The following are excerpts from messages from John Baez about this
project. E-mail addresses deleted to reduce spam. Posted with
permission. The quoted text was written by Dan Christensen.
From: baez
Subject: Re: Spin foam continuum limits
To: jdc
Date: Fri, 15 Sep 2000 18:39:45 -0700 (PDT)
Hi -
Here's the basic idea: first one
needs to get good at computing 10j-symbols (a function of 10
spins). Then one needs to take a 4-manifold (the simpler the
better), chop it up into 4-simplices, label the triangles with
spins, compute a 10j symbol for each 4-simplex, multiply them up
for all the 4-simplices, and sum over all labellings. The sum
over labellings is infinite, so really one would do a Monte Carlo
calculation where you randomly pick spins to label the triangles,
and sum over a lot of these random choices. Or else put an upper
bound on the spins being summed over.
In short, you compute
sum_j f(j)
where j runs over labellings of triangles by spins, and f(j)
is the product of 10j symbols over all 4-simplices.
This is not yet interesting.
But THEN you calculate some interesting quantities from these labellings
and instead of merely doing the above sums, you compute weighted
averages of these interesting quantities. In other words, you
compute
sum_j g(j) f(j) / sum_j f(j)
where g is some interesting function of the spins j. If the sums
are infinite the numerator and denominator may separately diverge,
but hopefully the ratio makes sense. That's when you start learning
stuff about physics.
Best,
jb
From: baez
Subject: Re: Spin foam continuum limits
To: jdc
Date: Sat, 16 Sep 2000 11:39:59 -0700 (PDT)
Hi -
> > In short, you compute
> >
> > sum_j f(j)
> >
> > where j runs over labellings of triangles by spins, and f(j)
> > is the product of 10j symbols over all 4-simplices.
> >
> > This is not yet interesting.
>
> With bounded spins, this sounds easy, and that's probably closely
> related to why it's not interesting. :-)
No, that's not why it's not interesting. This quantity is the "partition
function with infrared cutoff" Z(M,J) where M is the triangulated manifold
and J is the upper bound on the spins. Its use is to serve as a normalizing
constant for more interesting calculations, like this:
> > sum_j g(j) f(j) / sum_j f(j)
> >
> > where g is some interesting function of the spins j.
It's just like calculating the center of mass of an 1-dimensional
object by doing
integral x f(x) dx / integral f(x) dx
The denominator simply serves to turn the function f(x) into a
probability distributio f(x)/ integral f(x) dx.
In other words, we're calculating a kind of weighted average of g.
> This also looks easy if the spins are bounded above. But I expect
> that what you are hinting at is that to understand the "correct value"
> of the ratio, a more sophisticated approximation must be done.
No, you just take the ratio for a fixed upper bound on the spins,
say J, and then see what happens as you increase J - with luck, there
it will converge to something.
The reason it's not as easy as you think is this: say you chop a 4-manifold
into 4-simplices with a total of 20 triangular faces, and the upper bound
on the spins is 4.5 (so j = 0, 1/2, 1, ..., 9/2). Then the sum I wrote down
has 10^20 terms in it, and for each term you must do a fairly elaborate
calculation. I don't think even Beowulf can handle this monster. And
this is a puny little spacetime whose total size is on the order of the
Planck length.
> Also, what's an example of an interesting g?
Pretty much any geometrical quantity you might be interested in
for a Riemannian 4-manifold can be turned into a g (in a not-completely-
systematic way). The problem will be finding informative yet still
easily computable ones. Correlations between the area enclosed by
a loop and the holonomy around a loop are one possibility.
> Should one also sum over the various triangulations of the manifold?
> When I first read your message, I thought it said that one had to do
> this, so I worked out an algorithm for generating all triangulations
> of a given manifold.
There are different theories: you can fix a triangulation of a manifold,
or sum over triangulations of a manifold, or sum over triangulations of
all manifolds, or sum over an even larger class of simplicial sets.
> Do you have any idea what
> upper bounds on the spins would be reasonable, and how many
> 4-simplices the triangulations would typically have?
Basically as many as you can handle. :-)
> Are there lower
> dimensional analogues that would also be of interest?
Less interesting but perhaps good warmup since there are lots of
theories where one knows ahead of time what the answers should be!
> I'm going to do some further reading about this and also some rough
> calculations to determine feasibility before deciding whether to
> proceed.
Read Hamber and Williams' paper, cited in TWF. It uses a different
model of quantum gravity but many of the same ideas apply. Also the
paper by Renate Loll, cited in TWF, which is a review article on
discrete approaches to quantum gravity. This is very good.
Gotta run, unfortunately...
Best,
jb
From: baez
Subject: Re: Spin foam continuum limits
To: jdc
Date: Sat, 16 Sep 2000 14:33:17 -0700 (PDT)
Hi -
I'm back...
Here are those references:
Renate Loll, Discrete approaches to quantum gravity in four dimensions,
preprint available as
http://xxx.lanl.gov/abs/gr-qc/9805049
also available as a webpage on Living Reviews in Relativity at
http://www.livingreviews.org/Articles/Volume1/1998-13loll/
and:
Herbert W. Hamber and Ruth M. Williams, Newtonian potential in
quantum Regge gravity, Nucl. Phys. B435 (1995), 361-397.
The latter is nice because it shows how a numerical calculation
was used to determine the "running" of Newton's gravitational
constant G in quantum gravity. This may give ideas on what to
actually compute. On the other hand, the notion of "running
coupling constants" may seem obscure until you know a bit about
renormalization. It did to me, at least, until I wrote this:
http://math.ucr.edu/home/baez/renormalization.html
and this issue of TWF (see below).
Best,
jb
[see link to week 139]
From: baez
Subject: Re: Spin foam continuum limits
To: jdc
Date: Sun, 17 Sep 2000 13:23:40 -0700 (PDT)
Hi -
> You mention one way to make an
> approximation, by using a uniform distribution on the set of bounded
> above labellings. This is of course equivalent to the (non-uniform)
> distribution on *all* labellings which is 0 on those labellings
> with a large spin. Another method which occurs to me is to use a
> distribution on all spins which decays in some way as the spins become
> large, but I see no reason why this would be better. It would be fun
> to compare, though.
Yes, it might matter, and it would be nice to know if it does.
In all quantum field theory problems I've ever heard of, it turns out
that the answer doesn't depend on the details of how you impose
ultraviolet or infrared cutoffs, e.g. sharply or smoothly.
> > Less interesting but perhaps good warmup since there are lots of
> > theories where one knows ahead of time what the answers should be!
>
> That sounds perfect. Do you have a reference for this?
Probably my big fat review article on spin foams, gr-qc/9905087,
is the best place to start. There's a very simple theory called
BF theory which is easily formulated as a spin foam model in any
dimension, but probably most interesting in 2d, 3d and 4d. I talk
about it a lot here and derive a lot of stuff from scratch. This
paper has an annotated bibliography which can help you dig deeper.
> I didn't understand much of what Loll wrote, but it's clear that a lot
> of people have done a lot of computations. He makes no mention of
> spin foam models, though. Has anyone done any numerical computations
> of these models?
Not that I'm aware of. Lee Smolin has a grad student who was supposed
to be doing this, but I don't think he actually did. The people who
understand and like spin foams tend to be sufficiently mathematical
that they keep getting seduced by the lure of proving theorems instead
of doing numerical work, which is a pity in a way. Maybe you're so
much of a mathematician that you wouldn't mind a little numerical work! :-)
Best,
jb
From: baez
Subject: Re: Spin foam continuum limits
To: jdc
Date: Thu, 28 Sep 2000 14:23:37 -0700 (PDT)
Hi -
> Is there a good reference for how to compute (or at least
> define) the 10j symbols?
Yes, though it'll take a while to turn this information into code.
Try:
J.\ W.\ Barrett and L.\ Crane, Relativistic spin networks
and quantum gravity, {\sl J.\ Math.\ Phys.\ }{\bf 39} (1998) 3296--3302.
J.\ W.\ Barrett, The classical evaluation of relativistic spin
networks, {\sl Adv.\ Theor.\ Math.\ Phys.\ }{\bf 2} (1998) 593--600.
J.\ W.\ Barrett, R.\ M.\ Williams. The asymptotics of an
amplitude for the 4-simplex, {\sl Adv.\ Theor.\ Math.\ Phys.\ }{\bf 3}
(1999), 209--214.
and also a paper I'm almost done writing with John Barrett.
The "amplitude for a 4-simplex" is the same as the 10j symbols
by the way.
> (And will you be discussing this in the
> thread on sci.physics.research?)
Yes, but in the Lorentzian case. I think you may want to start
with the Riemannian case for computational purposes, so I've given
you references to that case. After I describe the Lorentzian one
you might want to post an article reminding me to describe how it goes
in the Riemannian case.
Best,
jb