1. The zeta function equals zero when s
is a negative even integer, but it also 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
gq-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.
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 S^2. 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:
omega(delta W, delta W') = (k/2pi) integral delta W ^ delta W'
Here omega 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 delta W and delta 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 couple of papers 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.
Preprint 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 pi gamma sqrt(j(j+1))
for some constant gamma.
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 pi 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, gamma 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 pi gamma 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 hbar 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 pi gamma
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 epsilon 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 epsilon, 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 epsilon 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 pi gamma sqrt(j(j+1)) = 4 pi gamma sqrt(3)
to the horizon 2-sphere. Since the total area is close to A this
means that there are about A/(4 pi gamma sqrt(3)) punctures. Then
for each puncture we can pick m = -1/2 or m = 1/2. This gives
N = 2^{A/4 pi gamma sqrt(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 pi gamma sqrt(3)) A
Believe it or not, this crude estimate asymptotically approaches the
correct answer as A approaches infinity.
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
gamma = ln 2 / pi sqrt(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 gamma to match the
Bekenstein-Hawking result!"
However, this is not true: we can redo the calculation including
electromagnetism, and the same gamma 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!
(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
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html