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While I try to limit myself to mathematical physics in This Week's Finds, I can't always keep it from spilling over into other subjects... so if you're not interested in computers, just skip down to reference 8 below. A while back on sci.physics Matt McIrvin pointed out that the closest thing we have to the computer of old science fiction - the underground behemoth attended by technicians in white lab coats that can answer any question you type in - is AltaVista. I agree wholeheartedly.
In case you are a few months or years behind on the technological front, let me explain: these days there is a vast amount of material available on the World-Wide Web, so that the problem has become one of locating what you are interested in. You can do this with programs known as search engines. There are lots of search engines, but my favorite these days is AltaVista, which is run by DEC, and seems to be especially comprehensive. So these days if you want to know, say, the meaning of life, you can just go to
1) AltaVista, http://www.altavista.digital.com/
type in "meaning of life", and see what everyone has written about it. You'll be none the wiser, of course, but that's how it always worked in those old science fiction stories, too.
The intelligence of AltaVista is of course far less than that of a fruit fly. But Matt's comment made me think about how now, as soon as we develop even a rudimentary form of artificial intelligence, it will immediately have access to vast reams of information on the Web... and may start doing some surprising things.
An example of what I'm talking about is the CYC project, Doug Lenat's $35 million project, begun in 1984, to write a program with common sense. In fact, the project plans to set CYC loose on the web once it knows enough to learn something from it.
2) CYC project homepage, http://www.cyc.com/
The idea behind CYC is to encode "common sense" as about half a million rules of thumb, declarative sentences which CYC can use to generate inferences. To have any chance of success, these rules of thumb must be organized and manipulated very carefully. One key aspect of this is CYC's ontology - the framework that lets it know, for example, that you can eat 4 sandwiches, but not 4 colors or the number 4. Most of the CYC code is proprietary, but the ontology will be made public in July of this year, they say. One can already read about aspects of it in
3) Douglas B. Lenat and R.V. Guha, Building Large Knowledge-Based Systems: Representation and Inference in the Cyc Project, Addison-Wesley, Reading, Mass., 1990.
My network of spies informs me that many hackers are rather suspicious of CYC. For an interesting and somewhat critical account of CYC at one stage of its development, see
4) Vaughan Pratt, CYC Report, http://boole.stanford.edu/pub/cyc.report
Turning to something that's already very practical, I was very pleased when I found one could use AltaVista to do "backlinks". Certainly the World-Wide Web is in part inspired by Ted Nelson's visionary but ill-starred Xanadu project.
5) Project Xanadu, http://xanadu.net/the.project
Backlinking is one of the most tricky parts of Ted Nelson's vision, one often declared infeasible, but one upon which he has always insisted. Basically, the idea is that you should always be able to find all the documents pointing to a given document, as well as those to which it points. This allows commentary or annotation: if you read something, you can read what other people have written about it. My spies inform me that the World-Wide Web Committee is moving in this direction, but it is exciting that one can already do "backlinks browsing" with the help of a program written by Ted Kaehler:
6) Ted Kaehler's backlinks browser, http://www.foresight.org/backlinks1.3.1.html
Go to this page at the start of your browsing session. Follow the directions and let it create a new Netscape window for you to browse in. Whenever you want backlinks, click in the original page, and click "Links to Other Page". This launches an AltaVista search for links to the page you were just looking at.
It works quite nicely. I hope you try it, because with backlinking the Web will become a much more interesting and useful place, and the more people who know about it, the sooner it will catch on. For more discussion of backlinking, see
7) Backlinking news at the Foresight Institute, http://www.foresight.org/backlink s.news.html
Robin Hanson's ideas on backlinking, http://www.hss.caltech.edu/~hanson/findcritics.html
I thank my best buddy Bruce Smith for telling me about CYC, Project Xanadu, and Ted Kaehler's backlinks browser.
Now let me turn to some mathematics and physics.
8) Francesco Fucito, Maurizio Martellini and Mauro Zeni, The BF formalism for QCD and quark confinement, preprint available as hep-th/9605018.
9) Ioannis Tsohantjis, Alex C Kalloniatis, and Peter D. Jarvis, Chord diagrams and BPHZ subtractions, preprint available as hep-th/9604191.
These two papers both treat interesting relationships between topology and quantum field theory - not the "topological quantum field theory" beloved of effete mathematicians such as myself, but the pedestrian sort of quantum field theory that ordinary working physicists use to study particle physics. So we are seeing an interesting cross-fertilization here: first quantum field theory got applied to topology, and now the resulting ideas are getting applied back to particle physics.
Why don't we see quarks roaming the streets freely at night? Because they are confined! Confined to the hadrons in which they reside, that is. We mainly see two sorts of hadrons: baryons made of three quarks, like the proton and neutron, and mesons made of a quark and an antiquark, like the pion or kaon. Why are the quarks confined in hadrons? Well, roughly it's because if you grab a quark inside a hadron and try to pull it out, the force needed to pull it doesn't decrease as you pull it farther out; instead, it remains about constant. Thus the energy grows linearly with the distance, and eventually you have put enough energy into the hadron to create another quark-antiquark pair, and pop - you find you are holding not a single quark but a quark together with a newly born antiquark, that is, a meson! What's left is a hadron with a newly born quark as the replacement for the one you tried to pull out!
That's a pretty heuristic description. In fact, particle physicists do not usually grab hadrons and try to wrest quarks from them with their bare hands. Instead they smash hadrons and other particles at each other and study the debris. But as a rough sketch of the theory of quark confinement, the above description is not completely silly.
There are various interesting things left to do, though. One is to try to get those quarks out by means of sneaky tricks. The only way known is to heat a bunch of hadrons to ridiculously high temperatures, preferably at ridiculously high pressures. I'm talking temperatures like 2 trillion degrees, and densities comparable to that of nuclear matter! This should yield a "quark-gluon plasma" in which quarks can zip around freely at enormous energies. That's what the folks at the Relativistic Heavy Ion Collider are doing - see "week76" for more.
This should certainly keep the experimentalists entertained. On the other hand, theorists can have lots of fun trying to understand more deeply why quarks are confined. We'd like best to derive confinement in some fairly clear and fairly rigorous way from quantum chromodynamics, or QCD - our current theory of the strong force, the force that binds the quarks together. Unfortunately, mathematical physicists are still struggling to formulate QCD in a rigorous way, so they can't yet turn to the exciting challenge of proving that confinement follows from QCD. And we certainly don't expect any simple way to "exactly solve" QCD, since it is complicated and highly nonlinear. So what some people do instead is computer simulations of QCD, in which they approximate spacetime by a lattice and do a lot of number-crunching. They usually use a fairly measly-sounding grid of something like 16 x 16 x 16 x 16 sites or so, since currently calculations take too long when the lattice gets much bigger than that.
Numerical calculations like these have a lot of potential. In "week68", for example, I talked about how people found numerical evidence for the existence of a "glueball" - a hadron made of no quarks, just gluons, the gluon being the particle that carries the strong force. This glueball candidate seems to match the features of an observed particle! Also, people have put a lot of work into computing the masses of more familiar hadrons. So far I believe they have concentrated on mesons, which are simpler. Eventually we should in principle be able to calculate things like the mass of the proton and neutron - which would be really thrilling, I think. Numerical calculations have also yielded a lot of numerical evidence that QCD predicts confinement.
Still, one would very much like some conceptual explanation for confinement, even if it's not quite rigorous. One way people try to understand it is in terms of "dual superconductivity". In certain superconductors, magnetic fields can only penetrate as long narrow tubes of magnetic flux. (For example, this happens in neutron stars - see "week37".) Now, just as electromagnetism consists of an "electric" part and a "magnetic" part, so does the strong force. But the idea is that confinement is due to the electric part of the strong force only being able to penetrate the vacuum in the form of long narrow tubes of field lines. The electric and magnetic fields are "dual" to each other in a precise mathematical sense, so this is referred to as dual superconductivity. Quarks have the strong force version of electric charge - called "color" - and when we try to pull two quarks apart, the strong electric field gets pulled into a long tube. This is why the force remains constant rather than diminishing as the distance between the quarks increases.
A while back, 't Hooft proposed an idea for studying confinement in terms of dual superconductivity and certain "order" and "disorder" observables. It seems this idea has languished to some extent due to a lack of necessary mathematical infrastructure. For the last couple of years, Martellini has been suggesting to use ideas from topological quantum field theory to serve this role. In particular, he suggested treating Yang-Mills theory as a perturbation of BF theory, and applying some of the ideas of Witten and Seiberg (who related confinement in the supersymmetric generalization of Yang-Mills theory to Donaldson theory). In the paper with Fucito and Zeni, they make some of these ideas precise. There are still some potentially serious loose ends, so I am very interested to hear the reaction of others working on confinement.
I have not studied the paper of Tsohantjis, Kalloniatis, and Jarvis in any detail, but people studying Vassiliev invariants might find it interesting, since it claims to relate the renormalization theory of φ^3 theory to the mathematics of chord diagrams.
10) Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, q-alg/9606009.
The "canonical" and "crystal" bases associated to quantum groups, studied by Kashiwara, Lusztig, and others, are exciting to me because they indicate that the quantum groups are just the tip of a still richer structure. Whenever you see an algebraic structure with a basis in which the structure constants are nonnegative integers, you should suspect that you are really working with a category of some sort, but in boiled-down or "decategorified" form.
Consider for example the representation ring R(G) of a group G. This is a ring whose elements are just the isomorphism classes of finite- dimensional representations of G. Addition in R(G) corresponds to taking the direct sum of representations, while multiplication corresponds to taking the tensor product. Thus for example if x and y are irreducible representations of G - or "irreps" for short - and [x] and [y] are the corresponding basis elements of R(G), the product [x][y] is equal to a linear combination of the irreps appearing in x tensor y, with the coefficients in the linear combination being the multiplicities with which the various irreps appear in x tensor y. These coefficients are therefore nonnegative integers. They are an example of what I'm calling "structure constants".
What's happening here is that the ring R(G) is serving as a "decategorified" version of the category Rep(G) of representations of the group G. Alsmost everything about R(G) is just a decategorified version of something about Rep(G). For example, R(G) is a monoid under multiplication, while Rep(G) is a monoidal category under tensor product. R(G) is actually a commutative monoid, while Rep(G) is a symmetric monoidal category - this being jargon for how the tensor product is "commutative" up to a nice sort of isomorphism. In R(G) we have addition, while in Rep(G) we have direct sums, which category theorists would call "biproducts". And so on. The representation ring is a common tool in group theory, but a lot of the reason for working with R(G) is simply that we don't know enough about category theory and are too scared to work directly with Rep(G). There are also good reasons for working with R(G), basically because it is simpler and contains less information than Rep(G).
We can imagine that if someone handed us a representation ring R(G) we might eventually notice that it had a nice basis in which the structure constants were nonnegative integers. And we might then realize that lurking behind it was a category, Rep(G). And then all sorts of things about it would become clearer....
Something similar like this seems to be happening with quantum groups! Ignoring a lot of important technical details, let me just say that quantum groups turn out have nice bases in which the structure constants are nonnegative integers, and the reason is that lurking behind the quantum groups are certain categories. What sort of categories? Categories of "Lagrangian subvarieties of the cotangent bundles of quiver varieties". Yikes! I don't think I'll explain that mouthful! Let me just note that a quiver is itself a cute little category that you cook up by taking a graph and thinking of the vertices as objects and the edges as morphisms, like this:
o->-o->-o->-o->-o
If you do this to a graph that's the Dynkin diagram of a Lie group - see "week62" and the weeks following that - then the fun starts! Dynkin diagrams give Lie groups, but also quantum groups, and now it turns out that they also give rise to certain categories of which the quantum groups are decategoried, boiled-down versions.... I don't understand all this, but I certainly intend to, because it's simply amazing how a world of complex symmetry emerges from these Dynkin diagrams.
For more on this stuff try the paper by Crane and Frenkel referred to in "week38" and "week50". It suggests some amazing relationships between this stuff and 4-dimensional topology....
Let me conclude by reminding you where I am in "the tale of n-categories"
and where I want to go next. So far I have spoken mainly of 0-categories,
1-categories, and 2-categories, with lots of vague allusions as to how
various patterns generalize to higher n. Also, I have concentrated mainly
on the related notions of equality, isomorphism, equivalence, and
adjointness. Equality, isomorphism and equivalence are the most natural
notions of "sameness" when working in 0-categories, 1-categories, and
2-categories, respectively. Adjointness is a closely related but more
subtle and exciting concept that you can only start talking about once
you get to the level of 2-categories. People got tremendously excited
by it when they started working with the 2-category Cat of all small
categories, because it explained a vast number of situations where you
have a way to go back and forth between two categories, without the
categories being "the same" (or equivalent). Another exciting thing
about adjointness is that it really highlights the relation between
2-categories and 2-dimensional topology - thus pointing the way
to a more general relation between n-categories and n-dimensional topology.
From this point of view, adjointness is all about "folds":
(In physics, "doubling back" is related to the notion of antiparticles
as particle moving backwards in time, and appears in the Feynman diagrams
for annihilation and creation of particle/antiparticle pairs. For
those familiar with the category-theoretic approach to Feynman diagrams,
the stuff in "week83" about dual vector spaces should suffice to make
this connection precise.)
Next I will talk about another 2-dimensional concept, the concept
of "joining" or "merging":
M: f2 => f
If we have such a thing, we can draw it as a traditional 2-categorical
diagram:
To continue reading the `Tale of
n-Categories', click here.
© 1996 John Baez
\ /
\ /
\/
/\
/ \
/ \
and their ability to cancel:
/\ | |
/ \ | |
/ \ | |
| \ / = |
| \ / |
| \/ |
| /\ |
| / \ |
| / \ |
\ / | = |
\ / | |
\/ | |
This concept of "doubling back" or "backtracking" is a
very simple and powerful one, so it's not surprising that it
is prevalent throughout mathematics and physics. It is an
essentially 2-dimensional phenomenon (though it occurs in higher
dimensions as well), so it can be understood most generally in the
language of 2-categories.
\ /
\ /
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This is probably even more powerful than the concept of "folding":
it shows up whenever we add numbers, multiply numbers, or in many
other ways combine things. The 2-categorical way to understand
this is as follows. Suppose we have an object x in a 2-category,
and a morphism f: x → x. Then we can ask for a 2-morphism
x
/ \
/ \
f/ \f
/ M \
/ \
x----->-----x
f
(where we should label the M with a big double arrow pointing down from
f2 to f), or dually as a "string diagram"
f f
\ /
\ /
|M
|
f
Regardless of how you draw it, the 2-morphism M: f2 => f represents
a process turning two copies of f into one. And as we'll see, all
sorts of fancy ways mathematicians have of studying this sort of
process - "monoids", "monoidal categories", and "monads" - are
special cases of this sort of situation.
baez@math.removethis.ucr.andthis.edu
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