# June 4, 1995 {#week55}

I recently went to a workshop on canonical quantum gravity in Warsaw,
organized by Jerzy Kijowski and Jerzy Lewandowski, and I learned some
interesting things. I'll talk about some of them in this issue, and
some in the next.

Conferences are a funny thing. On science newsgroups on the net, there
is very little talk about conferences. This is probably because the
people who really understand conferences are too busy flying from one
conference to the next to post to newsgroups very often. Academic
success is in part measured by the number of conference invitations one
receives, the prestige of the conferences, and the type of invitation.
For example, a big plenary lecture on an impressive stage, preceded by a
little warmup where someone explains how great you are, counts for
infinitely many talks in those parallel sessions where dozens of people
get 10 minutes each to explain their work before the moderator begins to
make little coughs indicating that it's time for the next one, while
all the while people drift in and out in a feeble attempt to find the
really interesting talks. Still, giving any sort of talk is regarded as
better than giving none, so academics spend a lot of time doing this
sort of thing.

One of the great dangers of being a successful academic, in fact, is
that one may get invited to so many conferences that one never has time
to think. Winning the Nobel prize is purported to be the kiss of death
in this respect. Of course, it's a universal platitude that the real
thinking at conferences gets done not during the talks, but informally
in small groups. But the funny thing is that at most conferences people
are so worn out after going to a day's worth of talks that they have
limited energy for serious conversation afterwards: they usually seem
more interested in finding the good local restaurants and scenic
attractions. If people could have conferences with no lectures
whatsoever, or maybe one a day, it would probably be more productive.
But the idea that a bunch of people could figure something out just by
sitting around and chatting informally is absolutely foreign to our
conception of "work". People expect to receive money from bureaucrats
to go to conferences, but to convince a bureaucrat that you are deserve
the money, you need to give a lecture, so of course all conferences have
too many lectures.

Turning back towards Warsaw, a city with a marvelous mathematical
history, I am reminded of Gian-Carlo Rota's biographical sketch of
Stanislaw Ulam, in which (as a master of irony) he talks about how lazy
Ulam was: all he wanted to do was sit around in cafes and come up with
interesting conjectures and research programs, and leave it to others to
work them out. And this in turn reminds me of the Scottish Cafe, where
Polish mathematicians used to hang out and write on the tablecloths,
until the owner provided them with a notebook, in which many famous
conjectures were formulated, and I believe prizes like bottles of wine
were offered for their solutions. Was the Scottish Cafe in Warsaw? \[No,
Lwow.\] Does it still exist? I completely forgot to check while I was
there. The Banach Center, in which the conference participants stayed,
comes from a later stratum of Polish mathematical history; it was built
after the war, and one room still contains a portrait of Lenin. I know
that because a film crew used it to shoot a scene for a historical
movie!

Anyway, I enjoyed this conference in Warsaw quite a bit, because a lot
of people working on the loop representation of quantum gravity were
there, and I managed to have a fair number of serious conversations.
Before going into what I learned there, I should say that I just found a
fun thing for people to read who are interested in quantum gravity, but
are not necessarily specialists:

1) Gary Au, "The quest for quantum gravity", available as `gr/qc-9506001`.

This consists mainly of interviews with Chris Isham, Abhay Ashtekar and
Edward Witten. What's nice is that the interviews are conducted by
someone who knows physics. The questions and answers are technical
enough to convey some of the real substance of the subject, while still
(I hope) non-technical enough so that you don't have to be an expert
to get a lot out of them. Isham talks mainly about the "problem of
time" in quantum gravity, Ashtekar talks mainly about the loop
representation of quantum gravity, and Witten talks about string theory.

Anyway, Ashtekar and a bunch of other good people were at this Warsaw
conference, which is why I went. The main topics of conversation were
spin networks and their use in studying the area and volume operators in
quantum gravity. As I explained earlier in ["Week 43"](#week43),
one may very roughly think of a spin network as a graph whose edges are
labelled with "spins" $0$,$1/2$,$1$,$3/2$, and so on, and who vertices are
labelled with certain gadgets called "intertwining operators" (which
in the simplest case are just the Clebsch-Gordon coefficients you learn
about when studying angular momentum in quantum mechanics). Penrose
introduced these as abstract graphs (see ["Week 22"](#week22) and
["Week 41"](#week41)), as a kind of substitute for thinking of
space as a manifold, but more recently Rovelli and Smolin started
thinking of them as graphs embedded into 3d space, and saw that these
were a really natural way to describe states of quantum gravity: even
better than loops, because they form an orthonormal basis! Actually, it
was mainly me who proved in a really rigorous way that they form an
orthonormal basis, but Rovelli and Smolin had already been doing
calculations using this idea for a while. One thing they computed was
the eigenvalues of the observables in quantum gravity corresponding to
the area of a surface in space, or the volume of a region.

Now there are all sorts of technical caveats and subtleties that I
don't want to get into here, but in a really rough sort of sense, what
their answers suggest is that IF the loop representation of quantum
gravity is right, and we are on the right track about how it works, then
the area of surfaces comes in certain (not integer, but discrete)
multiples of the Planck length squared, and the volume of regions comes
in multiples of the Planck length cubed. Note: that was a big "IF".
This is especially interesting because it doesn't arise by assuming
from the start that spacetime has a discrete structure. In fact, their
computations assume spacetime is a continuous manifold. Nonetheless this
discreteness pops out. It's not completely surprising: after all,
Schrodinger's equation for the hydrogen atom is a perfectly
"continuous" sort of thing, a partial differential equation, but the
energy of the bound states winds up being a discrete sort of thing.
Still, it's sort of exciting and new.

An interesting thing happened at the conference. Renate Loll, who works
on the loop representation of gauge theories and also lattice gauge
theory, has recently developed a lattice formulation of quantum gravity
closely modelled after the loop representation:

2) Renate Loll, "Nonperturbative solutions for lattice quantum gravity", preprint available as [`gr-qc/9502006`](https://arxiv.org/abs/gr-qc/9502006).

This has the wonderful feature that it's perfectly rigorous and also
one can start using computers to start calculating things with it. For
example, the most subtle aspect of the loop representation of quantum
gravity is the Wheeler-DeWitt equation
$$H\psi=0$$
where $H$ is an operator called the "Hamiltonian constraint". More on
this later; my point here is just that physical states of quantum
gravity need to satisfy this equation. Getting $H$ to be well-defined is
tricky when space is a continuum, but in Loll's lattice version of
theory (which is an approximation to the full continuum theory) she has
already done this, so one can now start trying numerically to find
solutions and see what they look like. She has also found some explicit
solutions.

*Also*, she did some work on the volume operator in her lattice
approach, and came up with a result in contradiction to Rovelli and
Smolin's paper on the subject (cited in ["Week 43"](#week43)).
They had said that states corresponding to trivalent spin networks
--- spin networks with only 3 edges at each vertex --- could have
nonzero volume. But using her version of the theory she computed that
trivalent states --- states with only 3 nonzero spins at the edges of
the lattice incident to any vertex --- all had zero volume, and that
she needed at least 4 nonzero spins to get volume! The volume operator,
in case you're wondering, acts as a certain sum over vertices: each one
winds up contributing a certain finite amount of volume, which the
theory allows you to compute.

This led to a whole lot of discussion and scribbling on the blackboards
of the Banach center. I found it truly delightful to see all these
physicists drawing pictures of spin networks and doing graphical
computations just the way a knot theorist like Kauffman does all the
time. It was as if the universe had this spin network aspect to it, and
everyone was finally starting to catch on. Either that or mass delusion!
I hadn't quite gotten the hang of how to compute these volume operators
before, so it was a great chance to learn: one person would do a
computation, then someone else would do it a different way and get a
different answer, then someone else would do it yet another way and get
yet another answer, and so on, so you could ask lots of questions
without seeming too dumb. Even I did a computation after a while, and
got zero volume for at least a certain class of trivalent vertices. The
votes in favor of trivalent vertices having zero volume kept piling up.
Finally Smolin noticed that he and Rovelli had made a sign mistake. This
is incredibly easy to do, since there are lots of tricky sign
conventions in spin network theory. Fundamentally these are due to the
fact that spin-$1/2$ particles are fermions... but I don't think people
fully understand the physical implications of this. (There is also a
marvelous category-theoretic explanation of it, but I fear that if I go
into that all the physicists will stop reading. Maybe some other time.)
Rovelli and Smolin got pretty depressed about this for a while, but I
tried to reassure them that only people who write really interesting
papers ever get anybody to find the mistakes.

So perhaps we know a little more about the meaning of volume in a
quantum theory of spacetime.

Spin networks are very beautiful and simple things. To learn about them,
in addition to the various papers listed in the "weeks" above, one can
now turn to Rovelli and Smolin's paper:

3) C. Rovelli and L. Smolin, "Spin networks in quantum gravity", preprint available in LaTeX form as `gr/qc-9505006`.

If you are more of a mathematician, or less of an expert on quantum
gravity, you might also try a review article I wrote about them, which
starts with a quick summary of what the heck canonical quantum gravity
is about, why it's hard to do, and why the loop representation seems to
help:

4) J. Baez, "Spin networks in nonperturbative canonical quantum gravity", preprint available in LaTeX form as `gr-qg/9504036`, or via ftp from `math.ucr.edu`, as the file [`net.tex`](http://math.ucr.edu/home/baez/net.tex) in the directory `baez`.

Now so far I have been trying to make things sound simple, but here I
should point out that when one talks about "states of quantum gravity"
there are at least three quite different things one might mean. This is
because the loop representation follows Dirac's general philosophy of
quantizing systems with constraints, with some extra twists here and
there. As I've repeatedly explained (e.g. ["Week 43"](#week43)),
Einstein's equation for general relativity has 10 components, and if
you split spacetime up into space and time (more or less arbitrarily
--- there's no "best" way) 4 of these can be seen as constraints
that the metric on space and its first time derivative must satisfy (at
any given time), while the remaining 6 describe how the metric on space
evolves in time (which makes sense, because the metric has 6
components). When you follow Dirac's procedure for quantizing the
equations what you do is this. First you forget about the constraint and
get a big space of states, the "kinematical state space". There are
lots of mathematical choices involved here, but Ashtekar and Lewandowski
came up with a particular nice way of doing this rigorously, and one
calls this space of states "$L^2$ of the space of $\mathrm{SU}(2)$ connections
modulo gauge transformations with respect to the Ashtekar-Lewandowski
generalized measure". Spin networks form an orthonormal basis of this
Hilbert space. All the stuff about area and volume operators above
refers to operators on this space.

Then, however, you need to deal with the constraints. Now 3 of the 4
constraints simply amount to requiring that your states be invariant
under all diffeomorphisms of space, so these are usually dealt with
first, and called the "diffeomorphism constraint". Imposing these
constraints are a bit tricky; naively one would first guess that this
"diffeomorphism- invariant state space" is just a subspace of the
original kinematical state space, but actually it's not quite so
simple. In any event, there are also spin network states at the
diffeomorphism-invariant level, corresponding not to *particular*
embeddings of graphs in space, but to diffeomorphism equivalence classes
thereof. This again has been used by Rovelli, Smolin and others for a
while now, but it was first rigorously shown in the following paper:

5) Abhay Ashtekar, Jerzy Lewandowski, Don Marolf, Jose Mourao, and Thomas Thiemann, "Quantization of diffeomorphism invariant theories of connections with local degrees of freedom", to appear in the November 1995 _Jour. Math. Phys._ special issue on diffeomorphism-invariant field theory, preprint available as [`gr-qc/9504018`](https://arxiv.org/abs/gr-qc/9504018).

This paper is nice in part because it doesn't assume you already have
read every previous paper about this stuff; instead, it describes the
general plan of the loop representation before constructing the
diffeomorphism- invariant spin network states. Also, buried in an
appendix somewhere, it gives nice conceptual formulas for the area and
volume operators, which serve as a complement to Rovelli and Smolin's
explicit computations of their matrix elements in terms of the spin
network basis.

Anyway, after taking care of the diffeomorphism constraint, one finally
needs to take care of the Hamiltonian constraint, meaning one needs to
find states satisfying the Wheeler-DeWitt equation. This is the hardest
thing to make rigorous, and the most exciting aspect of the whole
subject, because it expresses the fact that "physical states" of
quantum gravity are invariant under diffeomorphisms of space-TIME, not
just space. There is much more to say about this, but I won't go into
it here.

Now besides Loll and Rovelli and Smolin, all the authors of the above
paper except Mourao were at the conference in Warsaw, so there was a
large contingent of spin network fans around, not even counting some
other folks whose work I will get to in a while. This is why I was so
eager to go there, especially because my own talk was on a rather
esoteric subject which only these experts could be expected to give a
darn about. Namely....

The breakthrough of Ashtekar and Lewandowski, when it came to making the
loop representation rigorous, involved working with piecewise
real-analytic loops rather than piecewise smooth loops. (Actually
Penrose suggested this idea.) This is because piecewise smooth loops can
intersect in crazy ways, like in a Cantor set, which nobody could figure
out how to handle. But the price of this breakthrough was that one had
to assume the 3-manifold representing space was real-analytic, and
things then only work nicely for real-analytic diffeomorphisms, as
opposed to smooth ones. This always bugged me, so I have been working
away for about a year trying to deal with smooth loops, and finally I
got smart and teamed up with Steve Sawin, and we recently figured out
how to get things to work with smooth loops (at least a bunch of things,
like the Ashtekar-Lewandowski generalized measure). Our paper will be
out pretty soon, but for now anyone who wants a taste of the
mathematical technology involved should look at:

6) Steve Sawin, "Path integration in two-dimensional topological quantum field theory", to appear in the October 1995 _Jour. Math. Phys._ issue on diffeomorphism-invariant field theory, preprint available as `gr/qc-9505040`.

Loop representation ideas are applicable not only to canonical quantum
gravity but also to path integrals in gauge theory, because in both
cases one is doing integrals over a space of connections mod gauge
transformations. Here Sawin uses them to give a rigorous formulation of
2d TQFTs in terms of path integrals. There aren't many unitary 2d
TQFTs, and all of them are isomorphic to $2$-dimensional quantum gravity
with the usual Einstein-Hilbert action, with different values of the
coupling constant, or else direct sums of such theories.

Next "week" I'll talk about cool new idea Smolin has about TQFTs,
quantum gravity, and Bekenstein's bound on the entropy of a physical
system in terms of its surface area.