Also available at http://math.ucr.edu/home/baez/week164.html
January 13, 2001
This Week's Finds in Mathematical Physics (Week 164)
John Baez
What are the top ten questions for physics in this millennium?
The participants of the conference Strings 2000 chose these:
1. Are all the (measurable) dimensionless parameters that characterize
the physical universe calculable in principle or are some merely
determined by historical or quantum mechanical accident and uncalculable?
2. How can quantum gravity help explain the origin of the universe?
3. What is the lifetime of the proton and how do we understand it?
4. Is Nature supersymmetric, and if so, how is supersymmetry broken?
5. Why does the universe appear to have one time and three space dimensions?
6. Why does the cosmological constant have the value that it has, is it
zero and is it really constant?
7. What are the fundamental degrees of freedom of Mtheory (the theory
whose lowenergy limit is elevendimensional supergravity and
which subsumes the five consistent superstring theories) and does
the theory describe Nature?
8. What is the resolution of the black hole information paradox?
9. What physics explains the enormous disparity between the gravitational
scale and the typical mass scale of the elementary particles?
10. Can we quantitatively understand quark and gluon confinement in
Quantum Chromodynamics and the existence of a mass gap?
For details see:
1) Physics problems for the next millennium,
http://feynman.physics.lsa.umich.edu/strings2000/millennium.html
I think most of these questions are pretty good if one limits physics
to mean the search for new fundamental laws, rather than interesting
applications of the laws we know. I would leave out question 7, since
it's too concerned with a particular theory, rather than the physical
world itself. I'd instead prefer to ask: "What physics underlies the
Standard Model gauge group SU(3) x SU(2) x U(1)?"
Of course, this business of limiting "physics" to mean "the search
for fundamental laws" annoys condensed matter physicists like Philipp
Anderson, since it excludes everything they work on. He writes:
My colleagues in the fashionable fields of string theory and
quantum gravity advertise themselves as searching desperately
for the 'Theory of Everything", while their experimental colleagues
are gravid with the "God Particle", the marvelous Higgson which
is the somewhat misattributed source of all mass. (They are also
after an understanding of the earliest few microseconds of the Big
Bang.) As Bill Clinton might remark, it depends on what the meaning
of "everything" is. To these savants, "everything" means a list of
some two dozen numbers which are the parameters of the Standard Model.
This is a set of equations which already exists and does describe very
well what you and I would be willing to settle for as "everything".
This is why, following Bob Laughlin, I make the distinction between
"everything" and "every thing". Every thing that you and I have
encountered in our real lives, or are likely to interact with in
the future, is no longer outside of the realm of a physics which is
transparent to us: relativity, special and general; electromagnetism;
the quantum theory of ordinary, usually condensed, matter; and, for
a few remote phenomena, hopefully rare here on earth, our almost
equally cutanddried understanding of nuclear physics. [Two
parenthetic remarks: 1) I don't mention statistical mechanics
only because it is a powerful technique, not a body of facts;
2) our colleagues have done only a sloppy job so far of deriving
nuclear physics from the Standard Model, but no one really doubts
that they can.]
I am not arguing that the search for the meaning of those two
dozen parameters isn't exciting, interesting, and worthwhile:
yes, it's not boring to wonder why the electron is so much
lighter than the proton, or why the proton is stable at least
for another 35 powers of ten years, or whether quintessence exists.
But learning why can have no real effect on our lives, spiritually
inspiring as it would indeed be, even to a hardened old atheist
like myself.
For the rest of his remarks, see:
2) What questions have disappeared?, The World Question Center,
http://www.edge.org/documents/questions/q2001.html
Personally, I would be wary of asserting that a piece of knowledge
"can have no real effect on our lives" unless we are limiting the
discussion to shortterm effects  not the next millennium. There's
also, potentially, quite a lot more to fundamental physics than
determining the values of a couple dozen constants. This remark
of Anderson's sounds dangerously reminiscent of Michelson's comment,
shortly before the turn of the 20th century, that henceforth physics
might consist of nothing but measuring various constants to an
increasing number of decimal places. (Yes  irony of ironies, this
was Michelson of MichelsonMorley fame.) But I don't think physics
should be construed to mean only the search for "fundamental laws".
That neglects too much fun stuff! It would be nice to see the
condensed matter theorists' list of problems for the next millennium,
for example.
On to something a bit more mathematical....
Careful readers of This Week's Finds will remember Diarmuid Crowley
from "week151". This week he visited U. C. Riverside and talked
about the topology of 7 and 15dimensional manifolds. He also
told me the following cool things.
You may recall from "week163" that the Poincare homology 3sphere is
a compact 3manifold that has the same homology groups as the ordinary
3sphere, but is not homeomorphic to the 3sphere. I explained
how this marvelous space can be obtained as the quotient of SU(2) = S^3
by a 120element subgroup  the double cover of the symmetry group of
the dodecahedron. Even better, the points in S^3 which lie in this
subgroup are the centers of the faces a 4d regular polytope with 120
dodecahedral faces. That's pretty cool. But here's another cool way
to get the Poincare homology sphere:
E8 is the biggest of the exceptional Lie groups. As I explained in
"week64", the Dynkin diagram of this group looks like this:
ooooooo


o
Now, make a model of this diagram by linking together 8 rings:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Imagine this model as living in S^3. Next, hollow out all these rings:
actually delete the portion of space that lies inside them! We now have
a 3manifold M whose boundary dM consists of 8 connected components, each
a torus. Of course, a solid torus also has a torus as its boundary. So
attach solid tori to each of these 8 components of dM, but do it via this
attaching map:
(x,y) > (y,x+2y)
where x and y are the obvious coordinates on the torus, numbers between
0 and 2pi, and we do the arithmetic mod 2pi. We now have a new 3manifold
without boundary... and this is the Poincare homology sphere.
We see here a strange and indirect connection between E8 and the
dodecahedron. This is not the only such connection! There's also
the "McKay correspondence" (see "week65") and a way of getting the
E8 root lattice from the "icosians" (see "week20").
Are these three superficially different connections secretly just
different views of the same grand picture? I'm not sure. I think
I'd know the answer to part of this puzzle if I better understood
the relation between ADE theory and singularities.
But Diarmuid Crowley told me much more. The Poincare homology sphere
is actually the boundary of a 4manifold, and it's not hard to say what
this 4manifold is. I just gave you a recipe for cutting out 8 solid
tori from the 3sphere and gluing them back in with a twist. Suppose
we think of 3sphere as the boundary of the 4ball D^4, and think of
each solid torus as part of the boundary of a copy of D^2 x D^2, using
the fact that
d(D^2 x D^2) = S^1 x D^2 + D^2 x S^1.
Then the same recipe can be seen as instructions for gluing 8 copies
of D^2 x D^2 to the 4ball along part of their boundary, getting a new
4manifold with boundary. If you ponder it, you'll see that the
boundary of this 4manifold is the Poincare homology 3sphere.
Now, this is actually no big deal, at least for folks who know some
4dimensional topology. But Crowley like higherdimensional topology,
and what he told me is this: the whole story generalizes to higher
dimensions! Instead of starting with this picture of linked 1spheres
in the 3sphere:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
start with an analogous pattern of 8 nspheres linked in the (2n+1)sphere.
Do all the same stuff, boosting the dimensions appropriately... and you'll
get an interesting (2n+1)dimensional manifold dM which is the boundary of
a (2n+2)dimensional manifold M.
When n is *odd* and greater than 1, this manifold dM is actually an
"exotic sphere". In other words, it's homeomorphic but not diffeomorphic
to the usual sphere of dimension 2n+1.
Now, exotic spheres of a given dimension form an abelian group G
under connected sum (see "week141"). This group consists of two parts:
the easy part and the hard part. The easy part is a normal subgroup N
consisting of the exotic spheres that bound parallelizable smooth
manifolds. The size of this subgroup can be computed in terms of
Bernoulli numbers and stuff like that. The hard part is the quotient
group G/N. This is usually the cokernel of a famous gadget called
the "Jhomomorphism". I say "usually" because this is known to be true
in most dimensions, but in certain dimensions it remains an open question.
Anyway: the easy part N is always a finite cyclic group, and this is
*generated* by the exotic sphere dM that I just described!
For example:
In dimension 7 we have G = N = Z/28, so there are 28 exotic spheres in
this dimension (up to orientationpreserving diffeomorphism), and they
are all connected sums of the exotic 7sphere dM formed by the above
construction.
In dimension 11 we have G = N = Z/992, so there are 992 exotic spheres,
and they are all connected sums of the exotic 11sphere dM formed by the
above construction.
In dimension 15 we no longer have G = N. Instead we have N = Z/8128
and G = Z/8128 + Z/2. There are thus 16256 exotic spheres in this
dimension, only half of which are connected sums of the exotic 15sphere
dM formed by the above construction.
And so on.
While we're on the subject of exotic 15spheres, I can't resist
mentioning this. I explained in "week141" how to construct a bunch
of exotic 7spheres (24 of them, actually) using the quaternions.
Once you understand this trick, it's natural to wonder if you can
construct exotic 15spheres the same way, but using octonions instead
of quaternions. Well, you can:
3) Nobuo Shimada, Differentiable structures on the 15sphere and
Pontrjagin classes of certain manifolds, Nagoya Math. Jour. (12) 1957, 5969.
I should also explain what I really like about the above stuff.
In topological quantum field theory, people like to get 3manifolds
by "surgery on framed links". The idea is to start with a framed
link in the 3sphere, use the framing to thicken each component
to an embedded solid torus, cut out these solid tori, and reattach
them "the other way", using the fact that S^1 x S^1 is the boundary
of both S^1 x D^2 and D^2 x S^1. We can get any compact oriented
3manifold this way.
The above construction of the Poincare homology sphere was just an
example of this, where the link was
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
and each component had two twists in the framing as we go around,
as compared to the standard "blackboard" framing. This is why there
was that mysterious number "2" in my formula for the attaching map.
Whenever we describe a 3manifold using "surgery on framed links"
this way, there's an important matrix where the entry in the ith
row and jth column is the linking number of the ith component and
the jth component of our framed link, with the diagonal entries
standing for the "selflinking" numbers of the components, that is,
the number of twists their framings have. This matrix is important
because it also describes the "intersection form" on the 2nd homology
group of a simplyconnected 4manifold M whose boundary dM is the
3manifold we're describing.
For example, in the case of the Poincare homology sphere, this matrix
is called the E8 Cartan matrix:
2 1 0 0 0 0 0 0
1 2 1 0 0 0 0 0
0 1 2 1 0 0 0 0
0 0 1 2 1 0 0 0
0 0 0 1 2 1 0 1
0 0 0 0 1 2 1 0
0 0 0 0 0 1 2 0
0 0 0 0 1 0 0 2
The Dynkin diagram simply summarizes this matrix in pictorial form.
I already described the 4manifold M whose boundary is the Poincare
homology sphere; now you know its intersection form.
Anyway, what I find exciting is that all this stuff generalizes
to higher dimensions if we restrict attention to manifolds that
have trivial homotopy groups up to a certain point! For example,
it works for compact oriented smooth 7manifolds that have trivial
pi_1 and pi_2. Any such manifold can be obtained by doing surgery
on some framed 3spheres embedded in S^7. Just as 1spheres can
link in 3d space since 1+1 is one less than 3, 3spheres can link
in 7d space since 3+3 is one less than 7. We again get a matrix of
linking numbers. As before, this matrix is also an intersection
form: namely, the intersection form on the 4th homology group of an
8manifold M whose boundary dM is the 7manifold we're describing.
Moreover, this matrix is symmetric in both the 3manifold example
and the 7manifold example, since it describes an intersection
pairing on an *evendimensional* homology group.
Even better, all the same stuff happens in manifolds with enough
trivial homotopy groups in dimension 11, and dimension 15... and
all dimensions of the form 4n1. And what's *really* neat is that
these higherdimensional generalizations are in some ways simpler
than the 3d story. The reason is that a 1sphere can be knotted
in 3space in really complicated ways, but the higherdimensional
generalizations do not involve such complicated knotting. The
framing aspects can be more complicated, since there's more to
framing an embedded sphere than just an integer, but it's not all
*that* complicated.
So maybe I can learn some more 3d topology by first warming up with
the simpler 7d case....
Finally, I'd like to list a few articles that I've been meaning to
read, but haven't gotten around to. I hope to read them sometime
*this* millennium! I'll quote the abstracts and make a few comments.
4) Jack Morava, Cobordism of symplectic manifolds and asymptotic expansions,
a talk at the conference in honor of S.P. Novikov's 60th birthday,
available at math.SG/9908070.
The cobordism ring of symplectic manifolds defined by V.L. Ginzburg
is shown to be isomorphic to the Pontrjagin ring of complexoriented
manifolds with free circle actions. This suggests an interpretation
of the formal group law of complex cobordism, in terms of a
compositionlaw on semiclassical expansions. An appendix discusses
related questions about cobordism of toric varieties.
I started trying to explain the relation between formal group laws
and complex oriented cohomology theories in "week150", because I'm
quite puzzled about the deep inner meaning of this relation. This
paper might be the key to this mystery!
5) Detlev Buchholz, Current trends in axiomatic quantum field theory,
available as hepth/9811233.
In this article a nontechnical survey is given of the present
status of Axiomatic Quantum Field Theory and interesting future
directions of this approach are outlined. The topics covered are
the universal structure of the local algebras of observables, their
relation to the underlying fields and the significance of their
relative positions. Moreover, the physical interpretation of the
theory is discussed with emphasis on problems appearing in gauge
theories, such as the revision of the particle concept, the
determination of symmetries and statistics from the superselection
structure, the analysis of the short distance properties and the
specific features of relativistic thermal states. Some problems
appearing in quantum field theory on curved spacetimes are also
briefly mentioned.
I've been falling behind on new developments in axiomatic quantum field
theory. Lots of cool stuff is happening, I hear. This might help me
catch up.
6) Matt Visser, The reliability horizon, available as grqc/9710020.
The "reliability horizon" for semiclassical quantum gravity
quantifies the extent to which we should trust semiclassical
quantum gravity, and gives a handle on just where the "Planck
regime" resides. The key obstruction to pushing semiclassical
quantum gravity into the Planck regime is often the existence of
large metric fluctuations, rather than a large backreaction.
This seems like a very sensible enterprise: determining just where
semiclassical calculations are likely to break down, and quantum gravity
effects to become important. Why haven't I read this? It's obviously
worthwhile!
7) Bianca Letizia Cerchiai and Julius Wess, qDeformed Minkowski Space
based on a qLorentz Algebra, available as math.QA/9801104.
The Hilbert space representations of a noncommutative qdeformed
Minkowski space, its momenta and its Lorentz boosts are
constructed. The spectrum of the diagonalizable space elements
shows a latticelike structure with accumulation points on the
lightcone.
The qdeformed Lorentz algebra plays a role in quantum gravity with
nonzero cosmological constant, but it also shows up in noncommutative
geometry. Are the two roles related? I don't know! This is on my list
of puzzles to ponder.
The people applying the qdeformed Lorentz algebra to noncommutative
geometry want to develop the theory of qdeformed Minkowski space, see
if it makes the infinities in quantum field theory go away, and see what
physical predictions it makes. It makes spacetime discrete in a very
pretty way; that I know from Julius Wess' talk in Schladming a few years
back (see "week129"). One of the more interesting ways to apply
noncommutative geometry to physics would be to develop the theory of
qdeformed Minkowski space, see if it makes the infinities in quantum
field theory go away, and see what physical predictions it makes. It
makes spacetime discrete in a very pretty way; that I know from Julius
Wess' talk in Schladming. But I should learn more about this, and not
just because Bianca Letizia Cerchiai is a very nice person who invited
my girlfriend and I to lunch at her parents' apartment in Milan.... oh,
now I'm feeling *terribly* guilty for not reading her paper! How nasty
of me! I'd better print it out and read it as soon as I go into the
office!
In fact, now that I think of it, I've had at least *some* dealings with
*all* the authors of these papers. And now I'm publicly admitting I
haven't read some of their most interesting papers! Ugh! At least
this admission may shame me into reading them now...
Bye.

On sci.physics.research, Aaron Bergman clarified something about
these millennial physics problems:
>John Baez wrote:
>> Aaron Bergman (abergman@Princeton.EDU) wrote:
>>>John Baez wrote:
>>>> Of course, this business of limiting "physics" to mean "the
>>>> search for fundamental laws" annoys condensed matter
>>>> physicists like Philipp Anderson, since it excludes everything
>>>> they work on.
>
>>> One should note that Gross explicitly says  there's a Realaudio
>>> of the talk online  that this is a very narrowminded list that
>>> excludes fundamental questions in other fields. It's not really
>> intended to be a universal list.
>
>> Good! It's too bad the text of the webpage doesn't make that
>> clearer. I'm appending your comment to the version of "week164"
>> on my website, assuming you don't object.
>
>Sure. Or you can just refer them to the transparencies and the
>talk. For those who don't want to bother listening to the whole
>thing, start listening at about 7:30 mins into the RealAudio
>stream:
>
> http://feynman.physics.lsa.umich.edu/cgibin/s2ktalk.cgi?questions
>
>It's on transparency 4 which is why I mentioned,
>
>>> And Witten is coming back.
>
>> You mean he's not staying in LA? Can't take the winters out here?
>
>I won't speculate on the reasons, but his grad students have said
>that he's coming back to the Institute.
>
>Aaron
>
>Aaron Bergman
>

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 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