Also available at http://math.ucr.edu/home/baez/week170.html
August 8, 2001
This Week's Finds in Mathematical Physics (Week 170)
John Baez
I've been travelling around a lot lately. For a couple of weeks I was
in Turkey, resisting the lure of the many internet cafes. I urge you
all to visit Istanbul when you get a chance! Fascinating music fills
the streets. There are a lot of nice bookstore-cafes on Istiklal
Caddesi near Taksim Square, and a huge number of musical instrument
shops at the other end of this street, down near Tunel Square. I bought
a nice doumbek at one of these shops, and looked at lots of ouzes, sazes
and neys, none of which I can play. It's also imperative to check out
the Grand Bazaar, the mosques, and the Topkapi Palace - the harem there
has most beautiful geometric tiling patterns I've ever seen. I'm not
sure why that's true; perhaps this is where the sultans spent most of
their time.
The mathematics of tilings is a fascinating subject, but that's not what
I'm going to talk about. After my trip to Turkey, I went to a conference
at Stanford:
1) Conference on Algebraic Topological Methods in Computer Science,
Stanford University, http://math.stanford.edu/atmcs/index.htm
There were lots of fun talks, but I'll just mention two.
The talk most related to physics was the one by my friend Dan Christensen,
who spoke on "Spin Networks, Spin Foams and Quantum Gravity", describing
a paper he is writing with Greg Egan on efficient algorithms for computing
Riemannian 10j symbols. Dan is a homotopy theorist at the University of
Western Ontario, and Greg is my favorite science fiction writer. They're
both interested in quantum gravity, and they're both good at programming.
Together with some undergraduate students of Dan's, the three of us are
starting to study the Riemannian and Lorentzian Barrett-Crane models of
quantum gravity with the help of computer simulations. But to get anywhere
with this, we need to get good at computing "10j symbols".
Huh? "10j symbols"??
Well, as with any quantum field theory, the key to the Barrett-Crane model
is the partition function. In the Riemannian version of this theory, you
compute the partition function as follows. First you take your 4-dimensional
manifold representing spacetime and triangulate it. Then you label all the
triangles by spins j = 0, 1/2, 1, 3/2, etcetera. Following certain specific
formulas you then calculate a number for each 4-simplex, a number for each
tetrahedron, and a number for each triangle, using the spin labellings. Then
you multiply all these together. Finally you sum over all labellings to
get the partition function. The only tricky part is the convergence of
this sum, which was proved by Perez:
2) 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.
The most interesting aspect of all this is the formula giving numbers for
4-simplices. A 4-simplex has 10 triangular faces all of which get labelled
by spins, and the formula says how to compute a number from these 10 spins -
the so-called "10j symbol".
How do you compute 10j symbols? One approach involves representation theory,
or in lowbrow terms, multiplying a bunch of matrices. Unfortunately, if you
go about this in the most simple-minded obvious fashion, when the spins
labelling your triangles are all about equal to j, you wind up needing to
work with matrices that are as big as N x N, where
N = (2j+1)^{12}.
If you do this, already for j = 1/2 you are dealing with square matrices
that are 2^12 by 2^12. This is too big to be practical!
In computer science lingo, this algorithm sucks because it uses O(j^{12})
time and also O(j^{12}) space. You might think it was O(j^{24}), but
it's not that bad... however, it's still very bad!
Luckily, Dan and Greg have figured out a much more efficient algorithm,
which uses only O(j^6) time and O(j) space. Alternatively, with more caching
of data, they can get O(j^5) time and O(j^3) space, or maybe even better.
Using an algorithm of this sort, Dan can compute the 10j symbol for spins
up to 55. For all spins equal to 55, the calculation took about 10 hours
on a normal desktop computer. However, for computing partition functions
it appears that small spins are much more important, and then the computation
takes milliseconds.
(Actually, for computing partition functions, Dan is not using a desktop:
he is using a Beowulf cluster, which is a kind of supercomputer built out
of lots of PCs. This works well for partition functions because the
computation is highly parallelizable.)
John Barrett has also figured out a very different approach to computing
10j symbols:
3) John W. Barrett, The classical evaluation of relativistic spin networks,
Adv. Theor. Math. Phys. 2 (1998), 593-600. Also available as math.QA/9803063.
In this approach one computes the 10j symbols by doing an integral over the
space of geometries of a 4-simplex - or more precisely, over a product of
5 copies of the 3-sphere, where a point on one of these 3-spheres describes t
he normal vector to one of the 5 tetrahedral faces of the 4-simplex.
Dan and Greg have also written programs that calculate the 10j symbols by
doing these integrals. The answers agree with their other approach.
We've already been getting some new physical insights from these
calculations. If you write down the integral formula for the Riemannian
10j symbols, a stationary phase argument due to John Barrett and Ruth
Williams suggests that, at least in the limit of large spins, the dominant
contribution to the integral for the 10j symbol comes from 4-simplices whose
face areas are the 10 spins in your 10j symbols:
4) John W. Barrett and Ruth M. Williams, The asymptotics of an amplitude
for the 4-simplex, Adv. Theor. Math. Phys. 3 (1999), 209-215. Also available
as gr-qc/9809032.
However, Dan and Greg's calculations suggest instead that the dominant
contribution comes from certain "degenerate" configurations. Some of
these correspond to points on the product of 5 copies of the 3-sphere that
are close to points of the form (v,v,v,v,v) - or roughly speaking,
4-simplices whose 5 normal vectors are all pointing the same way. Others
come from sprinkling minus signs in this list of vectors. Heuristically,
we can think of these degenerate configurations as extremely flattened-out
4-simplices.
For simplicity, we have concentrated so far on studying the 10j symbols
in the case when all 10 spins are equal. In this case we can show that the
only nondegenerate 4-simplex with these spins as face areas is the regular
4-simplex (all of whose faces are congruent equilateral triangles). Greg
used stationary phase to compute the contribution of this regular 4-simplex
to Barrett's integral formula for the 10j symbols, and it turned out that
asymptotically, for large j, this contribution decays like j^{-9/2}. On the
other hand, Dan's numerical computations of the 10j symbol suggests that it
goes like j^{-2}. This suggests that for large j, the contribution of the
regular 4-simplex is dwarfed by that of the degenerate 4-simplices.
Greg has gotten more evidence for this by studying the integral formula for
the 10j symbols and estimating the contribution due to degenerate 4-simplices.
This estimate indeed goes like j^{-2} for large j.
There is a lot more to be understood here, but plunging ahead recklessly, we
can ask what all this means for the physics of the Barrett-Crane model. For
example: is the dominant contribution to the partition function going to come
from spacetime geometries with lots of degenerate 4-simplices?
I think that's a premature conclusion, because we already have evidence that 4
-simplices with large face areas are not contributing that much compared to
those with small face areas when we compute the partition function as a sum
over spin foams. In other words, it seems that in the Riemannian Barrett-Crane
model, spacetime is mostly made of lots of small 4-simplices, rather than a
few giant ones. If so, the tendency for the giant ones to flatten out may
not be so bad.
Of course the really important thing will be to study these questions for
the Lorentzian theory, but it's good to look at the Riemannian theory too.
Another talk on a subject close to my heart was given by Noson Yanofsky.
It was based on these papers of his, especially the last:
5) Noson S. Yanofsky, Obstructions to coherence: natural noncoherent
associativity, Jour. Pure Appl. Alg. 147 (2000), 175-213. Also
available at math.QA/9804106.
The syntax of coherence. To appear in Cahiers Top. Geom. Diff..
Also available at math.CT/9910006.
Coherence, homotopy and 2-theories. To appear in K-Theory.
Also available at math.CT/0007033.
One of the cool things Yanofsky has done is to study what happens when
we categorify Lawvere's concept of an "algebraic theory". I've already
explained this idea of "algebraic theory" in "week53" and "week136", so
I'll just quickly recap it here:
The notion of "algebraic theory" is just a slick way to study sets
equipped with extra algebraic structure. We call a category C with
finite products an "algebraic theory" if its objects are all of the
form 1, X, X^2, X^3, ... for some particular object X. We call a
product-preserving functor F: C -> Set a "model" of the theory. And we
call a natural transformation between such functors a "homomorphism"
between models. This gives us a category Mod(C) consisting of models
and homomorphisms between them, and it turns out that many categories
of algebraic gadgets are of this form: the category of monoids, the
category of groups, the category of abelian groups, and so on.
Since algebraic theories are good for studying sets with extra algebraic
structure, we might hope that by categorifying, we could obtain a concept
of "algebraic 2-theories" which is good for studying *categories* with
extra algebraic structure. And it's true! In 1974, John Gray defined an
"algebraic 2-theory" to be a 2-category C with finite products, all of whose
objects are of the form 1, X, X^2, X^3, ... for some particular object X.
Define a "model" of this 2-theory to be a product-preserving 2-functor
F: C -> Cat. And define a "homomorphism" between models to be a pseudonatural
transformation between such 2-functors.
Huh? "Pseudonatural"??
Sorry, now things are getting a bit technical: the right thing going between
2-functors is not a natural transformation but something a bit weaker called
a "pseudonatural transformation", where the usual commuting squares in the
definition of a natural transformation are required to commute only up to
certain specified 2-isomorphisms, which in turn satisfy some coherence laws
described here:
6) G. Maxwell Kelly and Ross Street, Review of the elements of 2-categories,
Springer Lecture Notes in Mathematics 420, Berlin, 1974, pp. 75-103.
However, you don't need to understand the details right now. There is also
something going between pseudonatural transformations called a "modification",
and this gives us "2-homomorphisms" between homomorphisms between models of
our algebraic theory. Thanks to these there is a 2-category Mod(C) consisting
of models of our 2-theory homomorphisms between those, and 2-homomorphisms
between those.
Some examples might help! For example, there's a 2-theory C called the
"theory of weak monoidal categories". Models of C are weak monoidal
categories, homomorphisms are monoidal functors, and 2-homomorphisms are
natural transformations, so Mod(C) is the usual 2-category of monoidal
2-categories. There's a similar 2-theory C' called "the theory of strict
monoidal categories", for which Mod(C') is the usual 2-category of strict
monoidal categories.
(Hyper-technical note for n-category mavens only: in both examples here,
monoidal functors are required to preserve unit and tensor product only
*up to coherent natural isomorphism*. This nuance is what we get from
working with pseudonatural rather than natural transformations. Without
this nuance, some of the stuff I'm about to say would be false.)
Now, whenever we have a product-preserving 2-functor between 2-theories,
say F: C -> C', we get an induced 2-functor going the other way,
F*: Mod(C') -> Mod(C).
For example, there's a product-preserving 2-functor from the theory of
weak monoidal categories to the theory of strict monoidal categories,
and this lets us turn any strict monoidal category into a weak one.
Now in this particular example, F* is a biequivalence, which is the nice
way to say that the 2-categories Mod(C) and Mod(C') are "the same" for all
practical purposes. And in fact, saying that this particular F* is a
biequivalence is really just an ultra-slick version of Mac Lane's theorem -
the theorem we use to turn weak monoidal categories into strict ones.
Now, Mac Lane's theorem is the primordial example of a "strictification
theorem" - a theorem that lets us turn "weak" algebraic structures on
categories into "strict" ones, where lots of isomorphisms, like the
associators in the monoidal category example, are assumed to be equations.
This suggests that lots of coherence theorems can be stated by saying that
2-functors of the form F* are biequivalences.
So: is there a super-general strictification theorem where we can start
from any 2-theory C and get a "strictified" version C' together with an
F: C -> C' such that F* is a biequivalence?
As a step in this direction, Yanofsky has cooked up a model category of
algebraic 2-theories, in which F: C -> C' is a weak equivalence precisely
when F* is a biequivalence.
Huh? "Model category"??
Well, if you don't know what a "model category" is, you're in serious
trouble now! They're a concept invented by Quillen for generalizing
the heck out of homotopy theory. Try reading his book:
7) Daniel G. Quillen, Homotopical Algebra, Springer Lecture Notes in
Mathematics, vol. 43, Springer, Berlin, 1967.
or for something newer:
8) Mark Hovey, Model Categories, American Mathematical Society Mathematical
Surveys and Monographs, vol 63., Providence, Rhode Island, 1999.
or else:
9) Paul G. Goerss and John F. Jardine, Simplicial Homotopy Theory,
Birkhauser, Boston, 1999.
(By the way, Jardine was one of the organizers of this Stanford conference,
along with Gunnar Carlsson. He told me he had created a hypertext version
of this book, but has not been able to get the publisher interested in it.
Sad!)
Anyway, in the framework of model categories, the problem of "strictifying"
an algebraic structure on categories then amounts to finding a "minimal
model" of a given 2-theory C - roughly speaking, a weakly equivalent
2-theory with as little flab as possible. The concept of "minimal model"
is important in homotopy theory, but apparently Yanofsky is the first to
have given a general definition of this concept applicable to any model
category. Yanofsky has not shown that every algebraic 2-theory admits a
minimal model, but this seems like a fun and interesting question.
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all ignorance toboggans into know
and trudges up to ignorance again.
e.e.cummings, 1959
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