Posted: Aug 30, 2010 5:25 pm
by twistor59
Post 9a The New Dynamics

There is a large volume of work on LQG now following a new line of approach. It goes roughly like this:

"We've come a long way down the road of quantizing the classical picture of gravity. It has given us a very promising model of the quantum three geometry, but as yet there is no unique approach to the dynamics. Instead, we will take as a starting point the spin-network Hilbert space picture, and try to construct a "natural" dynamics on it and see where it leads us. The hope is that it will give something that has classical GR as its low energy limit".

So we now forget about the fact that the spin network graphs were once thought of as embedded in a three space Σ. They're now abstract graphs, describing purely combinatorial relations (incidently this is exactly what Penrose was thinking in his original invention of spin networks). We still have the mechanisms for constructing the Hilbert space from them - this only makes reference to the spin network, not to its embedding. We can still construct the area and volume operators on them.

The nodes represent "grains" of space. Nodes connected by a link represent grains which are adjacent. The area of the boundary surface between adjacent grains is determined by the spin on the link.

Elements of the Hilbert space are of course superpositions of basis states which look nothing like classical spacetime. Two grains of space in an element of the Hilbert space will in general be in a superposition of being adjacent and not adjacent ! To select states which resemble classical three geometries, use is made of the idea of coherent states.

Mechanisms have been given for constructing such states, which are in some sense "peaked" around a discretized classical 3 geometry. Discrete versions of differential geometry have been around for some time.

In conventional quantum theory, dynamics represents an evolution from a state on a spacelike three surface at time t1 to a state on a spacelike three surface at time t2. There is a complex number (an "amplitude") which, when squared, represents the probability of this evolution. This amplitude may be computed by a Feynman path integral.

In the background-independent quantum gravity that we're trying to formulate, there isn't really such a time evolution. The "time evolution" moves points around the three space along integral curves of the Hamiltonian flow. It's just another constraint. So what do we do ?

The answer is that we define an amplitude for each state on Σ where Σ is a boundary to some region R of spacetime. This amplitude map


is the (quantum) dynamics. This seems to be the appropriate generalisation of the conventional picture, where the amplitude is


and |φ> is the initial state on Σ1 and |ψ> is the final state on Σ2. Here the boundary consists of two disconnected components and things are assumed to die away at spatial infinity. Obviously we can't make assumptions like this in the background independent case, because since the whole thing is diffeomorphism invariant, we don't have a notion of spatial infinity any more. If you need your intuition on this renormalizing (as I did), I recommend this discussion.

So the dynamics will be defined by a linear functional W on H, the Hilbert space of states| ψ> on the boundary Σ of the spacetime region R. The associated probability is

P(ψ) = |<W|ψ>|2

In Rovelli's words, we will attempt to describe this (motivated by the Feynman path integral approach) as a sum over "histories of states"

|<W|ψ>| = Sum over σ W(σ)

where σ is a sequence of states (think spinfoam) with ψ as boundary state, and W(σ) is an amplitude for a given state. The task is therefore to build the map W.

The answer turns out to be the spinfoam sum:

<W|ψ> = Σ Π d(jf) Π Wv(σ)

The sum is over spinfoams σ which have ψ as boundary
The first product is over the faces f of the spinfoam
The second product is over the vertices of the spinfoam
d(jf) is the dimension of the representation assigned to face f

Wv(σ) = <Wv| ψv>

where ψv is a natural local spin network surrounding the vertex v (how to construct this local spin network is discussed here).

Finally, <Wv| ψv> = (fγ ψv) (𝟙)

𝟙 just denotes the identity SL(2, ℂ) element on the rep attached to each of the edges of the vertex v.

fγ is the function that I introduced at the end of the last post.

Next post I'll try to summarise in English rather than mathematics some of this wibble that I've been accumulating.