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 M-theory (the theory whose low-energy limit is eleven-dimensional 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 cut-and-dried 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 short-term effects - not the next millennium. 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 15-dimensional manifolds. He also told me the following cool things.

You may recall from "week163" that the Poincare homology 3-sphere is a compact 3-manifold that has the same homology groups as the ordinary 3-sphere, but is not homeomorphic to the 3-sphere. I explained how this marvelous space can be obtained as the quotient of SU(2) = S3 by a 120-element subgroup - the double cover of the symmetry group of the dodecahedron. Even better, the points in S3 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:

Now, make a model of this diagram by linking together 8 rings:
   /\   /\   /\   /\   /\   /\   /\ 
  /  \ /  \ /  \ /  \ /  \ /  \ /  \
 /    \    \    \    \    \    \    \
/    / \  / \  / \  / \  / \  / \    \ 
\    \ /  \ /  \ /  \ /  \ /  \ /    / 
 \    \    \    \    \ /\ \    \    /
  \  / \  / \  / \  / \  \ \  / \  /
   \/   \/   \/   \/ / \/ \ \/   \/
                    /      \ 
                    \      /
                     \    /
                      \  /
Imagine this model as living in S3. Next, hollow out all these rings: actually delete the portion of space that lies inside them! We now have a 3-manifold 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 2π, and we do the arithmetic mod 2π. We now have a new 3-manifold 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 4-manifold, and it's not hard to say what this 4-manifold is. I just gave you a recipe for cutting out 8 solid tori from the 3-sphere and gluing them back in with a twist. Suppose we think of 3-sphere as the boundary of the 4-disk D4, and think of each solid torus as part of the boundary of a copy of D2 x D2, using the fact that

d(D2 x D2) = S1 x D2 + D2 x S1.

Then the same recipe can be seen as instructions for gluing 8 copies of D2 x D2 to the 4-ball along part of their boundary, getting a new 4-manifold with boundary. If you ponder it, you'll see that the boundary of this 4-manifold is the Poincare homology 3-sphere.

Now, this is actually no big deal, at least for folks who know some 3-dimensional topology. But Crowley likes higher-dimensional topology, and what he told me is this: the whole story generalizes to higher dimensions! Instead of starting with this picture of linked 1-spheres in the 3-sphere:

   /\   /\   /\   /\   /\   /\   /\ 
  /  \ /  \ /  \ /  \ /  \ /  \ /  \
 /    \    \    \    \    \    \    \
/    / \  / \  / \  / \  / \  / \    \ 
\    \ /  \ /  \ /  \ /  \ /  \ /    / 
 \    \    \    \    \ /\ \    \    /
  \  / \  / \  / \  / \  \ \  / \  /
   \/   \/   \/   \/ / \/ \ \/   \/
                    /      \ 
                    \      /
                     \    /
                      \  /
start with an analogous pattern of 8 n-spheres 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 "J-homomorphism". 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 orientation-preserving diffeomorphism), and they are all connected sums of the exotic 7-sphere 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 11-sphere 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 15-sphere dM formed by the above construction.

And so on.

While we're on the subject of exotic 15-spheres, I can't resist mentioning this. I explained in "week141" how to construct a bunch of exotic 7-spheres (24 of them, actually) using the quaternions. Once you understand this trick, it's natural to wonder if you can construct exotic 15-spheres the same way, but using octonions instead of quaternions. Well, you can:

3) Nobuo Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. Jour. (12) 1957, 59-69.

I should also explain what I really like about the above stuff. In topological quantum field theory, people like to get 3-manifolds by "surgery on framed links". The idea is to start with a framed link in the 3-sphere, 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 S1 x S1 is the boundary of both S1 x D2 and D2 x S1. We can get any compact oriented 3-manifold 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 3-manifold 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 "self-linking" 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 simply-connected 4-manifold M whose boundary dM is the 3-manifold 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 4-manifold 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 7-manifolds that have trivial π1 and π2. Any such manifold can be obtained by doing surgery on some framed 3-spheres embedded in S7. Just as 1-spheres can link in 3d space since 1+1 is one less than 3, 3-spheres 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 8-manifold M whose boundary dM is the 7-manifold we're describing. Moreover, this matrix is symmetric in both the 3-manifold example and the 7-manifold example, since it describes an intersection pairing on an even-dimensional 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 4n-1. And what's really neat is that these higher-dimensional generalizations are in some ways simpler than the 3d story. The reason is that a 1-sphere can be knotted in 3-space in really complicated ways, but the higher-dimensional 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 as math.SG/9908070.

The cobordism ring of symplectic manifolds defined by V.L. Ginzburg is shown to be isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This suggests an interpretation of the formal group law of complex cobordism, in terms of a composition-law 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 hep-th/9811233.

In this article a non-technical 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 gr-qc/9710020.

The "reliability horizon" for semi-classical quantum gravity quantifies the extent to which we should trust semi-classical quantum gravity, and gives a handle on just where the "Planck regime" resides. The key obstruction to pushing semi-classical quantum gravity into the Planck regime is often the existence of large metric fluctuations, rather than a large back-reaction.
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, q-Deformed Minkowski Space based on a q-Lorentz Algebra, available as math.QA/9801104.

The Hilbert space representations of a non-commutative q-deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone.
The q-deformed 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 q-deformed Lorentz algebra to noncommutative geometry want to develop the theory of q-deformed 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"). 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...


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 
>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 Bergman

© 2001 John Baez