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 Γ 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 Γ 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.
Ψ
Γ, j, v
Here the j
i label the edges----i = 1,...,n---with reps.
And the v
r label the nodes----r = 1,...,m---with multilinear forms.
Here Γ 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 Ψ 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 Ψ
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:
H
i is the finite dimensional Hilbert space which the irreducible representation j
i acts on.
All that "irreducible" means is that H
i is not any bigger than it has to be----it doesn't have a nontrivial subspace left invariant---everything in it moves under the j
i action, except the zero vector.
Now we assume that all the edges of a graph Γ,embedded in the manifold M, have been labeled with irred. reps j
i 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 I
p that tells which edges γ
i meet there.
And the set of reps will be {j
i for i ε I
p}
The crafty Upadhya, with Rovelli looking over his shoulder, tells us to take the tensor product of all the finite dimensional hilbert spaces {H
i for i ε I
p} and to define H
p to be the subspace invariant under the combined action {j
i for i ε I
p}.
Upadhya discusses how to ensure that this subspace H
p 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 Γ j
i, v
p
or more simply Γ, j, v
where j
i is an irreducible rep labeling each edge γ
i, and
and v
p, is a chosen for each node p from the basis of H
p.
Now at last we can write the new quantum gravity state
based on the labeled graph Γ, j, v
Ψ
Γ, 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
Ψ
Γ, 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.
