# The rotations, SO(3), in Loop Quantum Gravity

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this is a collective effort to hit the easy parts of LQG
for anyone who has shown an interest in this approach to
quantum gravity to add constructive input

Lubos Motl has been a critic of LQG and has just published
a clear presentation of it, plus made an important contribution to and he recommends a certain 7 page paper by Rovelli and Upadhya as "an efficient review of Loop Quantum Gravity".

Only 7 pages plus a couple of appendices and it introduces the subject! they call it a LQG "primer".

he also recommends Thiemann LivingReviews for "a more extensive" presentation. I looked at the Rovelli-Upadhya primer and was very impressed by the clarity and conciseness.
I think it is really the best paper to begin with of all I have seen
(except it is 1998 so it uses SU(2) instead of SO(3))

arXiv:gr-qc/9806079

And Motl himself is exceptionally clear and concise in style, so
his recommending Rovelli-Upadhya as a way into the subject carries a lot of weight with me.

Motl:
arXiv:gr-qc/0212096

I don't take Motl's bad-mouthing LQG seriously because he is bright and obviously very interested all of a sudden in LQG
and he knows why and he contributes a solid mathematical result.
His actions speak louder than his heckling.

Motl's paper is also quite short, only 26 pages in all, and if you read it you see immediately that we need to review some things about SO(3).

So I am hoping this thread can do some of that, and also
recap some of the stuff treated in the previous two LQG threads.

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Here is how Rovelli-Upadhya 1998 LQG primer begins its development. Except that I substitute SO(3) in for SU(2):

[[Let M be a fixed three-dimensional compact smooth manifold. ...
Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

We denote by A the space of smooth so(3) valued 1-forms A on M.

The space A equipped with the supremum norm is a topological space. We denote by L the space of
continuous functions on A. Equipped with the pointwise topology, L is a topological vector space...]]

Rovelli is a great guy but he never wrote this IMHO. It must have been Upadhya.

So then Upadhya, if it is he, introduces the cylindrical functions---a special subset of L

And before you know it he has an inner product defined on the cylindrical functions. And presto there is a Hilbert space.
It is all very brief and easy and "comme il faut". No confusion, waste effort or false moves.

Then, in short order, he gets an orthonormal basis for the hilbert space and (within 7 pages) defines the AREA operator on the Hilbert space. Where was this paper when I was trying to read Thiemann! OK Thiemann is very good and careful and thorough but this is good too in a different way.

Now the riddle in this approach up until this year has always been the Immirzi number. That number not being pinned down has been considered a crisis in LQG. Suddenly this year Olaf Dreyer appears to have found out that it is 1/8.088.
Motl says this is the size of the "bare" G as compared with the
macroscopic Newtonian G we are familiar with, and Motl is very interested in this (which is a favorable omen) and he points out as Dreyer also did that it means you have to look at SO(3)

Now as good luck would have it this is the worlds simplest least complex most intuitive Lie group and its Lie algebra is also as simple as can be. So this is actually entry-level! Someone who is curious to learn about classical Lie groups can actually step in here and make it their first adventure along those lines. This is extremely lucky explanation-wise. So I will try to pull together some basic facts.

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SO(3) is a compact Lie group of dimension 3.

Its Lie algebra so(3) is the space of real skew-symmetric 3x3 matrices
with bracket [A,B] = AB - BA.

The Lie algebra so(3) can be identified with R3
the 3-tuples of real numbers by a vectorspace isomorphism
called the"hat map"

v = (v1,v2,v3) goes to v-hat, which is a skew-symmetric matrix
meaning its transpose its its NEGATIVE, and you just stash the three numbers into such a matrix like:

+0 -v3 +v2
+v3 +0 -v1
-v2 +v1 +0

v-hat is a matrix and apply it to any vector w and
you get vxw.

Everybody in freshman year got to play with v x w
the cross product of real 3D vectors (in May I remember someone wrote to PF about freshman physics and rotations and v x w)
and R3 with ordinary vector addition and cross product v x w is kind of the ancestral Lie algebra from whence all the others came.

And the hat-map is a Lie algebra isomorphism

EULER'S THEOREM (just following CalTech's Marsden which is the vanilla ice cream of Lie groups)

Every element A in SO(3) not equal to the identity is a rotation
thru an angle &theta; about an axis w.

SO SO(3) IS JUST THE WAYS YOU CAN TURN A BALL---it is the group of rotations (this will drive chroot nuts since he is used to much harder stuff, its really basic)

THE EIGENVALUE LEMMA is that if A is in SO(3) one of its
eigenvalues has to be equal to 1.
The proof is just to look at the characteristic polynomial which is of degree three and consider cases.

Proof of Euler is just to look at the eigenvector with eigenvalue one----pssst! it is the axis of the rotation. It takes three sentences to prove.

A CANONICAL MATRIX FORM to write elements of SO(3) in
is

+1 +000 +000
+0 +cos&theta; -sin&theta;
+0 +sin&theta; cos&theta;

For typography I have to write 0 as +000
to leave space for the cosine and sine under it
maybe someone knows how to write handsomer matrices?

EXPONENTIAL MAP
Let t be a number and w be a vector in R3
Let |w| be the norm of w (sqrt sum of squares)
Let w^ be w-hat, the hat-map image of w in so(3), the Lie algebra. Then:

exp(tw^) is a rotation about axis w by angle t|w|

It is just a recipe to cook up a matrix giving any amount of rotation around any axis you want.

Hope no one thinks it is bad taste to go over such elementary stuff. I think it is nice and I believe that historically it is the
paradigm of Lie groups and Lie algebras---a kind of ancestor of the huge proliferation of groups in physics we have today.

Anyway, Dreyer found that this ancestor belongs in LQG
---instead of the more usual SU(2). So now I or somebody has to tell that story and say what
SO(3) does in Loop Quantum Gravity

hint: connections...and how tangent vectors get rotated by parallel translation around loops

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rutwig
Quite interesting overview. Although the facts about so(3) and su(2) are quite standard, I don't know whether other members will find it as evident as it is (basing on past experience). We will see if anyone gives some answer.

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Originally posted by rutwig
Quite interesting overview. Although the facts about so(3) and su(2) are quite standard, I don't know whether other members will find it as evident as it is (basing on past experience). We will see if anyone gives some answer.

Yes rutwig, we will see!

and I think you might approve of my adding here that
SO(3) is diffeomorphic to the real projective sphere RP3

proof: about any axis there are always two senses in which
one can turn the ball to get the same effect.

wait, one should be more deliberate----they are both diffeomorphic in fairly obvious ways to the solid ball in R3 with antipodal points on the boundary identified. QED

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it now seems quite possible that the LQG people have discovered the quantum of area and the size of the bare gravitational constant G---many people I guess, including
an outspoken critic Lubos Motl, now see this.

What I am describing is something else: namely that I am just beginning to see how "fast" the theory is. One can go thru it like a hot knife thru butter and in a few pages one can get to the premier results---the area operator and the Immirzi number 1/8.088.
It is a "streamlined" mathematical theory and this (according to an age-old prejudice going back perhaps to Pythagoras )
could be a good omen.
It seems that when the theory was first being developed it was not so efficient.

Here is a partial recap of the initial segment, following Rovelli-Upadhya's primer. I refer to them collectively as Upadhya because I like the name.

Let M be a fixed three-dimensional compact smooth manifold. ...
Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

We denote by A the space of smooth so(3) valued 1-forms A on M.

The space A equipped with the supremum norm is a topological space. We denote by L the space of
continuous functions on A. Equipped with the pointwise topology, L is a topological vector space.

CYLINDRICAL FUNCTIONS

Let f be a function defined on SO(3)n----the cartesian product of many copies of the group of rotations. Confidentially it is going to play the role of "trace"---a numerical function defined on matrices by summing the diagonal. But we want it to be defined more generally----on an n-tuple of group elements.

Let &Gamma; be a network of n piecewise analytic curves &gamma; meeting at nodes denoted "p" if we need to mention them. &Gamma; is simply a graph embedded in the 3D manifold M.

Now the cylindrical function &Psi;&Gamma;,f

is the following beautiful and sexy object. It is defined for every connection A in A.

So you choose a connection A for it to work on and what do you do? You run parallel transport on each leg of the graph.
That gives you n ways to roll the ball!

There are n legs "&gamma;" so by running "holonomy", which is a faintly pretentious word for parallel transport, you get n elements of SO(3). So...you just apply f(...) to them and presto you have a plain old number.

this is really boiling it down fast. You start with a delicate complicated thing A, a connection expressing the curvature of the manifold and useful for transporting tangents from one point to another.
And you hit this connection A with the cylindrical function
&Psi;&Gamma;,f and bang you have a plain old number, written &Psi;&Gamma;,f(A).

It is suggestive that these network-based cylindrical functions span the function space, L, described earlier ----to me privately it confirms my respect for Roger Penrose who had a hunch that networks were the essence of space---space is all the possible networks. He had that hunch a long time ago. Now I am seeing that some network-based &Psi; functions, which will turn out to be quantum states, span a certain interesting linear space. Excuse me if I sound excited---I am, just now.

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Originally posted by jeff
I wouldn't mind seeing that paper. Where did you get it?

From arXiv-----

arXiv:gr-qc/0212096

Jeff you have undoubtably already seen his website---he is a cool guy with a sense of humor---anybody else who hasnt yet, just google.

Dont be put off by the title of his paper!

"An analytical computation of asymptotic Schwarzschild quasinormal frequencies."

It is really a clearsighted assessment of LQG from an outsider and sometimes critic.

maybe I will get links later to the arXiv paper and his site, but I think you (Jeff) most likely can go from here

I got them and edit them in here:

http://www.arxiv.org/abs/gr-qc/0212096

http://www.arxiv.org/abs/gr-qc/9806079

personally I like olives, or the branches anyway
this might be a favorable combination of people
jeff, sauron, instanton, rutwig, chroot---- if they stay around
or at least persist in dropping in

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to the extent that it is possible to ask help from rutwig this might be a point where we could use advice

In the Upadhya primer it is 1998 and they use SU(2) and
the function space L
is complex valued functions on the connections
and each cylindrical function &Psi; is defined
using some function f taking values in the complex numbers.

The inner product on cyl. functions is easy to define
using the haar measure of the group,
and one gets a complex hilbert space.

But here we don't have to use complex valued functions if
we don't want to (but very likely for future convenience we
will want to!). Still, for the initial construction this could be a real Hilbert space.

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Extending the initial segment to inner products:

Here is a partial recap of the initial segment, following Rovelli-Upadhya's primer. I refer to them collectively as Upadhya because I like the name.

Let M be a fixed three-dimensional compact smooth manifold. ...
Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

We denote by A the space of smooth so(3) valued 1-forms A on M.

The space A equipped with the supremum norm is a topological space. We denote by L the space of
continuous complex valued functions on A. Equipped with the pointwise topology, L is a topological vector space.

CYLINDRICAL FUNCTIONS

Let f be a function defined on SO(3)n----the cartesian product of many copies of the group of rotations. Confidentially it is going to play the role of "trace"---a numerical function defined on matrices by summing the diagonal. But we want it to be defined more generally----on an n-tuple of group elements.

Let &Gamma; be a network of n piecewise analytic curves &gamma; meeting at nodes denoted "p" if we need to mention them. &Gamma; is simply a graph embedded in the 3D manifold M.

Now the cylindrical function &Psi;&Gamma;,f
is the following beautiful and sexy object. It is defined for every connection A in A.

&Psi;&Gamma;,f(A) = f(... U(&gamma;i, (A)...)

You choose a connection A for it to work on and what do you do? You run parallel transport on each leg of the graph.
That gives you n group elements
Ui = U(&gamma;i, A)
Then you apply the function f to those n group elements
To make it a little easier on the eyes,
&Psi;&Gamma;,f(A) = f(...Ui...)

So given any network with n legs, you run parallel transport on each leg and get n elements of SO(3) and apply f(...) to them and voila a complex number.

Networks really can feel any connection A out and tell everything about it. Because doing parallel transport around loops shows what any connection is made of. So these network-based &Psi; functions are a fully adequate to generate the function space. Actually the only drawback is that they are more than you need. They span but arent linearly independent. So a later refinement "spin networks" is a way of selecting out a linearly independent basis for the space.

DEFINING THE INNER PRODUCT OF TWO CYLINDRICAL FUNCTIONS

Now suppose we have two of these psi functions

&Psi;&Gamma;1,f(A) = f(...Ui...)

&Psi;&Gamma;2,g(A) = g(...Ui...)

We merge the two graphs into a combined piecewise analytic graph &Gamma; with some larger number of legs, say n.
We define a new f and g on SO(3)n to just be equal to the old f and g where they were defined and otherwise zero. It is a trivial construction just to get an integral to be defined. So then the inner product is this:

(&Psi;&Gamma;1,f, &Psi;&Gamma;2,g) =

&int; f*(...Ui...)g(...Ui...) dU1...dUn

There is a uniform measure on SO(3) which is just what you think it is, uniformly spread out, that is called "Haar" measure and the integral is a straightforward multiple integral with that uniform measure on the group.

**********

Now we have the inner product and the Hilbert space of quantum states, except for some technical proceedures to get a nicer basis.
So we should pause and think about where we are going, which is the area operator.

The area and volume operators have discrete spectrum which is to say that there are steps of area, for example, and that it is quantized. And Olaf Dreyer found out that the size of the step is 4 ln(3) times the conventional Planck unit area. Have to keep this target in sight.

I'll try posting this to see how it looks

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Originally posted by jeff
Marcus,

Just so there's no confusion it's me steinitz. I just didn't like being called "steinitz".

I found a paper yesterday published in january of 2003 in which a by now well-known but inconclusive argument is made that SU(2) can be restored without hurting lubos's result. But for some stupid reason today I can't find it. If you have any time between doing all that typing, see if you can locate it. I think it's important.

Jeff,

any keyword would help, any author's name or word in the title
you think might help find it? hope it turns up.
was it discussed by anyone on Usenet? might find it thru
a usenet thread. thank you for mentioning it. don't think
I will go hunting for it myself but wait to see if it turns up here at PF---either that or some keyword clues

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We should get the goal in clearer focus
Here is a short quote from near the front of the thread
this post is going to spell stuff out very methodically and
deliberately----many if not all readers will have noticed
these things already----but I will still shift down into low gear
to include as many potential readers as I can.

Originally posted by marcus

...Now the riddle in this approach up until this year has always been the Immirzi number. That number not being pinned down has been considered a crisis in LQG. Suddenly this year Olaf Dreyer appears to have found out that it is 1/8.088.
Motl says this is the size of the "bare" G as compared with the
macroscopic Newtonian G we are familiar with,...

The two numbers that LQG has unexpectedly come up with are this 8.088 and the area quantum 4 ln3

the numbers are exact predictions---I just happen to write the 8.088 as if it were approximate. The physical meaning of the 4 ln 3 number is that area is quantized in steps which are that number times the conventional Planck area.

If you multiply 4 ln 3 by 8.088 and divide by sqrt 2, you get 25.13. This 25.13 is the 8 pi that appears in the Einstein equation of GR. So these two numbers 8.088 and 4 ln 3 are related by the theory. Originally these numbers seem to have arisen semiclassically from Bekenstein/Hawking black hole entropy and Hod's black hole vibration frequencies, later refined by Motl. Black holes have always motivated quantum gravity development ever since Bekenstein 1974. To sum up:

1/8.088 GNewton is the bare gravity constant Gbare

4 ln 3 Plancklength2 is the smallest step of area

The LQG area formula (in form given by Motl) is as follows. There is an orthogonal basis for the Hilbert space which consists of a countable set of spin network states.
Given any 2D surface S in the 3D manifold M we can see which of the spin networks intersect the surface and sum over those that intersect with it! Typically an intersection contributes sqrt 2 to the summation &Sigma;. What people call the LQG area formula is essentially this, with a symbol for the Immirzi parameter in place of the dorky 1/8.088 which it is my irritating custom to write.

AreaS = 8pi (1/8.088) GNewton &Sigma; sqrt 2

I guess you could also write this as

AreaS = 8pi Gbare &Sigma; sqrt 2

And also 8pi (1/8.088) sqrt 2 = 4 ln 3.

I should say that although I am reporting what seems to me innovative work by people I respect--a good many things here
baffle me and the idea of quantizing area runs contrary to my intuition and prejudices. Good luck to any who can understand these matters

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rutwig

Originally posted by marcus
In the Upadhya primer it is 1998 and they use SU(2) and
the function space L
is complex valued functions on the connections
and each cylindrical function &Psi; is defined
using some function f taking values in the complex numbers.
The inner product on cyl. functions is easy to define
using the haar measure of the group,
and one gets a complex hilbert space.
But here we don't have to use complex valued functions if
we don't want to (but very likely for future convenience we
will want to!). Still, for the initial construction this could be a real Hilbert space.

When I have a close look to the reference given I will probably give a more concise answer, but the main reason to work usually on the complex is because of the representation theory of su(2). Since it is the compact form of usual sl(2,C), all complex representations are recovered from it. Now these representations of su(2)=so(3) (isomrophisma at Lie algebra level) are also complex, and this is the classical source for the failure of the exponential mapping. If the representation is integer-spin, then we can find a transformation such that the induced representation is also real, but for half-spin-integer representations this is no longer true, and the corresponding real representation is of double size. So for example the spin j=1/2 representation of su(2) is complex, and to obtain the corresponding real one we have to consider a four dimensional space. However, thanks to some classical results, if we work on C we can later adapt all to the real case, without being forced to make the previous distinction. This is why usually the complex is more comfortable.

P.S: If there is some question about convergence, the thing is more serious, because conclusions on R can easily become false.

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a little eureka about labeling the graphs

rutwig thanks, your occasional comment is more helpful than you may realize.

I had a small eureka this morning about how to do "spin networks" (essentially embedded graphs labeled by representations of the group) in the SO(3) case.

One can simply use the double cover and consider the SO(3) representations used to label the edges as "pulled back" to be SU(2) representations.

So the labels are merely drawn from a particular restricted class of labels. And then one can proceed as before to construct the quantum state corresponding to that particular network.

I have nothing more of any concern to say at the moment and my wife is insisting that I go get gas for the car----that is, she is reminding me that it is necessary that this be done.

But I will try to get back soon and carry the exposition a little further.

BTW what is the basic reason to move to "labeled network" states, when we already have a nice collection of "cylindrical functions" defined earlier, which span the hilbert space. The answer is this, I think:

Network labeling is a clever way of using the graph &Gamma; to restrict the choices of the function f. In the previous construction
of cylindrical functions, f was allowed to be essentially anything.
That introduces a lot of redundancy.

Now we are going to fix on a graph &Gamma; and set up some rules for labeling it and consider all the different ways we can label it (conforming with those rules, which restrict the possibilities) and then derive a numerical valued function from each legal labeling. This way we get an efficient selection of functions and avoid the redundancy.

So the "labeled network" (aka "spin network") states will
turn out not only to span the Hilbert space but to provide an orthogonal basis. I will have to edit this later.

***************************

Extending the initial segment to inner products and then to "spin" or labeled networks:

Here is a partial recap of the initial segment, following Rovelli-Upadhya's primer.

Let M be a fixed three-dimensional compact smooth manifold. ...
These things always have an analytic structure available if one needs it, so we might sometimes use piecewise analytic embeddings. But the basic idea is just a compact smooth manifold. Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3).

We denote by A the space of smooth so(3) valued 1-forms A on M.

The space A equipped with the supremum norm is a topological space. We denote by L the space of continuous complex valued functions on A. Equipped with the pointwise topology, L is a topological vector space.

CYLINDRICAL FUNCTIONS ("cylindrical" is just a time-honored customary terminology, they don't look at all cylindrical to me but I have to conform to other people's usages)

Let &Gamma; be a network of n piecewise analytic curves &gamma; meeting at nodes denoted "p" if we need to mention nodes. &Gamma; is simply a graph embedded in the 3D manifold M.

Let f be a function defined on SO(3)n, the cartesian product of many copies of the group of rotations. It is incredibly general---pretty much any nicely behaved function defined on n-tuples of group elements.

Let's define a quantum state of gravity (!)
The cylindrical function &Psi;&Gamma;,f
is the following beautiful and sexy object, defined for every connection A in A.

&Psi;&Gamma;,f(A) = f(... U(&gamma;i, A)...)

You choose a connection A for it to work on and what do you do? You run parallel transport on each leg of the graph.
That gives you n group elements
Ui = U(&gamma;i, A)
Then you apply the function f to those n group elements
To make it a little easier on the eyes,

&Psi;&Gamma;,f(A) = f(...Ui...)

So given any network with n legs, you run parallel transport on each leg and get n elements of SO(3) and apply f(...) to them and voila a complex number.

Networks can feel any connection A out and tell everything about it, because doing parallel transport around loops tests out the connection and detects curvature. These network-based &Psi; functions represent enough information to generate the function space. Actually the only drawback is that they are more than you need. They span but arent linearly independent. So a later refinement, "spin" or labeled networks, is a way of selecting out a linearly independent basis for the space.

DEFINING THE INNER PRODUCT OF TWO CYLINDRICAL FUNCTIONS

Now suppose we have two of these psi functions

&Psi;&Gamma;1,f(A) = f(...Ui...)

&Psi;&Gamma;2,g(A) = g(...Ui...)

We merge the two graphs into a combined piecewise analytic graph &Gamma; with some larger number of legs, say n.
We define a new f and g on SO(3)n to just be equal to the old f and g where they were defined and otherwise zero. It is a trivial construction just to get an integral to be defined. So then the inner product is this:

(&Psi;&Gamma;1,f, &Psi;&Gamma;2,g) =

&int; f*(...Ui...)g(...Ui...) dU1...dUn

There is a uniform measure on SO(3) which is just what you think it is, uniformly spread out, that is called "Haar" measure and the integral is a straightforward multiple integral with that uniform measure on the group.

**********
"SPIN" OR LABELED NETWORKS

Hurkyl helped me get to my present stage of understanding of the kind of labeled network &Gamma;, j, v which is described here. (H. not responsible, however, for errors.)

We now have the inner product and the Hilbert space of quantum states, but we need to go through a technical proceedure to get a nicer basis.

A "spin" or labeled network is a graph which is designed and equiped to self-destruct, when you give it a connection, and yield a number. It collapses by a great crashing tensor contraction.
It has to be set up right to do this.

We fix on a graph &Gamma; and consider all the possible ways we can "color" the edges and nodes. The edges will be labeled (or "colored" as they sometimes say) with representations of the group and the nodes will be labeled with multilinear forms on the representation spaces.

There is going to be a new psi function defined on the configuration space which is A the space of connections.

&Psi;&Gamma;, j,v

Here the ji label the edges----i = 1,...,n---with reps.
And the vr label the nodes----r = 1,...,m---with multilinear forms.

This is all just a plot to obtain a number! We are going to grab a connection A out of the configuration space A and evaluate &Psi; on it. The connection A will give us a group element by running parallel transport along any edge. Then the rep will interpret that group element as a linear operator on a vector space (the space of the representation).

Each node with valence k will give us a k-linear form. And the whole works consisting of the operators and multilinear form will collapse down to a number. So there will be a way to evaluate &Psi;

HURKYLS TRIANGLE EXAMPLE

Consider a network which is simply a triangle with three nodes, each of valence 2, and three edges. Suppose the nodes are labeled with 2-linear forms (3 x 3 matrices) L,M,N and that the representations, applied to the group elements resulting from parallel transport by the connection, give linear operators X,Y,Z.
Then the tensor contraction of the whole shebang is

trace(LXMYNZ)

Maybe we should consider the labeling of the graph in a bit more detail, but this is the rough idea anyway.

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Originally posted by marcus

This short summary of LQG follows Rovelli-Upadhya's primer.
However because of Dreyer's result SO(3) is preferred to SU(2)
in some places.

This installment recaps the preceding and tries to go a bit farther. I have edited it to incorporate some valuable clarification by Hurkyl.

Let M be a fixed three-dimensional compact smooth manifold.
Such things have an analytic structure available if one needs it, so we might sometimes use piecewise analytic embeddings. But the basic idea is just a compact smooth manifold. Let A be an SO(3) connection on M: that is, A is a smooth 1-form with values in so(3), the Lie algebra of SO(3). We denote by A the space of smooth so(3) valued 1-forms A on M.

The space A equipped with the supremum norm is a topological space. We denote by L the space of continuous complex valued functions on A. Equipped with the pointwise topology, L is a topological vector space. Using "cylindrical functions" an inner product has been defined, making a Hilbert space of quantum states of gravity.

LABELED NETWORK STATES

A labeled network, in this context, is a graph which is equiped to self-destruct, when you give it a connection, and yield a number. It collapses by a great crashing tensor contraction.
It has to be set up right to do this.

We are going to fix on a graph &Gamma; and set up some rules for labeling it. We consider all the different ways it can be labeled conforming with those rules, which restrict the possibilities and allow us to derive a numerical valued function. This way we get an efficient selection of quantum states of gravity and avoid the redundancy.

The labeled network states will turn out not only to span the Hilbert space but to provide an orthogonal basis.

The edges of the graph &Gamma; will be labeled (or "colored" as they sometimes say) with representations of the group and the nodes will be labeled with multilinear forms on the representation spaces.

There is going to be a new psi function defined on the configuration space which is A the space of connections.

&Psi;&Gamma;, j, v

Here the ji label the edges----i = 1,...,n---with reps.
And the vr label the nodes----r = 1,...,m---with multilinear forms.
Here &Gamma; has m nodes and n edges.

This is all just a plot to obtain a number! We are going to grab a connection A out of the configuration space A and evaluate &Psi; on it. The connection A will give us a group element by running parallel transport along any edge. Then the rep will interpret that group element as a linear operator on a vector space (the space of the representation).

Each node with valence k will give us a k-linear form. And the whole works consisting of the operators and multilinear form will collapse down to a number. So there will be a way to evaluate &Psi;

HURKYLS TRIANGLE EXAMPLE

Consider a network which is simply a triangle with three nodes, each of valence 2, and three edges. Suppose the nodes are labeled with 2-linear forms (3 x 3 matrices) L,M,N and that the representations, applied to the group elements resulting from parallel transport by the connection, give linear operators X,Y,Z.
Then the tensor contraction of the whole shebang is

trace(LXMYNZ)

A FIXED CHOICE OF REPRESENTATION MACHINERY

A choice of irreducible representations of the group is made once and for all at the outset---finite dimensional vectorspaces with inner product, with linear operators (matrices) to represent elements of the group.

These can be unrelated to the Hilbert space of quantum states. But the finite dimensional vectorspaces on which the reps act are themselves Hilbert spaces, with inner product. So we have one big Hilbert space of quantum states defined on the connections---and also a whole bunch of little finite dimensional Hilbert spaces defined on the side, which are just machinery to crank out numbers with.

The whole reason for this is that going around loops with parallel transport ROTATES tangent vectors (that is what curvature is about) and we need ways to boil rotations down to plain old numbers so we can define numerical valued functions on our space of connections. At least for the moment, that is why this extra "irreducible representations" machinery is sitting around.

Any representation of SO(3) can be considered a representation of SU(2) by the covering map.

A little notation:
Hi is the finite dimensional Hilbert space which the irreducible representation ji acts on.
All that "irreducible" means is that Hi is not any bigger than it has to be----it doesn't have a nontrivial subspace left invariant---everything in it moves under the ji action, except the zero vector.

Now we assume that all the edges of a graph &Gamma;,embedded in the manifold M, have been labeled with irred. reps ji and we proceed to the nodes. We look at some k-valent node in the graph, call it p, and the k edges that meet a p. There will be a subset of indices Ip that tells which edges &gamma;i meet there.
And the set of reps will be {ji for i &epsilon; Ip}

The crafty Upadhya, with Rovelli looking over his shoulder, tells us to take the tensor product of all the finite dimensional hilbert spaces {Hi for i &epsilon; Ip} and to define Hp to be the subspace invariant under the combined action {ji for i &epsilon; Ip}.

Upadhya discusses how to ensure that this subspace Hp is non-trivial and he assumes that an orthonormal basis has been chosen for it ahead of time once and for all. That probably should have been mentioned at the beginning.

Now we can write a labeled graph &Gamma; ji, vp
or more simply &Gamma;, j, v
where ji is an irreducible rep labeling each edge &gamma;i, and
and vp, is a chosen for each node p from the basis of Hp.

Now at last we can write the new quantum gravity state
based on the labeled graph &Gamma;, j, v

&Psi;&Gamma;, j, v (A)

this is a numerical valued function of connections A where
you get the number by a gigantic orgasmic tensor contraction.

The recipe for this well-nigh catastrophic tensor contraction is to first run A on each edge to get a group element. And then apply the label (a group rep) to get a linear operator. So now each leg of the graph has an operator sitting on it.

And each node, you suddenly notice, has a toad sitting on it (no I mean a k-linear form). You snap your fingers and everything begins eating everything else. The edges disappear as their operators apply themselves to the multilinear forms---producing new multilinear forms---and the nodes disappear as those are eaten in turn by other operators. The network "contracts" or consumes itself until finally the only thing left is a number. This number is the value of the function

&Psi;&Gamma;, j, v

on the connection A.

Care must be taken to set up the graph properly so there are no loose ends that might cause uneaten scraps to be left over! In fact some of the review articles tell you to manipulate the graph first so all the nodes are "tri-valent" -----have 3 edges meeting at them. According to Upadhya one may eliminate all univalent and bivalent nodes without loss of generality. And the labeling around trivalent nodes should satisfy "Clebsh-Gordan conditions" which ensure there is one and only one possible choice of vector at such a node. But rather than get into such fine detail, I want to stop here with the unrigorous and figurative image of the labeled network, once it has been provided with a connection to use in parallel transport along its links, consuming itself and producing a number.

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Staff Emeritus
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I wouldn't be surprised if I misunderstood the paper, but didn't it say that the labels for each node were multilinear forms? The labels on the nodes would be vectors only if the node had only one edge incident with it... in general a node with order n would be labelled with an n-linear form.

For example, suppose the graph is simply a cycle with 3 nodes. Each node has 2 edges incident upon it, so it would be labelled with a bilinear form. If L, M, and N are matrices representing the bilinear forms at each node, and X, Y, and Z are matrices representing parallel transport in the connection A along the edge joining the nodes labelled L & M, M & N, and N & L respectively, then:

&Psi;(A) = trace(LXMYNZ)

It is just a scheme for yielding a number, but it's not as simple as you seem to make it.

edit: I wrote this before I saw your most recent post

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You are right that the nodes are labeled by multilinear forms!
Yet for some reason in several places Upadhya refers to the
labels on the nodes as "vectors". This must mean that he
thinks of the multilinear forms as a vector space and is able
to call a large tensor product a "vector" without his hair falling out. Your post helps much to clarify. Glad to know at least
someone read my latest attempt to cover their LQG primer.

Originally posted by Hurkyl
I wouldn't be surprised if I misunderstood the paper, but didn't it say that the labels for each node were multilinear forms? The labels on the nodes would be vectors only if the node had only one edge incident with it... in general a node with order n would be labelled with an n-linear form.

For example, suppose the graph is simply a cycle with 3 nodes. Each node has 2 edges incident upon it, so it would be labelled with a bilinear form. If L, M, and N are matrices representing the bilinear forms at each node, and X, Y, and Z are matrices representing parallel transport in the connection A along the edge joining the nodes labelled L & M, M & N, and N & L respectively, then:

&Psi;(A) = trace(LXMYNZ)

It is just a scheme for yielding a number, but it's not as simple as you seem to make it.

edit: I wrote this before I saw your most recent post

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Hurkyl, this usage of "vector" when one ought
to say multilinear form is bad and I should edit it out
and give you credit.

Before I get around to it, you may have other suggestions!

Let me know of any paragraphs that you think should be
improved or scrapped.

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Gold Member
Yah, the set of n-linear forms with the usual addition and scalar multiplication forms a vector space, so it is accurate to call them vectors, if confusing! It's probably better to call them multilinear forms to avoid confusion until we're more comfortable with the idea.

I'm going to do some paraphrasing of the first initial ideas, both because I think I can put it into simpler concepts, and to make sure I really understand what I think I do.

For the sake of simplicity, let's presume that the graph in question, &Gamma;, lies entirely inside a single coordinate chart. If j is a link in our graph, then the idea of parallel transport along j simply becomes the rotation a traveller experiences (with respect to the coordinate chart) as he traverses the link.

Now, if f is a function of n rotations and our graph has n links, then we can form the pair:

&Psi;&Gamma;,f(A)

which simply applies the function f to the rotations corresponding to the individual links. Of course, f has to be suitably defined so switching coordinate charts doesn't change the resulting value.

Hypothesis: Can we completely ditch the manifold at this point and abstract the idea of a connection merely to something that acts on edges to give rotations?

I'm not up on representation theory, so I don't have too much to say about the spin networks... but the idea of "consuming" the graph to compute the big tensor product was quite informative. I was having a hard time swallowing just what it could mean, but iteratively performing pieces of the product as you shrink the graph made more sense (and I think I can go from there to explicitly writing the effect that the graph manipulations have on the actual product).

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Originally posted by jeff
Marcus, what's the code for using upper and lower case indices in posts?

I believe one just says [ sub ] and writes whatever one wants and then [ / sub ]

So that one can put whatever one wants, upper or lowercase, in for the subscript.

But I think that there is only one level of subscripts. And likewise for super. Let me know if you discover more, please.

I am writing spaces in [ sup ] to fool it but you eliminate the spaces to get it to work.

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Originally posted by jeff
Different levels of subscript and superscript are achieved by nesting as in the following examples:

R[ sup]j[ sub]i[/sub][/sup] gives Rji

R[ sub]j[ sub]i[/sub][/sub] gives Rji

thanks Jeff
BTW do you happen to know a code for tensor product
that would work here at PF?

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Originally posted by jeff
Yeah, in fact I struck the motherlode of unicode characters.

With perhaps one borderline inconsequential technical qualification, this idea is actually correct! How did this bit of insight occur to you, because you guessed it without much basis, and then treat your guess as fact without further justification. Again, note that this is not criticism, it just threw me for a loop.

Thanks for trying with the unicode!

I am actually extracting from several different surveys and primers (mostly by Rovelli, Baez, Thiemann) as well as this
At the moment the most useful is the LivingReviews article by Rovelli alone.

Though combining sources, the basic outline I am trying to follow at least up to this point is the Rovelli-Upadhya primer----supplemented with
whatever insight I can get from other sources.

As introduced in the first post, this is open to collaboration. So far there has been helpful input about LQG from Sauron, Instanton (in other related threads) and Hurkyl.
Instanton (in another thread) gave a thumbnail sketch of how LQG is developed. Chroot has commented.
I would be happy if the work of trying to go thru the key review articles and introductory expositions (primers) was widely shared.
Probably no one reads these threads except people who are trying to do the same thing, namely teach themselves some LQG.

The main thing is not to waste time criticising. If you think I have a wrong interpretation of some page of Rovelli then make your own interpretation and post it in a "no-fault" way. The readers can choose for themselves or mix their own blend of interpretations.

Hard enough to make progress here, don't want to waste thread arguing.

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the mentor already said stop bickering
you seem to want to trash LQG threads
by empty criticism (picking nits out of context)

Originally posted by jeff
Any object, not just tangent vectors (which by the way won't in general remain tangent to the curve along which they're transported).

this is condescending. I know this and was not talking about tangent vectors of the curve. tangent vectors to the manifold get mapped into tangent vectors

the remark seems pointless

Actually, this is just a rotation about one axis. The full matrix involves three angles of rotation and is considerably more complicated. A good place to look it up is in any good classical mechanics text.

I was giving Corollary 9.2.8 on page 291 of Marsden's textbook.
His chapter 9 is an excellent treatment of Lie groups. Your remark is incredible. Of course I know it is rotation around one axis. That is the point of Euler's theorem---that an orthogonal matrix with det=1 can be expressed as rotation around one fixed axis.
(Theorem 9.2.6 in Marsden)

Your point has no sense that I can see---takes what I was describing out of context so it is not comprehensible---and seems to be insulting. why tell me irrelevant detail that I know already in a way that seems to contradict?

Lie group elements are obtained not from the exponentiation of vectors but of the generators of the associated lie algebra, these being square matrices, as is necessary for exponentiation to make sense mathematically.

Yes I know this, what is the point of your saying it?

Oh, I see. You did not read what I wrote. Marsden defines a square matrix v^
or "vee-hat" by what he calls the "hat map" and exponentiates it.

Marsden teaches physics majors at CalTech their Lie groups and is a world-class author on dynamical systems. If he wants to define a "hat map" and thinks its useful I will go along with it---in fact it does seem like a good mapping between vectors and matrices to know! Apparently it is unfamiliar to you and you thought you had to tell me that exponentiation is
applied to the square matrices of the Lie algebra!
This is extremely insulting and condescending.

I think you must be being intentionally rude.
and I think you should control the impulse.
You have now trashed another Loop Quantum Gravity thread.

Holonomy quantifies the effect of parallel transport. [/B]

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Sauron
I liked to read these thread and i don´t know why it would stop.

I have a question, or two.

you have explained what a network is. Now i think it is a ogod idea to explain what it represents.

if we restric to pure gravity it represents an state of the gravity field (ok, its obvious9. It´s evolutions is given by the action of it of the canonical constraints.

Later it need´s to be adressed the question of which operators could act on hese spin network which would represnt physical quantities and the dificoulties to construct them.

But, what about the non gravity sector? Suposedly now the nodes and vertex would allow states which reprsent matter?, that´s is the reason the SO(2) vs SU(2) debate afther all.

The question is, which constraints would dictate the volution of hese spin networks. Would they be derived form the matter lagrangian?

Theretically yes, because they would somwhat represnt the energy-momentum side in the classical counter part.

Another thing, these regarding strings in LQG/LQG2.

Are totally imcopmatible both therys (Strings and LQG) in the gravity sector.
I guess not.If we aceept the spin-2 particle of strings as a graviton we could include it action in LQG as a variation to the spin network.

Let's say strings would be valid for nonperturbative aspects of quantum gravity but the wholle non perturbative theory would still be LQG.

I´ll explain why i suggest these. For point particles you have their interaction as gague particles. their inclussion in LQG doesn´t modfy it´s behaviour under gauge interactons. It just takes account of their gravitatonal behaviour.

The same would rule for strings. they would be included in the spin network which would dictate their non perturbative gravitational behaviour but it wouldn´t affect their gauge interaction´s, which would include in these case a perturbative gravitational behaviour.

Under these scheme, if at last both -string and LQG- prove to be mathematically coherent theories it is raised the question if point particles and extended objects are incopatible. That is, you could have that matter actually is made of both, point particles and extended objects.

From the stringy side these could be adrressed allowing pontlike particles and triying to see how to extend the polyakov prescription for the path integral to include them. Well, later we would see how to include them in M-theroy and String field theorys.

Like a wya to resume. If LQG is a valid theory strings -not beeing imprescindible any more- could be revised as an anlisys of the viability and utility of extended objects in QFT. These would be a natural way to see heings. Afther all, if gravity is quantizable there is no obvious reason why nature prefers 0-extended over n-extended objects and we would need to include all of them in a theroy which aims to be a TOE.

I didn´t go into any math detaill here because the generality of the questions forbides to do so. Well, of course to be honest i have no idea of how to do the math for the polyakov prescription which would include pointlike particles and strings. It is just an idea i have realized today. if i would have the math for it i guess it would be a good paper for hep .

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Hey Sauron!
I am having to run very hard to catch up with you!
other kinds of things (other fields) to the picture. I feel
confident that people will do that, if LQG survives experimental
tests in some form.

So I am focusing on trying to understand it simply as a quantum
theory of gravity------a manifold whose metric (or rather whose connection) is a quantum state.

It seems that the theory is nearing completion faster and more suddenly than one might have expected 4 or 5 years ago. And as it nears completion it begins to make definite predictions that can be checked by observation.

this is a turbulent and controversial stage.

My hunch (intuitive suspicion) is that you understand the issues around verifiability and testability of LQG as well or better than I do.

You have, I believe, read arXiv:hep-th/0303185 "How far are we from quantum gravity" which lists various predictions that may proved verifiable now or over the next few years.

I'd like to hear your reading of this. The first wave of any theory can get shot down, but if it is basically strong another modified version grows up to take its place. LQG is becoming testable---or so it would appear (this itself is surprising, given the extremity of Planck scale).

As you know I was very impressed by the determination of the Immirzi parameter by Dreyer with the help of several other people.

Would like to know your perspective on section 3.4 "characteristic predictions of LQG".

BTW I enjoy editing your posts---getting rid of typos---and it makes it more easy for me to read and think about. Hope this OK with you. Here is a partial example of corrected spelling etc.

Originally posted by Sauron
I liked to read these thread and i don´t know why it would stop.

I have a question, or two.

you have explained what a network is. Now i think it is a good idea to explain what it represents.

if we restrict to pure gravity it represents an state of the gravity field (ok, its obvious). Its evolution is given by the action on it of the canonical constraints.

Later it needs to be addressed the question of which operators could act on these spin network which would represnt physical quantities and the difficulties to construct them.

But, what about the non-gravity sector? Supposedly now the nodes and vertex would allow states which represent matter?, that is the reason for the SO(2) vs SU(2) debate after all.

The question is, which constraints would dictate the evolution of these spin networks. Would they be derived form the matter lagrangian?

Theoretically yes, because they would somwhat represent the energy-momentum side in the classical counterpart.

Another thing, these regarding strings in LQG/LQG2.

Are totally incompatible both theories (Strings and LQG) in the gravity sector.
I guess not. If we accept the spin-2 particle of strings as a graviton we could include it action in LQG as a variation to the spin network.

Let's say strings would be valid for nonperturbative aspects of quantum gravity but the whole non perturbative theory would still be LQG.

I´ll explain why i suggest these. For point particles you have their interaction as gauge particles. their inclusion in LQG doesn´t modfy its behaviour under gauge interactons. It just takes account of their gravitatonal behaviour...

Sauron
When you said you understood strings vis a vis "polchinski" what did you mean?

Anticipating your previsible questions here is my background in string theory:

Because i like to have an historical perspective i first studied two books in the "old" heterótical string. The correponding chapters in Brian Hatfield book "Quantum field theory of point particles and strings" and the Lüsth-theisen ´s "Lectures on string theory".

And also some articles of the same ecpoch, about WZW theory´s and the "black strings" based on it. And later i readed a 1994 doctoral thesis of a friend about dualitys and string cosmology.

LAter i readed the Michio Kaku book about strings and M-theory (i did so because i had readed it´s book in QFT previously). My main understanding of "second revolution" of string theory comes form it.

With these backgroound i used the two Polchinsky books sa reference books where to go deeper in some questions. Also i have used sometimes the articles posted in the superstringtheroy.com to fill some gaps in my understanding.

But as i gained familiarity with strigs i beguined to get knowledge of LQG (my first contact was a CERN courrier article of Nick Mavromatos about Liouville strings and experiments in quantum gravity). And since them i have found it more interesting and have given priority in learning it over improving my acquitance with strings. Ii admit that my undertanding of D-branes could be better than it is.

As i know you like these "who knows what" expositions i´ll give you some background. My basis in GR (beyond the two courses in the graduate studies) comes mainly from two books. The Wald´s one and a big book about black holes of which ai actually have no the exact reference, but i seriously recommend to anyone who wants to get familiarity with many not too publicited aspects about GR. I tend to belive i have a very good knowledge of GR. In part because these year i have spended a lot of time in another foros in debates about it. Triyoin , for example, to Van Flanders critics to RG.

My basic in QFT comes mainly form the Hatfield and Kaku books, which i readed carefully (mainly the first one) and also Cheng-Li book "gauge tehory of particle physics". Not beeing bad i think i understand it worst that RG, mainly because i am somewhat skeptical about the Feyman path integral. I learned, beyond the usual introduction in string books, supersimmetry form the Collin,Martin,Squires book "Particle physics and cosmology".

I readed the original articles aobut quantum gravity of Feyman nad deWitt, and some papers of the same time, by t´hoof, Sagnoty, Faddeev, etc. Also i readed to review articles about the earlier developments in quntum gravity wirten by Enrique alvarez in 1986 and another by Alvarez-Gaume and Vazquez-Mozo in 1994.

My basic in maths includes regular studies of graduate studies in the areas of geometry (my favourite book is the great "Manifolds and riemanina geometry" -or something similar, by Willian M Bootby, complemneted with the bishop goldberg "tensor analisys on manifolds"), basic algebraic and diferential topology and also measure theory and funtional analisys.

The two books of Charlesh Nash about topology for physics.
The Edwin H. Spanier´s book "Algebraic topology".
Some books of Milnor in diferentail topology.
The Goldberg book "curvature and homolgy" about Khaler manifolds.
Also i have used ocasonally as reference the Choquete-Brouehte "Analisys, manifodls and physics" and the 3 books form Dubrovin Novikov and Fomenko about geometry and topology.

I don´t claim to have a prfect knowledge of everything, of course, but i think i have a reasonable understanding of most of it.

I am graduate student in physics, have pased most hte exams to be graduate in maths and i was a year studying a doctorrate about "geometric quantizzation" in the math department of my facoulty, but i couldn´t finish it because of "burocratic" problems.

I hope now you have the right felling about what my understanding is and which degree of deepnest you can use to answer my questions.

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Originally posted by jeff on 6-13
Here's a question inspired by various posts above.

In LQG spin networks, are equal integer spin IRs (irreducible representations) of SU(2) and SO(3) interchangeable?

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quote:
--------------------------------------------------------------------------------
Originally posted by marcus
...a good many things here baffle me...
--------------------------------------------------------------------------------

No sh*t, but since it appears you started this thread in large part to lecture, this should have been at the top.

quote:
--------------------------------------------------------------------------------
Originally posted by marcus
Good luck to any who can understand these matters
--------------------------------------------------------------------------------

Thanks good friend, but I already understand them, and with a lot of hard work (and therapy), you (or one of your personalities), might one day be able to understand them too! Could happen, you just never know unless you try.

P.S. This post wouldn't have been condoned according to my previous foolishly literal interpretation of rule 5 of the site guidelines. So here's to you Greg for clearing that confusion up for me. Thanks buddy (What, too far?).

Last edited by jeff on 06-28-2003 at 06:48 PM