This Week's Finds in Mathematical Physics (Week 84)
John Baez
John Baez
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.
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/backlinks.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 phi^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":
\ /
\ /
\/
/\
/ \
/ \
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.
(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":
\ /
\ /
|
|
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
M: f^2 -> f
If we have such a thing, we can draw it as a traditional 2-categorical
diagram:
/\
/ \
f/ \f
/ M \
/ \
x---->-----x
f
or dually as a "string diagram"
f f
\ /
\ /
|M
|
f
Regardless of how you draw it, the 2-morphism M: f^2 -> 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.
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issue, go to
Philip Gibbs
John Baez wrote:
>
> 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.
Does this suggest that the process of quantisation might take us one
step further up the categorical ladder than previously thought.
In your paper on n-categories you talk about quantisation in category
terms
being a deformation of a symmetric monoidal category to give a braided
monoidal category and you suggest there may be n-category
generalisations
(if I understood you right). You have also talked about quantisation in
another context as a functor from a Hilbert space to a larger Hilbert
space
by making a Fock space (that was in categories.html).
Those cases seem to see quantisation as a construction taking you from
an n-category to another n-category, but if a quantum group is just a
decategorified form of something one step up the ladder could it turn
out
that quantisation takes a more general form which maps some class of
n-categories to (n+1)-categories?
>
> 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).
How far have people gone in generalising the idea of forming the
category of representations of a group? If a representation of a
group is a functor from the group to a Hilbert space can you define
more general categories of functors from a category? Is it
possible to replace tensor products with Cartesian products in this
context? Does it make sense to talk about the category of TQFT's on
a given cobordism category in a similar way for example?
I hope these questions make some sense. I don't understand these
castegory concepts very well and I appreciate your potted category
course in TWF enormously.
--
====================================================
Phil Gibbs p...@pobox.com http://pobox.com/~pg
john baez
In article <31D790...@pobox.com>,
Philip Gibbs <philip...@pobox.com> wrote:
>John Baez wrote:
>> 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.
>Does this suggest that the process of quantisation might take us one
>step further up the categorical ladder than previously thought.
Very good question! In other words, I haven't the foggiest notion as to
the answer, and I am dying to know. Lusztig's canonical basis
construction has a rather intimidating reputation, since it uses the
language of "perverse sheaves", which most of us peons haven't gotten
around to mastering. As a result, a lot more people know what the
canonical bases are than *why they really exist*. Answering your
question would require knowing why why they really exist. And that's
why I'm interested in Kashiwara's paper, since it seems to give a
different (and perhaps more comprehensible??) construction.
>In your paper on n-categories you talk about quantisation in category
>terms
>being a deformation of a symmetric monoidal category to give a braided
>monoidal category and you suggest there may be n-category
>generalisations
>(if I understood you right).
That's right. Ordinarily quantization is conceived of in terms of a
deformation of a commutative algebra to give a noncommutative algebra
(the deformation parameter being Planck's constant). Quantum groups
were a big suprise, and here one has a deformation of a symmetric
monoidal category (the category of representations of your group) to a
braided monoidal category (the category of representations of the
corresponding quantum group). The same sort of thing, in other words,
but one step up the n-categorical ladder. The crystal basis/canonical
basis stuff suggests that there is also a kind of deformation of a
symmetric monoidal 2-category to a braided monoidal 2-category going on!
But this is not just a further prolongation of the same sort of pattern,
since 1) if it were, we would be deforming a strongly involutory
2-category to a weakly involutory one, and 2) as far as I know, the
canonical basis stuff only works when the deformation parameter q is a
root of unity, so the sense in which we have a "deformation" is subtler.
Something very exciting and mysterious is going on here, that's for
sure.
>You have also talked about quantisation in
>another context as a functor from a Hilbert space to a larger Hilbert
>space by making a Fock space (that was in categories.html).
That's not so directly relevant here. You mean a functor from Hilb to
Hilb, by the way, taking each space to its Fock space.
>How far have people gone in generalising the idea of forming the
>category of representations of a group?
Very. Perhaps too.
>If a representation of a
>group is a functor from the group to a Hilbert space can you define
>more general categories of functors from a category?
You mean a representation of a group is a functor from that group
(regarded as a one-object category) to Hilb. In my paper with Jim Dolan
we stress the idea of a "representation of a category", namely a functor
from that category to Hilb. TQFTs are examples of this, perhaps the
first example that really caught the attention of physicists.
>Is it
>possible to replace tensor products with Cartesian products in this
>context?
I don't get what you mean, since we don't use the concept of tensor
product in the concept of "functor from a group to Hilb" or "functor from
a category to Hilb". You might be a bit mixed up here.
>Does it make sense to talk about the category of TQFT's on
>a given cobordism category in a similar way for example?
See above.
>I hope these questions make some sense. I don't understand these
>castegory concepts very well and I appreciate your potted category
>course in TWF enormously.
What you say makes sense, though you seem to be doing a little minor
level-slipping, speaking of "a Hilbert space" when you probably should
say "Hilb, the category of all Hilbert spaces", and "the category of
TQFTs" when you probably should say "a TQFT". Of course, level-slipping
is to n-category theory as losing your footing is to rock-climbing:
you can't really avoid it when you are pushing your limits. Glad you
like the course.
Goncalo Violante Rodrigues
Philip Gibbs (philip...@pobox.com) wrote partially:
: > 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).
: How far have people gone in generalising the idea of forming the
: category of representations of a group? If a representation of a
: group is a functor from the group to a Hilbert space can you define
: more general categories of functors from a category? Is it
: possible to replace tensor products with Cartesian products in this
: context? Does it make sense to talk about the category of TQFT's on
: a given cobordism category in a similar way for example?
You have 4 questions in here and they're all tied up intimatelly.
First ( in fact the third question ) the naming tensor product is pro
bably inducing you in error. A tensor product is just a name for a product
in a category that satisfies some axioms, mimicking those of the ordinary
tensor product in the category of vector spaces. And so, yes, the cartesian
product can be taken as " tensor product " but then, again in the case of
vector spaces, you end up with direct sum ( I'm thinking of the finite
dimensional case ) wich is not very interesting for a number of reasons,I
name just one: the unit object is the 0 space so if you want to take traces
( As in topology to fabricate quantum invariants ) you wind up with ...
nothing!
Second ( Fourth ), Yes, there is a category of TQFT's, the morphisms being
monoidal natural transformations ( possibly with additional conditions,
depending on one's taste ). This is a very interesting category in fact,
because it can be given ( much as in the classical case ) the structure of
a commutative categorial hopf ring ( this is a catchy name for the bug!
), wich can be seen as the categorial counterpart of the commutative
hopf algebra in ordinary algebra, in fact, the construction proceeds
along in much the same way, except that now you're in a 2-category.
I hope ( vainly?...) this already has enligtened you, but let me digress a
little bit more around the first question. As you probably know, a group can
be defined as an object in the category of sets ( this is, a set ) that
comes equipped with some maps, that satisfy some laws written down as
commutative diagrams. Taking this point of view you can shift the whole
definition to a general monoidal category, using now the tensor product
to formulate the definitios. Once again you can define a representation (
for example in the category of vector spaces ) as a functor from the one-
object category defined by the group.
For the end i'd like to say that this categorial business and TQFT's is
really, in James Joyce own happy coinage, " a funferal " :-)
john baez
>> 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.
>Does this suggest that the process of quantisation might take us one
>How far have people gone in generalising the idea of forming the
>category of representations of a group?
Very. Perhaps too.
>If a representation of a
>group is a functor from the group to a Hilbert space can you define
>more general categories of functors from a category?
You mean a representation of a group is a functor from that group
(regarded as a one-object category) to Hilb. In my paper with Jim Dolan
we stress the idea of a "representation of a category", namely a functor
from that category to Hilb. TQFTs are examples of this, perhaps the
first example that really caught the attention of physicists.
>Is it
>possible to replace tensor products with Cartesian products in this
>context?
I don't get what you mean, since we don't use the concept of tensor
product in the concept of "functor from a group to Hilb" or "functor from
a category to Hilb". You might be a bit mixed up here.
>Does it make sense to talk about the category of TQFT's on
>a given cobordism category in a similar way for example?
See above.