It's been about two months since the last issue of This Week's Finds, and I apologize for this. I've been very busy, and my limited writing energy has all gone into finishing up a review article on the octonions. I'm dying to talk about that... but first things first!
When I left off I was at Penn State, learning about the latest developments in quantum gravity. I told you how Martin Bojowald was using loop quantum gravity to study what came before the big bang... but I didn't mention that he'd written a nice little book on the subject:
1) Martin Bojowald, Quantum Geometry and Symmetry, Shaker Verlag, Aachen, 2000. Available at http://www.shaker.de/Online-Gesamtkatalog/Details.asp?ISBN=3-8265-7741-8
This does not cover his most recent work, in which his program is really starting to pay off... but it will certainly help you understand his recent work. He's doing lots of great stuff these days. In fact, he just came out with a paper yesterday:
2) Martin Bojowald, The semiclassical limit of loop quantum cosmology, available at gr-qc/0105113.
This explains how his new approach to quantum cosmology is related to the old "minisuperspace" approach. In the old approach, you just take some limited class of cosmologies satisfying the equations of general relativity and think of this class as a classical mechanics problem with finitely many degrees of freedom: for example, the size of the universe together with various numbers describing its shape. Then you quantize this classical system.
In this approach, you don't see any hint of spacetime discreteness on the Planck scale. But in Bojowald's approach, you do! What gives? He still starts with a limited class of cosmologies and quantizes that, but he does so using ideas taken from loop quantum gravity. This makes all the difference: now areas and volumes have discrete spectra of eigenvalues, and this saves us from the horrors of the singularity at the big bang. In fact, we can go back before the big bang, and find a time-reversed expanding universe on the other side!
But what's the relation between this new approach and the old one, exactly? Well, in loop quantum gravity, space is described using "spin networks", and area is quantized. Each edge of a spin network is labelled by some spin j = 0, 1/2, 1, ..., and when a spin-j edge punctures a surface, it gives that surface an area equal to
8 π γ sqrt(j(j+1))times the Planck length squared. Here γ is a constant called the "Immirzi parameter" - see "week112" and "week148" for more about that. Bojowald shows that you can recover the old approach to quantum cosmology from his new one by taking a limit in which the Immirzi parameter approaches zero while the spins labelling spin network edges go to infinity. In this limit, the spacings between the above areas go to zero - so the discrete spectrum of the "area operator" becomes continuous! Thus we lose the discrete geometry which is typical of loop quantum gravity.
I'm also excited by what's going on with spin foams lately. For one, my friend Dan Christensen is starting to do numerical calculations with the Riemannian Barrett-Crane model. I've discussed this model in "week113", "week120", and "week128", so I won't bore you with the details yet again. For now, let me just say that it's a theory of quantum gravity in which spacetime is a triangulated 4-dimensional manifold. There is also a Lorentzian version of this model, which is more physical, but it's trickier to compute with, so Dan has wisely decided to start by tackling the Riemannian version.
As you probably know, in quantum field theory, as in statistical mechanics, the partition function is king. So Dan Christensen is starting out by using a supercomputer to numerically calculate the partition function of a triangulated 4-sphere. He has some students helping him, and he's also gotten some help from Greg Egan....
Anyway: this partition function is a sum over all ways of labelling triangles by spins - but it's not obvious that the sum converges! For this reason Dan has begun by imposing a "cutoff", that is, an upper bound on the allowed spins. Physically this would be called an "infrared cutoff", since big spins mean big triangles. The question is: what happens as you let this cutoff approach infinity? Does the partition function converge or not?
Now, what's cool is that in November of last year, a fellow named Alejandro Perez claimed to have proven that it does converge:
3) Alejandro Perez, Finiteness of a spin foam model for euclidean quantum general relativity, Nucl. Phys. B599 (2001) 427-434. Also available at gr-qc/0011058.
I say "claimed", not because I doubt his proof, but because I still haven't checked it, and I should. But the great thing is: now we have both numerical and analytic ways of studying this spin foam model, and we can play them off against each other! This helps a lot when you're trying to understand a complicated problem.
Of course, the skeptics among you will say "Fine, but this is just Riemannian quantum gravity, not the Lorentzian theory. We're still not talking about the real world." And you'd be right! But luckily, there has also been a lot of progress on the Lorentzian Barrett-Crane model.
This version of the Barrett-Crane model is based on the Lorentz group instead of the rotation group. Because the representations of the Lorentz group are parametrized in a continuous rather than discrete way, in this version one computes the partition function as as an integral over ways of labelling the triangles by nonnegative real numbers. These numbers represent areas, so it seems that area is not quantized in this theory - but I should warn you, this is a hotly debated issue! We need to better understand how this model relates to loop quantum gravity, where area is quantized.
Anyway, when Barrett and Crane proposed the Lorentzian version of their model, it wasn't obvious that this integral for the partition function converged. Even worse, it wasn't clear that the integrand was well-defined! The basic ingredient in the integrand is the so-called "Lorentzian 10j symbol", which describes the amplitude for an individual 4-simplex to have a certain geometry, as specified by the areas of its 10 triangular faces. Barrett and Crane wrote down an explicit integral for the Lorentzian 10j symbol, but they didn't show this integral converges.
Last summer, in a fun-filled week of intense calculation, John Barrett and I showed that the integral defining the Lorentzian 10j symbols does in fact converge:
4) John Baez and John W. Barrett, Integrability for relativistic spin networks, available at gr-qc/0101107.
It took us until this January to write up those calculations. By April, Louis Crane, Carlo Rovelli, and Alejandro Perez had written a paper extending our methods to show that the partition function converges:
5) Louis Crane, Alejandro Perez, Carlo Rovelli, A finiteness proof for the Lorentzian state sum spin foam model for quantum general relativity, available as gr-qc/0104057.
So now we have a well-defined quantum gravity theory for a 4-dimensional spacetime with a fixed triangulation, and we can start studying it! The big question is whether it mimics general relativity at distance scales much larger than the Planck scale.
But enough of that. Now: octonions!
I've finally finished writing a survey of the octonions and their connections to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, the exceptional Lie groups, quantum logic, special relativity and supersymmetry:
6) John Baez, The octonions,
Also available at math.RA/0105155.
Let me just sketch some of the main themes. For details and precise statements, read the paper!
Octonions arise naturally from the interaction between vectors and spinors in 8-dimensional Euclidean space, but in superstring theory and other physics applications, what matters most is their relation to 10-dimensional Lorentzian spacetime. This is part of a pattern:
1) spinors in 1d Euclidean space are real numbers (R). 2) spinors in 2d Euclidean space are complex numbers (C). 3) spinors in 4d Euclidean space are quaternions (H). 4) spinors in 8d Euclidean space are octonions (O).(These numbers are just the dimensions of R, C, H and O.)
1) points in 3d Minkowski spacetime are 2x2 hermitian real matrices 2) points in 4d Minkowski spacetime are 2x2 hermitian complex matrices 3) points in 6d Minkowski spacetime are 2x2 hermitian quaternionic matrices 4) points in 10d Minkowski spacetime are 2x2 hermitian octonionic matrices(These numbers are 2 more than the dimensions of R, C, H and O.)
The octonions are also what lie behind the 5 exceptional simple Lie groups. The exceptional group G2 is just the symmetry group of the octonions. The other four exceptional groups, called F4, E6, E7 and E8, are symmetry groups of "projective planes" over:
1) the octonions, O 2) the complexified octonions or "bioctonions", C tensor O 3) the quaternionified octonions or "quateroctonions", H tensor O 4) the octonionified octonions or "octooctonions", O tensor Orespectively.
Warning: I put the phrase "projective planes" in quotes here because the last two spaces are not projective planes in the usual axiomatic sense (see "week145"). This makes the subject a bit tricky.
Now, it is no coincidence that:
1) spinors in 9-dimensional Euclidean space are pairs of octonions. 2) spinors in 10-dimensional Euclidean space are pairs of bioctonions. 3) spinors in 12-dimensional Euclidean space are pairs of quateroctonions. 4) spinors in 16-dimensional Euclidean space are pairs of octooctonions.(These numbers are 8 more than the dimensions of R, C, H and O.)
This sets up a relation between spinors in these various dimensions and the projective planes over O, C tensor O, H tensor O and O tensor O. The upshot is that we get a nice description of F4, E6, E7 and E8 in terms of the Lie algebras so(n) and their spinor representations where n = 9, 10, 12, 16, respectively.
It's all so tightly interlocked - I can't believe it's not trying to tell us something about physics! Just to whet your appetite for more, Just to whet your appetite for more, let me show you 7 quateroctonionic descriptions of the Lie algebra of E7:
e7 = isom((H tensor O)P2) = der(h3(O)) + h3(O)3 = der(O) + der(h3(H)) + (Im(O) tensor sh3(H)) = der(H) + der(h3(O)) + (Im(H) tensor sh3(O)) = der(O) + der(H) + sa3(H tensor O) = so(O + H) + Im(H) + (H tensor O)2 = so(O) + so(H) + Im(H) + (H tensor O)3I explain why these are true in the paper, but for now, let me just say what all this stuff means:
"isom" means the Lie algebra of the isometry group,
(H tensor O)P2 means the quateroctonionic projective plane with its god-given Riemannian metric,
"der" means the Lie algebra of derivations,
h3(O) is the exceptional Jordan algebra, consisting of 3x3 hermitian octonionic matrices,
h3(H) is the Jordan algebra of 3x3 hermitian quaternionic matrices,
Im(O) is the 7-dimensional space of imaginary octonions,
Im(H) is the 3-dimensional space of imaginary quaternions,
sh3(O) is the traceless 3x3 hermitian octonionic matrices,
sh3(H) is the traceless 3x3 hermitian quaternionic matrices,
sa3(H tensor O) is the traceless 3x3 antihermitian quateroctonionic matrices.
so(V) is the rotation group Lie algebra associated to the real inner product space V.
It is fun to compute the dimension of E7 using each of these 7 formulas and see that you get 133 each time!
I also give 6 bioctonionic descriptions of E6. Alas, I could not find 8 octooctonionic descriptions of E8, probably because this group is more symmetrical and in a curious sense simpler than the others.
Time for dinner.
© 2001 John Baez