Posted: Aug 27, 2010 7:59 am
Post8 – Diffeomorphism Invariance
Remember our plan for step by step imposition of the constraints, which will eventually lead us to the Hilbert space of physical states:

𝓗kin ->GaussConstraint->𝓗0kin ->DiffeomorphismConstraint->𝓗Diff -> HamiltonianConstraint->𝓗phys

In the last post, we imposed the Gauss constraint (i.e. SU(2) gauge invariance) which led us to 𝓗0kin, the set of spin-network states. One issue with 𝓗0kin is that it’s non separable. The graphs Γ we have been talking about are embedded in three-space. The non separability of 𝓗0kin can be traced back to the fact that it contains superpositions of states with different graphs. If I distort the embedding of the graph a tiny bit, I get a different graph, and hence generate different states from it.

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The solution to this is the imposition of the diffeomorphism constraint. Remember, a diffeomorphism is just a smooth map from the manifold into itself

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Imposition of the diffeomorphism constraint means that we end up with spin networks defined, not upon graphs Γ, but upon equivalence classes of graphs Γ under diffeomorphisms.

So I effectively regard two graphs as being the same if I can distort one into another by a diffeomorphism. For example, a circular loop is the same as a distorted circular loop, but not the same as if the loop has a knot in it.

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So our Hilbert space 𝓗Diff is spanned by the knotted spin networks. Note that by regarding things as identical under diffeomorphisms, we are moving away from regarding the spin networks as embedded in the three surface, and more towards regarding them as topological entities in their own right.

The final step in getting to the space of physical states is obtained by imposing the Hamiltonian constraint, which will lead us onto the idea of spin foams.