Loop derivative
Eric A. Forgy
I was just reading, "Loops, Knots, Guage Theories, and Quantum
Gravity" Gambini & Pullin. I got through the first 17 pages and found
this loop derivative to be totally mind boggling (in a good way!). I
did a search on the sci.physics.research archive for "loop derivative"
and the following link popped up:
In it, I found a little bit of a disturbing comment:
========================================
From: John Baez (ba...@galaxy.ucr.edu)
Subject: Re: The Wheeler-de Witt equation
Newsgroups: sci.physics.research
Date: 2000/04/13
[snip, regarding the loop derivative]
In early work on loop quantum gravity, folks assumed this derivative
existed for the kinematical states of interest. But then Ashtekar and
Lewandowski constructed a very nice Hilbert space of kinematical
states,
clearly "right" in many ways, but with the unfortunate property that
the
derivative does NOT exist.
Now loop quantum gravity is split into two broad schools, which one
could call the "northern" and "southern" schools. The northern school
uses the Ashtekar-Lewandowski Hilbert space of kinematical states, and
gives up on using loop derivatives. The southern school attempts to
make sense of loop derivatives, and works with a space of kinematical
states with no clear Hilbert space structure. The main exponents of
the southern school are Rodolfo Gambini, Jorge Pullin and their
collaborators, mostly from South America. The northern school
includes Abhay Ashtekar, Carlo Rovelli and Lee Smolin.
[snip]
Most of my work has been done on the nothern approach, but I think
it's too soon to say which approach is better. The southern approach
has a lot to do with the mathematics of "Vassiliev invariants" of
knots, and that raises some very interesting questions, like finding
a nice inner product on the space of Vassiliev invariants - but nobody
knows the answers to these questions. It could be that we are missing
some of the basic math needed to go further in the southern approach.
Maybe we'll figure this stuff out someday.
==============================================
After I read this, I got the sudden fear that what I was reading
wasn't really going to end up to be useful for anything. Is the
assessment above still the general concensus, i.e. are the schools
still split into the "southern" and "northern" approaches? The
material in Gambini & Pullin starting at page 10 with subsections
titled "Tensor character", "Commutation relations", "Bianchi
identities", "The Ricci identity", and "The loop derivative as a
generator of the group of loops" is absolutely beautiful!! It would be
a real shame if you have to throw this stuff out in order to have a
nicely defined Hilbert space. How is the progress coming on the
southern approach to come up with a nice Hilbert space? Are there
equally beautiful analogous properties as those subsection titles of
G&P found in the northern approach?
Eric
John Baez
Eric A. Forgy <fo...@uiuc.edu> wrote:
>I was just reading, "Loops, Knots, Guage Theories, and Quantum
>Gravity" Gambini & Pullin. I got through the first 17 pages and found
>this loop derivative to be totally mind boggling (in a good way!).
Good! And when your mind is sufficiently boggled, you'll be "loopy".
Once upon a time I wrote:
>Now loop quantum gravity is split into two broad schools, which one
>could call the "northern" and "southern" schools. The northern school
>uses the Ashtekar-Lewandowski Hilbert space of kinematical states, and
>gives up on using loop derivatives. The southern school attempts to
>make sense of loop derivatives, and works with a space of kinematical
>states with no clear Hilbert space structure. The main exponents of
>the southern school are Rodolfo Gambini, Jorge Pullin and their
>collaborators, mostly from South America. The northern school
>includes Abhay Ashtekar, Carlo Rovelli and Lee Smolin.
which disheartened Eric Forgy:
>After I read this, I got the sudden fear that what I was reading
>wasn't really going to end up to be useful for anything.
If you're worried about that, you really shouldn't be studying
quantum gravity... you should go to business school. :-)
>Is the
>assessment above still the general consensus, i.e. are the schools
>still split into the "southern" and "northern" approaches?
Yes, with a majority of work taking the "northern" approach, but
a lot of cooperation between the two approaches. By the way, the
center of gravity of the "southern" approach just moved further
south. Jorge Pullin, who had been at the Center for Gravitational
Physics and Geometry at Penn State - just moved to Louisiana, where
he was offered a big fancy position. The reason is that he also
works on LIGO, and one of the two LIGO detectors is based in Livingston,
Louisiana. Together with the fact that Lee Smolin seems almost
certain to move to the new Perimeter Institute in Canada, this
really changes the character of the CGPG. They'll have to hire
a new quantum gravity person. I'm really curious about who it
will be.
But I digress...
>The
>material in Gambini & Pullin starting at page 10 with subsections
>titled "Tensor character", "Commutation relations", "Bianchi
>identities", "The Ricci identity", and "The loop derivative as a
>generator of the group of loops" is absolutely beautiful!!
You're right - it's incredibly nice.
>It would be
>a real shame if you have to throw this stuff out in order to have a
>nicely defined Hilbert space.
Well, this technology is good for Yang-Mills, and in particular
electromagnetism, so even if it's not good for gravity, don't worry -
it won't go to waste. :-)
By the way, this stuff is also part of any decent education in
differential geometry - for mathematicians, not just physicists.
>How is the progress coming on the
>southern approach to come up with a nice Hilbert space?
The southern approach has made good progress in studying the
basic constraint equations in canonical quantum gravity - the
diffeomorphism and Hamiltonian constraints. In this respect it's
quite on par with the northern approach. The results are even
quite similar - the main good and bad features of Thiemann's
"northern" approach to the Hamiltonian constraint have "southern"
analogues.
The big drawback of the "southern" approach is that there's no
obvious candidate for an inner product on the various spaces that
one would want to be Hilbert spaces - the kinematical state space,
the diffeomorphism-invariant state space, and the physical state space.
This means one cannot discuss self-adjoint operators or compute their
expectation values.
>Are there
>equally beautiful analogous properties as those subsection titles of
>G&P found in the northern approach?
There's a very different sort of beautiful stuff in the northern
approach - the spin network basis, the quantization of area and
volume, etcetera.
The two approaches may represent two different "sectors" of quantum
gravity, that is, inequivalent representations of the same algebras
of observables. This phenomenon of different "sectors" is endemic
to quantum field theory, so there's no reason to be shocked by its
occurence in quantum gravity. The issue is just to identify which
sector is most relevant for the physics one is interested in.
Sectors in quantum field theory are mathematically very similar
to "phases" in statistical mechanics: the same basic theory can
describe ice, liquid water, and steam, and it's up to you to say
which you're interested in.
Of course, this is tough before you've worked out the inner product,
since you need that to compute expectation values of observables!
So, even if the southern school is just tackling a different sector
of quantum gravity, they have to work a bit more before they can tell
us what that sector is actually like, physically.
One can ask the mathematicians: is there a natural inner product
on the space of Vassiliev invariants? If you know one, contact
the southern school of loop quantum gravity.
Eric A. Forgy
> In article <3fa8470f.01062...@posting.google.com>,
> Eric A. Forgy <fo...@uiuc.edu> wrote:
> By the way, this stuff is also part of any decent education in
> differential geometry - for mathematicians, not just physicists.
Yeah, I can see why. Maybe in another 50 years or so we'll see it in
standard undergrad diff geo courses :)
I'm going back and working these properties out in gory detail on my
own regarding this loop derivative and I noticed something fishy. The
loop derivative F_{ab} is given by
P[dc o c] = [1 + 1/2 s^{ab} F_{ab}] P[c]
Then, when they derive the commutation relation, they keep terms up to
second order in s^{ab}, but how can you keep terms of order higher
than the original expansion?!?! If you want to keep terms of second
order in s^{ab} in ANY calculation, wouldn't you need to expand the
starting equation to higher orders to get anything meaningful out? For
instance, I'd expect a higher order expansion of the original equation
to be something like
P[dc o c] = [
1 +
1/2 s^{ab} F_{ab} +
1/8 s^{cd} s^{ef} (F_{cd} F_{ef} + F_{ef} F_{cd}) + ... ]
P[c]
However, this (more correct?) expansion would seem to lead to
different commutation relations, unless miraculously a bunch of terms
cancelled out.
I'm not sure if this would affect the Bianchi identity calculation,
since it seems to come from less than second order terms in s^{ab},
but the Ricci identity could conceivable be effected. It seems to me
that it SHOULD be effected to be correct because they have derived
F_{ab} = [D_a,D_b]
where D_a is the Mandelstram covariant derivative. But shouldn't you
end up with
F_{ab} = [D_a,D_b] - D_{[a,b]}
?? Although I'm not exactly sure what [a,b] would mean :) I can make a
guess though: [a,b] = a o b o a^{-1} o b^{-1}?
This stuff is great! I highly recommend it :)
Eric
PS: In a web search for "loop derivative", I found several interesting
things. One, the loop derivative appears in string theory, which isn't
too surprising. Two, there are things called "area derivatives" and
"volume derivatives". Take a look at some of the documents at:
http://www-dft.ts.infn.it/~ansoldi/curriculum/curriculum.eng.html#publications
Is there any nice reference that might discuss loop, area, and volume
derivatives all within the same context?
John Baez
>F_{ab} = [D_a,D_b]
>
>where D_a is the Mandelstram covariant derivative.
Since I don't have the book you're reading with me right now,
you'll have to tell me what sort of things a and b are, and
what a "Mandelstam" covariant derivative is. If you hadn't
written these curious formulas:
>F_{ab} = [D_a,D_b] - D_{[a,b]}
>[a,b] = a o b o a^{-1} o b^{-1}?
I would have guessed that a and b were subscripts running over
a basis of coordinate vector fields. Then D_a could be the plain
old fashioned covariant derivative in the a direction, in which
case
F_{ab} = [D_a,D_b]
would be the usual definition of the curvature tensor,
written out in coordinates. (Note that coordinate vector
fields have vanishing commutators, so there's no extra
term in this formula, the way there is in some other formulas
for the curvature tensor!)
But of course there is no such thing as a commutator [a,b]
of subscripts a and b. The fact that you're running
around writing formulas involving this thing, and talking
about the "Mandelstam" covariant derivative instead of just
the covariant derivative, makes me afraid I'm completely
misunderstanding your notation. Could you possibly be
using a and b to stand for paths, or loops? I'm confused!
Sorry to be of so little help.
>Is there any nice reference that might discuss loop, area, and volume
>derivatives all within the same context?
Not that I know of... but that doesn't mean much.