Posted: Aug 30, 2010 5:18 pm
Post 9: Dynamics

The last step in the quantization programme

𝓗Diff -> HamiltonianConstraint->𝓗phys

is the application of the Hamiltonian constraint. In my, admittedly not worth too much, opinion, this step does not go quite as well as the others. It is done by first writing the (classical) Hamiltonian in terms of the Ashtekar variables (connection and densitized triad). The expression is quite messy and looks like it will give some difficulty when replacing the expressions with operators.

This was done by Thiemann who re-expresses the nasty expression in terms of the volume of Σ and some Poisson brackets. This is then re-written in terms of the holonomy and flux operators. The operation of the Hamiltonian ends up looking like adding, to a given vertex, two new vertices and a new link.

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I haven't tried to closely follow the construction of the Hamiltonian this way, but:

1 Its action is known and finite, but
3 There is some arbitrariness in the order in which things are done.
4 It's not clear how to apply it to a complete spin network

These ambiguities 2-4 dont look too great, so the quantization procedure cannot be considered to be completed yet. It is still under investigation.

Spin Foams

One approach (which has been around for 10 years or so) to get past these (considerable) difficulties with the Hamiltonian in LQG is the development of the theory of spin foams.

If you think of a spin network moving forward in time, a link sweeps out a surface, and a node sweeps out a line (echoes of strings->branes anyone ?). As we said above, the Hamiltonian generates new nodes and links, so in the spacetime picture, a new surface gets added

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So you can see the motivation for studing the higher dimensional variant of spin networks. A spin foam can be thought of as a path in a Feynman path integral sum to detemine the amplitude for transition between an initial spin network state at time T1 and final spin network state at time T2. Just as in the derivation of the path integral, we can imagine a succession of timeslices and the Hamiltonian to operate on each one, generating a succession of intermediate states.

Although they were originally thought of as time evolutions of spin networks, spinfoams eventually came to be thought of as an independent approach in their own right, and became generalisations of spin networks in the sense that the "group theory stuff" (spins, representations, intertwiners) now gets attached to higher dimensional components (faces etc) instead of just to nodes and edges.

In this approach, the spin foams built with representations of either SO(3,1) (or equivalently SL(2,ℂ)) or SO(4) attached to their faces. In the SO(4) = SU(2)xSU(2) case (tensor product), a subset of representations is used, namely (j, j) j=1/2, 1, 3/2.... This leads to the "Barrett-Crane" spinfoam models.

More recently, a new class of spinfoam models, known as EPRL models (Engle, Livine, Pereira, Rovelli) have been introduced.

SL(2, ℂ) Spin Networks

In the previous set of posts, I outlined the "derivation" of the spin network Hilbert space kinematic formalism from the canonical quantization of gravity (using an ADM type formalism and the Ashetkar variables etc).

There is an alternative derivation, which leads to a slightly different picture which will be quite useful when discussing the dynamics. The derivation in question starts with a discrete GR on a 4d lattice (hence Rovelli refers to it as "covariant lattice quantization"). Look at the boundary of the lattice. This is a 3d manifold, and each tetrahedron in the boundary bounds a single simplex of the 4 manifold and each boundary triangle bounds just two tetrahedra.

Let Γ be the dual of the boundary triangulation. This is a 4-valent graph with nodes dual to the boundary tetrahedra and links dual to the boundary triangles. (See here). The gauge group is SL(2, ℂ) and the Hilbert space is

𝓗ΓSL(2, ℂ) = L2[SL(2, ℂ) L / SL(2, ℂ) N]

(there is something else called the "simplicity constraint" which comes into the definition of this Hilbert space, which I haven't really understood). The simplicity constraints mean that the classical theory corresponds to GR, not just any old url=http://en.wikipedia.org/wiki/BF_model]BF[/url] theory.

Now, there is a map from the old canonical LQG-style Hilbert space 𝓗Γ into 𝓗ΓSL(2, ℂ). To define it, use the Peter-Weyl decomposition (like we did a few posts back)

L2[SL(2, ℂ) L ] = sum over (pl, kl) of tensor product over links l of (𝓗*(pl,kl) tensor 𝓗(pl,kl) )

where SL(2, ℂ) unitary irreps are labelled by (p in ℝ, k in ℤ+).

Now if we pick an SU2 subgroup of SL(2, ℂ), then each SL(2, ℂ) irrep (p, k) decomposes as a sum of SU2 irreps. The sum is over the spins from k up to infinity. Neglecting everything except the lowest (k) spin rep in the sum allows us to define a map

Yγ : 𝓗Γ -> 𝓗ΓSL(2, ℂ)

SU2 spin-j things get mapped to (p=γj, k=k) SL(2, ℂ) things.

Apparently (!) this map respects the simplicity constraints, which means that the quantum theory is still GR.

If we compose this map Yγ with a projection onto the SL(2, ℂ) invariant states (i.e. we do the Gauss constraint for SL(2, ℂ) ), we get a very important map

fγ : 𝓗Γ -> 𝓗ΓSL(2, ℂ)

from SU(2) spin networks to SL(2, ℂ) spin networks.

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This is used in the "new dynamics" (next post).