From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.physics
Subject: Loop variables questions
Date: 17 Nov 92 06:58:33 GMT
Organization: University of California, Riverside

My friend Allen Knutson emailed the following to me, because in the
math department at Princeton (that place beloved to me) the news poster
is DEAD.   My comments are interwoven, marked with an initial JB:

Return-Path: <aknaton@math.Princeton.EDU>
Subject: Quantum GR questions
To: baez@ucrmath.ucr.edu
Date: Mon, 16 Nov 92 20:57:54 EST
Cc: aknaton@math.Princeton.EDU (Allen Knutson)

AK: I must have some deep misunderstanding about the A,R & S quantum gravity
stuff. I would ask you this on the net, but the nnposter is still broken,
dammit. Certainly feel free to answer on the net, if you don't want to
repeat yourself to other people asking about loopy QGR.

First question is about the loop representation of connections. If I
remember right, you said that to a connection and a representation of
the structure group one can associate a number, the trace of the 
holonomy around the loop, and with luck one can reconstruct the connection
uniquely up to gauge equivalence from knowing all these numbers. When
is this the case? Obviously the representation must be faithful.

JB:  Yes, clearly the representation must be faithful or it won't work.

Here's how the loop transform goes again folks.  This will be fairly heavy
going for those not versed in differential geometry, but it's never too late
to learn!

First of all, recall the notion of holonomy.  Suppose we are 
given a vector bundle E over the manifold M with a connection A.  Let
E_x denote the vector space sitting over some point x in M -- E_x is
called the "fiber" of the vector bundle E over the point x.  
Using the connection to parallel translate a vector in E_x around
a loop in M, each loop based at the point x in M gives rise to a holonomy, 
that is, a linear transformation of the vector space E_x.    If we take the
trace of this linear transformation we get a number.  This number doesn't
change if we do a gauge transformation on A.

Now the transform of a given connection A is the function on the space of loops
given by the trace of the holonomy.  So we may think of the loop transform
as a function from the space of connections modulo gauge transformations to
the space of functions on loops.

If for all loops this holonomy lies in a certain subgroup
G of End(E_x) (the space of linear transformations of E_x) we may say that
A is a G-connection.  Given any old Lie group G, we can define the
space of G-connections on E modulo gauge transformations (where we restrict
ourselves to G-valued gauge transformations).  The loop transform can be
regarded as a map from this space to the space of functions on the space 
of loops!  (Whew.)  

When is this one-to-one, AK askes.  I know it is for the defining rep of
SU(2) but not for SL(2,C).  Right now I am confused about the general criterion
for when it is.    I may be screwed up here, but part of what we need
is for the functions

tr(g^n)

to generate an algebra on G/[G,G] that separates points.  It is easy to see
that they do NOT for SL(2,C).  Take the matrices

1 1+a
0 1

These are conjugate for all a > 0 but not for a = 0.  (This is a good exercise -
they are conjugate by an element of SL(2,C), I mean.)   Thus no
continuous Ad-invariant function on SL(2,C) can separate the points

1 2
0 1

and 

1 0 
0 1

even though they are not conjugate in SL(2,C).  This is irritating but
it also implies that *no* continuous gauge-invariant function on the space
of connections (in any reasonable topology) can separate gauge equivalence
classes of connections for this gauge group. 

Hmm, I should reread R. Giles' Reconstruction of gauge potentials from 
Wilson loops, Phys Rev D24 (1981) 2160-2168.

AK: Second, if we have the numbers for knots, why do people want to know/have 
the right to ask for numbers on links? And why is the right answer to demand
the product of the other numbers?

JB: If one had a measure on the space of connections mod gauge transformations,
one could assign a number to any knot, by forming the trace of the holonomy
and then integrating over the space of connections mod gauge transformations.
One could also assign a number to any link, by forming the products of the
traces of the holonomies of each of the components of the link (which are 
knots).  People do both.

AK: Third, it is claimed (if I am reading right) that one of the QG constraints
amounts to saying "The loop functionals must give the same answer on two
isotopic loops". Say we cross a loop through itself, and look at how the
holonomy changes, i.e. smoothly I would have thought.

JB:  In this context (quantum gravity) the space of states can be viewed
as a certain space of "measures" on the space of connections modulo 
gauge transformations.  Measures on this space must satisfy two constraints to 
define states of quantum gravity in the canonical quantization approach:
diffeomorphism-invariance, and the Hamiltonian constraint.  

Now the loop transform can be extended to define a map from the space of
measures on the space of connections mod gauge transformations to the
space of functions on loops!  (First take the trace,
then integrate over the space of connections with respect to your measure.)
In quantum gravity one would like to say that the loop transform of
a diffeomorphism-invariant measure on the space of connections mod gauge
transformations is a link invariant, that is, only depends on the 
ambient isotopy class of the link.  This is true!  But the "measures"
that people are interested in, like the Chern-Simons path integral, are
not really measures in the honest sense.  One must generalize the notion
of a measure, much as one does in the case of linear field theories by
introducing the notion of a "distribution" -- in the sense of the book 
with Segal and Zhou.   I'm working on this now.

AK: Fourth, say we have a state s, i.e. a loop functional, and an area operator.
When we apply the area operator A to the state, we get another loop functional,
such that when we then evaluate As on a loop we count the number of 
intersections. Doesn't that mean that As isn't constant on loop classes,
and thus isn't a state? Oh dear, perhaps I shouldn't be asking this question,
since it's probably founded on so many misconceptions.

JB:  The area operators are not defined in the physical state space
of quantum gravity, in which the diffeomorphism-invariance constraint
has been taken into account!  They are defined in the space of all
"measures on the space of connections mod gauge transformations".

AK: Fifth, I notice that I see Ed Witten's name in the ends of many of these
papers; is it not true then that everybody at Princeton hates this approach
to GR? That'd certainly be encouraging.                         Allen K.

JB: No, it just shows that everyone drops Ed Witten's name.  You're at
Princeton - *you* see if everyone there hates loop variables!   

---
On an unrelated note, email from Alison Chaiken convinces me that my
inability to understand what she had written has nothing to do with
experimentalist/theorist stuff, but only my ignorance of solid state
physics.  Chastened, I will shortly run over to the library and hit the 
books... 
Newsgroups: sci.physics