September 6, 2003

This Week's Finds in Mathematical Physics (Week 198)

John Baez

I recently got back from a summer spent mostly in Hong Kong. It was interesting being there. Since I wasn't there long, most of my observations are pretty superficial. For example, they have a real commitment to public transportation. Not only is there a wonderful system of subways, ferries, buses and green minibuses where you can pay for your ride using a cool high-tech "octopus card", the local gangs run their own system of red minibuses. These don't run on fixed schedules, and they don't take the octopus card, but they seem perfectly safe, and they go places the others don't.

Another obvious feature is the casual attitude towards English, which is still widely used, but plays second fiddle to Cantonese now that the Brits have been kicked out. Menus feature strange items such as "mocked eel" and "mocked shark fin soup", which bring to mind the unsettling image of a cook ridiculing hapless sea creatures before cooking them. Also, perfectly nice people wear T-shirts saying things that wouldn't be wise where I come from, like

                         Lost Pig

                          I SEE 
                           WHY 
                         YOU SUCK

While in Hong Kong, I received a copy of a very interesting book:

1) David Corfield, Towards a Philosophy of Real Mathematics, Cambridge U. Press, Cambridge, 2003. More information and part of the book's introduction available at http://www-users.york.ac.uk/~dc23/Towards.htm

I should admit from the start that I'm completely biased in favor of this book, because it has a whole chapter on one of my favorite subjects: higher-dimensional algebra. Furthermore, Corfield cites me a lot and says I deserve "lavish praise for the breadth and quality of my exposition". How could I fail to recommend a book by so wise an author?

That said, what's really special about this book is that it shows a philosopher struggling to grapple with modern mathematics as it's actually carried out by its practitioners. This is what Corfield means by "real" mathematics. Too many philosophers of mathematics seem stuck in the early 20th century, when explicitly "foundational" questions - questions of how we can be certain of mathematical truths, or what mathematical objects "really are" - occupied some the best mathematicians. These questions are fine and dandy, but by now we've all heard plenty about them and not enough about other equally interesting things. Alas, too many philosophers seem to regard everything since Goedel's theorem as a kind of footnote to mathematics, irrelevant to their loftier concerns (read: too difficult to learn).

Corfield neatly punctures this attitude. He calls for philosophers of mathematics to follow modern philosophers of the natural sciences and focus more on what practitioners actually do:

[...] to the extent that we wish to emulate Lakatos and represent the discipline of mathematics as the growth of a form of knowledge, we are duty bound to study the means of production throughout its history. There is sufficient variation in these means to warrant the study of contemporary forms. The quaint hand-crafted tools used to probe the Euler conjecture in the early part of the nineteenth century studied by Lakatos in "Proofs and Refutations" have been supplanted by the industrial-scale machinery of algebraic topology developed since the 1930s.

[...] Straight away, from simple inductive considerations, it should strike us as implausible that mathematicians dealing with number, function and space have produced nothing of philosophical significance in the past seventy years in view of their record over the previous three centuries. Implausible, that is, unless by some extraordinary event in the history of philosophy a way had been found to filter, so to speak, the findings of mathematicians working in core areas, so that even the transformations brought about by the development of category theory, which surfaced explicitly in 1940s algebraic topology, or the rise of non-commutative geometry over the past seventy years, are not deemed to merit philosophical attention.

The book consists of chapters on loosely linked subjects, some of which seem closer to "real mathematics" than others. The chapters on "Communicating with automated theorem provers" and "Automated conjecture formation" are mildly depressing, given how poor computers are at spotting or proving truly interesting conjectures without lots of help from humans - at least so far. True, Corfield describes how in 1996 the automated theorem prover EQP was the first to crack the Robbins conjecture. This states that a Boolean algebra is the same as a set equipped with an commutative associative binary operation "or" together with a unary operation "not" for which one mind-numbing axiom holds, namely:

not(not(p or q) or not(p or not(q)) = p

Apart from the one on higher-dimensional algebra, the chapters that make me happiest are the ones on "The importance of mathematical conceptualisation" and "The role of analogy in mathematics".

The first is a marvelous study of the so-called "conceptual approach" in mathematics, which emphasizes verbal reasoning using broad principles over calculations using symbol manipulation. Some people are fond of the conceptual approach, while others regard it as "too abstract". Corfield illustrates this split using the debate over "groupoids versus groups", with the supporters of groupoids (including Grothendieck, Brown and Connes) taking the conceptual high road, but others preferring to stick with groups whenever possible. As a philosopher, Corfield naturally leans towards the conceptual approach.

The second is all about analogies. Analogies are incredibly important in mathematics. Some can be made completely precise and their content fully captured by a theorem, but the "deep" ones, the truly fruitful ones, are precisely those that resist complete encapsulation and only yield their secrets a bit at a time. Corfield quotes Andre Weil, who describes the phenomenon as only a Frenchman could - even in translation, this sounds like something straight out of Proust:

As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure.

The analogy he considers in most detail is a famous one between number fields and function fields, going back at least to Dedekind and Kummer. By a "number field", we mean something like the set of all numbers

a + b sqrt(-5)

P(z)/Q(z)                    

Sitting inside a number field we always have something called the "algebraic integers", which in the above example are the numbers

a + b sqrt(-5)

P(z)

So, an analogy exists. But the cool part is that there's a good generalization of "prime numbers" in the algebraic integers of any number field, invented by Kummer and called "prime ideals"... and prime ideals in the algebraic functions of a function field have a nice geometrical interpretation! In the example given above, they correspond to points in the complex plane!

The analogy between number fields and function fields has been pushed to yield all sorts of important results in number theory and algebraic geometry. In Weil's hands it led to the theory of adeles and the Weil conjectures. These in turn led to etale cohomology, Grothendieck's work on topoi, and much more. And the underlying analogy is still far from exhausted! But if we ever get it completely nailed down, then (in the words of Weil):

The day dawns when the illusion vanishes; intuition turns to certitude; the twin theories reveal their common source before disappearing; as the Gita teaches us, knowledge and indifference are attained at the same moment. Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us.

Anyway, I hope this book shows philosophers that modern mathematics poses many interesting questions apart from the old "foundational" ones. These questions can only be tackled after taking time to learn the relevant math... but what could be more fun than that?! I also hope this book shows mathematicians that having a well- informed and clever philosopher around makes math into a more lively and self-aware discipline.

(The same is true of physics, of course. I listed a few good philosophers of physics in "week190".)

Someday I'd like to say more about the analogy between number fields and function fields, because I'm starting to study this stuff with James Dolan... but it will take a while before I know enough to say anything interesting. So instead, let me say what's going on with spin foam models of quantum gravity.

I've already talked about these in "week113", "week120", "week128" and "week168". The idea is to calculate the amplitude for spacetime to have any particular geometry. An amplitude is just a complex number, sort of the quantum version of a probability. If you know how to calculate an amplitude for each spacetime, you can try to compute the expectation value of any observable by averaging its value over all possible geometries of spacetime, weighted by their amplitudes. When you do this to answer questions about physics at large distances scales, the amplitudes should almost cancel except for spacetimes that come close to satisfying the equations of general relativity. This is how quantum gravity should reduce to classical gravity at distance scales much larger than the Planck length.

But in a spin foam model, a spacetime geometry is not described by putting a metric on a manifold, as in general relativity. Instead, it's described in a somewhat more "discrete" manner. Only at distances substantially larger than the Planck length should it resemble a metric on a manifold.

How do you describe a spacetime geometry in a spin foam model? Well, first you take some 4-dimensional manifold representing spacetime and chop it into "4-simplices". A "4-simplex" is just the 4-dimensional analogue of a tetrahedron: it has 5 tetrahedral faces, 10 triangles, 10 edges and 5 vertices. Then, you label all the triangles in these 4-simplices by numbers. These describe the areas of the triangles. Here the details depend on which spin foam model you're using. In the Riemannian Barrett-Crane model, you label the triangles by spins j = 0, 1/2, 1, 3/2.... But in the Lorentzian Barrett-Crane model, which should be closer to the real world, you label them by arbitrary positive real numbers. Either way, a spacetime chopped up into 4-simplices labelled with numbers is called a "spin foam".

To compute an amplitude for one of these spin foams, you first use the labellings on the triangles and follow certain specific formulas to calculate a complex number for each 4-simplex, each tetrahedron, and each triangle. Then you multiply all these numbers together to get the amplitude!

In "week170", I mentioned some mysterious news about the Barrett-Crane model. At the time - this was back in August of 2001 - my collaborators Dan Christensen and Greg Egan were using a supercomputer to calculate the amplitudes for lots of spin foams. The hard part was calculating the numbers for 4-simplices, which are called the "10j symbols" since they depend on the labels of the 10 triangles. They had come up with an efficient algorithm to compute these 10j symbols, at least in the Riemannian case. And using this, they found that the 10j symbols were not coming out as an approximate calculation by Barrett and Williams had predicted!

Barrett and Williams had done a "stationary phase approximation" to argue that in the limit of a very large 4-simplex, the 10j symbols were asymptotically equal to something you'd predict from general relativity. This seemed like a hint that the Barrett-Crane model really did reduce to general relativity at large distance scales, as desired.

However, things actually work out quite differently! By now the asymptotics of the 10j symbols are well understood, and they're not given by the stationary phase approximation. If you want to see the details, read these papers:

2) John C. Baez, J. Daniel Christensen and Greg Egan, Asymptotics of 10j symbols, Class. Quant. Grav. 19 (2002) 6489-6513. Also available as gr-qc/0208010.

3) John W. Barrett and Christopher M. Steele, Asymptotics of relativistic spin networks, Class. Quant. Grav. 20 (2003) 1341-1362. Also available as gr-qc/0209023.

4) Laurent Freidel and David Louapre, Asymptotics of 6j and 10j symbols, Class. Quant. Grav. 20 (2003) 1267-1294. Also available as hep-th/0209134.

The physical meaning of this fact is still quite mysterious. I could tell you everyone's guesses, but I'm not sure it's worthwhile. Next spring, Carlo Rovelli, Laurent Freidel and David Louapre are having a conference on loop quantum gravity and spin foams in Marseille. Maybe after that people will understand what's going on well enough for me to try to explain it!

I'd like to wrap up with a few small comments about last Week. There I said a bit about a 24-element group called the "binary tetrahedral group", a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the "24-cell". The most important fact is that these are all the same thing! And I've learned a bit more about this thing from here:

5) Robert Coquereaux, On the finite dimensional quantum group H = M₃ + (M_2|1(Λ²))₀, available as hep-th/9610114 and at http://www.cpt.univ-mrs.fr/~coque/articles_html/SU2qba/SU2qba.html

Just to review: let's start with the group consisting of all the ways you can rotate a regular tetrahedron and get it looking the same again. You can achieve any even permutation of the 4 vertices using such a rotation, so this group is the 12-element group A₄ consisting of all even permutations of 4 things - see "week155". But it's also a subgroup of the rotation group SO(3). So, its inverse image under the double cover

SU(2) → SO(3)

has 24 elements. This is called the "binary tetrahedral group".

As usual, the algebra of complex functions on this finite group is a Hopf algebra. But the cool thing is, this Hopf algebra is closely related to the quantum group U_q(sl(2)) when q is a third root of unity - a quantum group used in Connes' work on particle physics because of its relation to the Standard Model gauge group!

In short: the plot thickens.

I'm not really ready to describe this web of ideas in detail, so I'll just paraphrase the abstract of Coquereaux's paper and urge you to either read this paper or look at his website:

We describe a few properties of the non-semisimple associative algebra H = M₃ + (M_2|1(Λ²))₀, where Λ² is the Grassmann algebra with two generators. We show that H is not only a finite dimensional algebra but also a (non-cocommutative) Hopf algebra, hence a "finite quantum group". By selecting a system of explicit generators, we show how it is related with the quantum enveloping algebra of U_q(sl(2)) when the parameter q is a cubic root of unity. We describe its indecomposable projective representations as well as the irreducible ones. We also comment about the relation between this object and the theory of modular representations of the group SL(2,Z/3), i.e. the binary tetrahedral group. Finally, we briefly discuss its relation with the Lorentz group and, as already suggested by A. Connes, make a few comments about the possible use of this algebra in a modification of the Standard Model of particle physics (the unitary group of the semi-simple algebra associated with H is U(3) x U(2) x U(1)).

Addenda: I got some interesting feedback from Martin Krieger and Noam Elkies.

Martin Krieger writes:

In the interchange on Corfield's book, and John Baez's discussion of it, there is a reference to Weil's Rosetta Stone analogy. The quotations come from a charming and deep letter Weil wrote in 1940 to his sister, Simone, from Bonne Nouvelle prison. In my book Doing Mathematics: Convention, Subject, Calculation, Analogy (World Scientific, 2003) that long letter is translated into English (see Appendix D, pp. 293-305). I also happen to have a discussion of the analogy, in chapter 5 (pp. 189-230), in connection with the Langlands program and with results in statistical mechanics of the Ising model.

Martin Krieger
University of Southern California
Los Angeles CA 90089-0626

You can now find Krieger's translation of this letter online, as long as you register with the American Mathematical Society (it's free):

6) Martin H. Krieger, A 1940 letter of Andre Weil on analogy in mathematics, AMS Notices 52 (March 2005), 334-341. Available at http://www.ams.org/notices/200503/200503-toc.html

Noam Elkies writes:

Hello again,

You write:

[...]

>I'd like to wrap up with a few small comments about last Week.
>There I said a bit about a 24-element group called the "binary 
>tetrahedral group", a 24-element group called SL(2,Z/3), and 
>the vertices of a regular polytope in 4 dimensions called the 
>"24-cell".  The most important fact is that these are all the 
>same thing!  And I've learned a bit more about this thing from 
>here:

[...]

Here's yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions - the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g' in the group |g-g'| is at most 2 so g-g' is not in 3A unless g=g'. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL₂(Z/3Z) - which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL₂(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 - which, however, does give us a map from SL₂(Z/5Z) to the symmetric group of order 5, using the action of SL₂(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL₂(Z/5Z) with the alternating group A₅.

Likewise the octahedral and icosahedral groups S₄ and A₅ can be found in PSL₂(Z/7Z) and PSL₂(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A₅ is also an index-6 subgroup of PSL₂(F₉), which yields the identification of that group with A₆.

NDE

© 2003 John Baez
baez@math.removethis.ucr.andthis.edu