Also available at http://math.ucr.edu/home/baez/week232.html
May 18, 2006
This Week's Finds in Mathematical Physics (Week 232)
John Baez
I'm at the Perimeter Institute now. It's great to see how it's
developed since I first saw their new building back in 2004 (see
"week208" for the story).
There's now a busy schedule of seminars and weekly colloquia, with
string theorists and loop quantum gravity people coexisting happily.
Their program of Superstring Quartets features some really hot bands,
like the Julliard and Emerson  unfortunately not playing while I'm
here. The Black Hole Bistro serves elegant lunches and dinners, there
are at least two espresso machines on each floor, and my friend
Eugenia Cheng will be happy to hear that they still have a piano
available (after 6 pm).
But don't get the impression that it's overly sophisticated:
there are also a couple of guys constantly playing foosball
in the Feynman Lounge.
Since I'm here, I should talk about quantum gravity  so I will.
But first, let's have the astronomy picture of the week.
This week it comes, not from outer space, but beneath the surface of
the South Pole:
1) Steve Yunck / NSF, Cerenkov light passing through the IceCube neutrino
detector, http://icecube.wisc.edu/gallery/detector_concepts/ceren_hires
This is an artist's impression of a huge neutrino observatory called
"IceCube". (Maybe they left out the space so the rap star by that name
doesn't sue them for trademark infringement, or go down there and shoot
them.)
IceCube is being built in the beautifully clear 18,000year old ice deep
beneath the AmundsenScott South Pole Station. When a highenergy
neutrino hits a water molecule, sometimes the collision produces a
muon zipping faster than the speed of light in ice. This in turn
produces something like a sonic boom, but with light instead of sound.
It's called "Cerenkov radiation", and it's the blue light in the picture.
This will be detected by an array of 5000 photomultiplier tubes  those
gadgets hanging on electrical cables.
One thing the artist's impression doesn't show is that IceCube is
amazingly large. The whole array is a cubic kilometer in size!
It will encompass the already existing AMANDA detector, itself
10,000 meters tall, shown as a yellow cylinder here with a neutrino
zipping through:
2) Darwin Rianto / NSF, Comparison of AMANDA and IceCube,
http://icecube.wisc.edu/gallery/detector_concepts/icecubeencomp_300
Even the very top of IceCube is 1.4 kilometers beneath the snowy
Antarctic surface, to minimize the effect of stray cosmic rays.
The station on top looks like this  not very cozy, I'd say:
3) Robert G. Stokstad / NSF, South Pole Station,
http://icecube.wisc.edu/gallery/antarctica/PC140287_300
I heard about IceCube from Adrian Burd, one of the oldtimers who
used to post a lot on sci.physics, a former cosmologist turned a
cosmologist turned oceanographer who recently visited Antarctica as
part of an NSFrun field course. He ran into some people working on
IceCube. It sounds like an interesting community down there! You can
read about it in their newspaper, the Antarctic Sun. For example:
4) Ice Cube turns up the heat, The Antarctic Sun, January 29, 2006,
http://antarcticsun.usap.gov/20052006/contentHandler.cfm?id=959
For more on IceCube and Amanda, these are fun to read:
5) Francis Halzen, Ice fishing for neutrinos,
http://icecube.berkeley.edu/amanda/icefishing.html
6) Katie Yurkiewicz, Extreme neutrinos, Symmetry, volume 1 issue 1,
November 2004, http://symmetrymagazine.org/cms/?pid=1000014
For some of AMANDA's results, including a map of the sky as
seen in neutrinos, try this:
7) M. Ackermann et al, Search for extraterrestrial point sources of
high energy neutrinos with AMANDAII using data collected in 20002002,
available as astroph/0412347.
For much more, try these:
8) AMANDA II Project, http://amanda.uci.edu/
9) Welcome to IceCube, http://icecube.wisc.edu/
And now, on to gravity.
You may have heard of the gravitational 3body problem. Well,
Richard Montgomery (famous from "week181") recently pointed out
this movie of the 60body problem:
10) Davide L. Ferrario, Periodic orbits for the 60body problem,
http://www.matapp.unimib.it/~ferrario/mov/index.html
60 equal masses do a complicated dance while always preserving
icosahedral symmetry! First 12 groups of 5 swing past each other,
then 20 groups of 3. If you want to know how he found these solutions,
read this:
11) Davide L. Ferrario and S. Terracini, On the existence of collisionless
equivariant minimizers for the classical nbody problem. Invent. Math.
155 (2004), 305362.
It's quite mathintensive  though just what you'd expect if you
know this sort of thing: they use the Gequivariant topology of loop
spaces, where G is the symmetry group in question (here the icosahedral
group), to prove the existence of actionminimizing loops with given
symmetry properties.
Next, I'd like to say a little about point particles in 3d quantum
gravity, and some recent work with Alissa Crans, Derek Wise and Alejandro
Perez on stringlike defects in 4d topological gravity:
12) John Baez, Derek Wise and Alissa Crans, Exotic statistics for strings
in 4d BF theory, available as grqc/0603085.
13) John Baez and Alejandro Perez, Quantization of strings and branes
coupled to BF theory, available as grqc/0605087.
(Jeffrey Morton is also involved in this project, a bit more on the
ncategory side of things, but that aspect is top secret for now.)
In "week222" I listed a bunch of cool papers on 3d quantum gravity,
but I didn't really explain them. What we're trying to do now is
generalize this work to higher dimensions. But first, let me start
by explaining the wonders of 3d quantum gravity.
The main wonder is that we actually understand it! The classical
version of general relativity is exactly solvable when spacetime has
dimension 3, and so is the quantum version. Most of the wonders I
want to discuss are already visible in the classical theory, where
they are easier to understand, so I'll focus on the classical case.
A nice formulation of general relativity in 3 dimensions uses a
"Lorentz connection" A and a "triad field" e. This is a gauge theory
where the gauge group is SO(2,1), the Lorentz group for 3d spacetime.
If we're feeling lowbrow we can think of both A and e as so(2,1)valued
1forms on the 3manifold M that describes spacetime. The action for
this theory is:
S = integral_M tr(e ^ F)
where F is the curvature of A. If you work out the equations of
motion one of them says that F = 0, so our connection A is flat.
The other, d_A e = 0, says A is basically just the LeviCivita
connection.
This is exactly what we want, because in the absence of matter,
general relativity in 3 dimensions says spacetime is *flat*.
A fellow named Phillipp de Sousa Gerbert came up with an interesting
way to couple point particles to this formulation of quantum gravity:
14) Phillipp de Sousa Gerbert, On spin and (quantum) gravity in 2+1
dimensions, Nuclear Physics B346 (1990), 440472.
He actually did it for particles with spin, but I'll just do the
spinzero case.
The idea is to fix a 1dimensional submanifold W in our 3manifold M
and think of it as the worldlines of some particles. Put so(2,1)valued
functions p and q on these worldlines  think of these as giving the
particles' momentum and position as a function of time.
Huh? Well, normally we think of position and momentum as vectors.
In special relativity, "position" means "position in spacetime",
and "momentum" means "energymomentum". We can think of both of
these as vectors in Minkowksi spacetime. But in 3 dimensions,
Minkowski spacetime is naturally identified with the Lorentz Lie
algebra so(2,1). So, it makes sense to think of q and p as elements
of so(2,1) which vary from point to point along the particle's
worldline.
To couple our point particles to gravity, we then add a term to
the action like this:
S = integral_M tr(e ^ F)  integral_W tr((e + d_A q) ^ p)
Now if you vary the e field you get a field equation saying that
F = p delta_W
Here delta_W is like the Dirac delta function of the worldline
W; it's a distributional 2form defined by requiring that
integral_W X = integral_M (X ^ delta_W)
for any smooth 1form X on W. This sort of "distributional differential
form" is also called a "current", and you can read about them in the
classic tome by ChoquetBruhat et al. But the main point is that
the field equation
F = p delta_W
says our connection on spacetime is flat except along the worldlines of
our particles, where the curvature is a kind of "delta function". This
is nice, because that's what we expect in 3d gravity: if you have a
particle, spacetime will be flat everywhere except right at the particle,
where it will have a singularity like the tip of a cone.
A cone, you see, is intrinsically flat except at its tip: that's why
you can curl paper into a cone without crinkling it!
So, our spacetime is flat except along the particles' worldlines, and
there it's like a cone. The "deficit angle" of this cone  the angle
of the slice you'd need to cut out to curl some paper into this cone 
is specified by the particle's momentum p.
Since delta functions are a bit scary, it's actually better to work with
an "integrated" form of the equation
F = p delta_W
The integrated form says that if we parallel transport a little tangent
vector around a little loop circling our particle's worldline, it gets
rotated and/or Lorentz transformed by the element
exp(p)
in SO(2,1). This will be a rotation if the particle's momentum p is
timelike, as it is for normal particles. Again, that's just as it
should be: if you parallel transport a little arrow around a massive
particle in 3d gravity, it gets rotated!
If p is timelike, our particle is a tachyon and exp(p) is a Lorentz
boost. And so on... we get the usual classification of particles
corresponding to various choices of p.
There are other equations of motion, obtained by varying other fields,
but all I want to note is the one you get by varying q:
d_A p = 0
This says that the momentum p is covariantly constant along the
particles' worldlines. So, momentum is conserved!
The really cool part is the relation between the Lie algebra element p
and the group element exp(p). Originally we thought of p as momentum 
but there's a sense in which exp(p) is the momentum that really counts!
First, exp(p) is what we actually detect by parallel transporting a
little arrow around our particle.
Second, suppose we let two particles collide and form a new one:
p p'
\ /
\ /
\ /



p"
Now our worldlines don't form a submanifold anymore, but if we keep
our wits about us, we can see that everything still makes sense, and
we get momentum conservation in this form:
exp(p") = exp(p) exp(p')
since little loops going around the two incoming particles can fuse to
form a loop going around the outgoing particle. Note that we're getting
conservation of the *groupvalued* momentum, not the Liealgebravalued
momentum  we don't have
p" = p + p'
So, conservation of energymomentum is getting modified by gravitational
effects! This goes by the name of "doubly special relativity":
15) Laurent Freidel, Jerzy KowalskiGlikman and Lee Smolin,
2+1 gravity and doubly special relativity, Phys. Rev. D69 (2004)
044001. Also available as hepth/0307085.
This effect is a bit less shocking if we put the units back in. I've
secretly been setting 4 pi G = 1, where G is Newton's gravitational
constant. If we put that constant back in  let's call it k instead
of 4 pi G  we get
exp(kp") = exp(kp) exp(kp')
or if you expand things out:
p" = p + p' + (k/2) [p,p'] + terms of order k^2 and higher
So, as long as the momenta are small compared to the Planck mass,
the usual law of conservation of momentum
p" = p + p'
*almost* holds! But, for large momenta this law breaks down  we
must think of momentum as groupvalued if we want it to be conserved!
I think this is incredibly cool: as we turn on gravity, the usual
"flat" momentum space curls up into a group, and we need to *multiply*
momenta in this group, instead of *add* them in the Lie algebra.
We can think of this group has having a "radius" of 1/k, so it's
really big and almost flat when the strength of gravity is small.
In this limit, multiplication in the group reduces to addition in
the Lie algebra.
I should point out that this effect is purely classical! It's still
there when we quantize the theory, but it only depends on the gravitational
constant, not Planck's constant. Indeed, in 3d quantum gravity, we
can build a unit of mass using just G and c: we don't need hbar. This
unit is the mass that curls space into an infinitely skinny cone! It
would be a bit misleading to call it "Planck mass", but it's the maximum
possible mass. Any mass bigger than this acts like a *negative* mass.
That's because the corresponding groupvalued momenta "wrap around"
in the group SO(2,1).
We also get another cool effect  exotic statistics. In the absence
of gravitational or quantum effects, when you switch two particles,
you just switch their momenta:
(p, p') > (p', p)
But in 3d gravity, you can think of this process of switching
particles as a braid:
\ /
\ /
\
/ \
/ \
and if you work out what happens to their groupvalued momenta,
say
g = exp(kp)
g' = exp(kp')
it turns out that one momentum gets conjugated by the other
as we switch them:
(g, g') > (gg'g^{1}, g)
So, the process of braiding two particles around each other has a
nontrivial effect on their momenta. In particular, if you braid two
In particular, if you braid two particles around other twice they
don't wind up in their original state!
Thus, our particles are neither bosons nor fermions, but "nonabelian
anyons"  the process of switching them is governed not by the
permutation group, but by the braid group. But again, if you expand
things out in powers of k you'll see this effect is only noticeable
for large momenta:
(p, p') > (p' + k[p,p'] + higher order terms..., p)
Summarizing, we see quantum gravity is lots of fun in 3 dimensions:
it's easy to introduce point particles, and they have groupvalued
momentum, which gives rise to doubly special relativity and braid
group statistics.
Now, what happens when we go from 3 dimensions to 4 dimensions?
Well, we can write down the same sort of theory:
S = integral_M tr(B ^ F)  integral_W tr((B + d_A q) ^ p)
The only visible difference is that what I'd been calling "e" is now
called "B", so you can see why folks call this "BF theory".
But more importantly, now M is an 4dimensional spacetime and W is an
2dimensional "worldsheet". A is again a Lorentz connection, which
we can think of as an so(3,1)valued 1form. B is an so(3,1)valued
2form. p is an so(3,1)valued function on the worldsheet W. q is an
so(3,1)valued 1form on W.
So, only a few numbers have changed... so everything works very
similarly! The big difference is that instead of spacetime having a
conical singularity along the worldline of a *particle*, now it's
singular along the worldsheet of a *string*.
When I call it a "string", I'm not trying to say it behaves like the
ones they think about in string theory  at least superficially, it's
a different sort of theory, a purely topological theory. But, we've
got these closed loops that move around, split and join, and trace out
surfaces in spacetime...
They can also braid around each other in topologically nontrivial
ways. So, we get exotic statistics as before, but now they are
governed not by the braid group but by the "loop braid group", which
keeps track of all the ways we can move a bunch of circles around in
3d space. Let's take our spacetime M to be R^4, to keep things
simple. So, our circles can move around in R^3... and there are two
basic ways we can switch two of them: move them around each other, or
pass one *through* the other.
If we just move them around each other, they might as well have been
point particles: we get a copy of the permutation group, and all we see
are ordinary statistics. But when we consider all the ways of passing
them through each other, we get a copy of the braid group!
When we allow ourselves both motions, we get a group called the "loop
braid group" or "braid permutation group"  and one thing Alissa Derek
and I did was to get a presentation of this group. This is an example
of a "motion group": just as the motion group of point particles in the
plane is the braid group, and motion group of point particles in R^3
is the permutation group, the motion group of strings in R^3 is the
loop braid group.
As before, our strings have groupvalued momenta: we can get an
element of the Lorentz group SO(3,1) by parallel transporting a
little tangent vector around a string. And, we can see how
different ways of switching our strings affect the momenta.
When we move two strings around each other, their momenta switch
in the usual way:
(g, g') > (g', g)
but when we move one through the other, one momentum gets conjugated
by the other:
(g, g') > (gg'g^{1}, g)
So, we have exotic statistics, but you can only notice them if you
can pass one string through another!
In the paper with Alejandro, we go further and begin the project of
quantizing these funny strings, using ideas from loop quantum gravity.
Loop quantum gravity has its share of problems, but it works perfectly
well for 3d quantum gravity, and matches the spin foam picture of this
theory. People have sort of believed this for a long time, but
Alejandro demonstrated this quite carefully in a recent paper with
Karim Noui:
15) Karim Noui and Alejandro Perez, Dynamics of loop quantum gravity
and spin foam models in three dimensions to appear in the proceedings
of the Third International Symposium on Quantum Theory and Symmetries
(QTS3), available as grqc/0402112.
The reason everything works so nicely is that the equations of motion
say the connection is flat. Since the same is true in BF theory in
higher dimensions, we expect that the loop quantization and spin foam
quantization of the theory I'm talking about now should also work well.
We find that we get a Hilbert space with a basis of "string spin networks",
meaning spin networks that can have loose ends on the stringy defects.
So, there's some weird blend of loop quantum gravity and strings going
on here  but I don't really understand the relation to ordinary string
theory, if any. It's possible that I can get a topological string theory
(some sort of welldefined mathematical gadget) which describes these
stringy defects, and that would be quite interesting.
But, I spoke about this today at the Perimeter Institute, and Malcolm
Perry said that instead of "strings" I should call these guys
(n2)branes, because the connection has conical singularities on
them, "which is what one would expect for any respectable (n2)brane".
I will talk to him more about this and try to pick his, umm, branes.
In fact I took my very first GR course from him, back when he was a
postdoc at Princeton and I was a measly undergraduate. I was too
scared to ask him many questions then. I'm a bit less scared now,
but I've still got a lot to learn. Tomorrow he's giving a talk about
this:
17) David S. Berman, Malcolm J. Perry, Mtheory and the string genus
expansion, Phys. Lett. B635 (2006) 131135. Also available as
hepth/0601141.

Quote of the Week:
I was sitting in a chair in the patent office in Bern when all of a
sudden a thought occurred to me. If a person falls freely, he will
not feel his own weight.  Albert Einstein

Addenda: Here's an email from Greg Egan, and my reply:
John Baez wrote:
> The really cool part is the relation between the Lie algebra
> element p and the group element exp(p). Originally we thought
> of p as momentum  but there's a sense in which exp(p) is the
> momentum that really counts!
Would it be correct to assume that the ordinary tangent vector p still
transforms in the usual way? In other words, suppose I'm living in a
2+1 dimensional universe, and there's a point particle with rest mass m
and hence energymomentum vector in its rest frame of p=m e_0. If I
cross its world line with a certain relative velocity, there's an
element g of SO(2,1) which tells me how to map the particle's tangent
space to my own. Would I measure the particle's energymomentum to be
p'=gp? (e.g. if I used the particle to do work in my own rest frame)
Would there still be no upper bound on the total energy, i.e. by making
our relative velocity close enough to c, I could measure the particle's
kinetic energy to be as high as I wished?
I guess I'm trying to clarify whether the usual Lorentz transformation
of the tangent space has somehow been completely invalidated for
extreme boosts, or whether it's just a matter of there being a second
definition of "momentum" (defined in terms of the Hamiltonian) which
transforms differently and is the appropriate thing to consider in
gravitational contexts.
In other words, does the cutoff mass apply only to the deficit angle,
and do boosts still allow me to measure (by nongravitational means)
arbitrarily large energies (at least in the classical theory)?
I replied:
Greg Egan wrote:
>John Baez wrote:
>>The really cool part is the relation between the Lie algebra
>>element p and the group element exp(p). Originally we thought
>>of p as momentum  but there's a sense in which exp(p) is the
>>momentum that really counts!
>Would it be correct to assume that the ordinary tangent vector p
>still transforms in the usual way?
Hi! Yes, it would.
>In other words, suppose I'm living in a 2+1 dimensional universe,
>and there's a point particle with rest mass m and hence
>energymomentum vector in its rest frame of p=m e_0. If I
>cross its world line with a certain relative velocity, there's
>an element g of SO(2,1) which tells me how to map the particle's
>tangent space to my own. Would I measure the particle's
>energymomentum to be p'=gp? (e.g. if I used the particle to
>do work in my own rest frame) Would there still be no upper
>bound on the total energy, i.e. by making our relative velocity
>close enough to c, I could measure the particle's kinetic energy
>to be as high as I wished?
To understand this, it's good to think of the momenta as
elements of the Lie algebra so(2,1)  it's crucial to the
game.
Then, if you have momentum p, and I zip past you, so you
appear transformed by some element g of the Lorentz group
SO(2,1), I'll see your momentum as
p' = g p g^{1}
This is just another way of writing the usual formula for
Lorentz transforms in 3d Minkowski space. No new physics
so far, just a clever mathematical formalism.
But when we turn on gravity, letting Newton's constant k
be nonzero, we should instead think of momentum as groupvalued,
via
h = exp(kp)
and similarly
h' = exp(kp')
Different choices of p now map to the same choice of h.
In particular, a particle of a certain large mass  the
Planck mass will turn out to act just like a particle
of zero mass!
So, if we agree to work with h instead of p, we are now
doing new physics. This is even more obvious when we decide
to multiply momenta instead of adding them, since multiplication
in SO(2,1) is noncommutative!
But, if we transform our groupvalued momentum in the correct
way:
h' = ghg^{1}
this will be completely compatible with our previous transformation
law for vectorvalued momentum!
>I guess I'm trying to clarify whether the usual Lorentz transformation
>of the tangent space has somehow been completely invalidated for
>extreme boosts, or whether it's just a matter of there being a second
>definition of "momentum" (defined in terms of the Hamiltonian) which
>transforms differently and is the appropriate thing to consider in
>gravitational contexts.
Good question! Amazingly, the usual Lorentz transformations still
work EXACTLY  even though the rule for adding momentum is new (now
it's multiplication in the group). We're just taking exp(kp) instead
of p as the "physical" aspect of momentum.
This effectively puts an upper limit on mass, since as
we keep increasing the mass of a particle, eventually it "loops
around" SO(2,1) and act exactly like a particle of zero mass.
But, it doesn't exactly put an upper bound on energymomentum,
since SO(2,1) is noncompact. Of course energy and momentum don't
take real values anymore, so one must be a bit careful with this
"upper bound" talk.
>In other words, does the cutoff mass apply only to the deficit
>angle, and do boosts still allow me to measure (by nongravitational
>means) arbitrarily large energies (at least in the classical theory)?
There's some sense in which energymomenta can be arbitrarily
large. That's because the space of energymomenta, namely SO(2,1),
is noncompact. Maybe you can figure out some more intuitive way
to express this.

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
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http://math.ucr.edu/home/baez/
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http://math.ucr.edu/home/baez/twfcontents.html
A simple jumpingoff 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