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Low Dimensional Topology

During this quantity, that's devoted to H. Seifert, are papers in line with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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If we geometrically quantize so(3)∗ , we obtain the direct sum of all the irreducible representations of SU(2): H∼ j. = j=0, 12 ,1,... Since this Hilbert space is a representation of SU(2), it has operators Jˆa on it satisfying the usual angular momentum commutation relations: [Jˆa , Jˆb ] = i abc Jˆc . We can think of H as the ‘Hilbert space of a quantum vector’ and the operators Jˆa as measuring the components of this vector. If we geometrically quantize (so(3)∗ )⊗4 , we obtain H⊗4 , which is the Hilbert space for 4 quantum vectors.

Each edge of γ intersects Σ at most once. For each vertex v of γ lying in Σ, we can divide the edges incident to v into three classes, which we call ‘upwards’, ‘downwards’, and ‘horizontal’. The ‘horizontal’ edges are those lying in Σ; the other edges are separated into An Introduction to Spin Foam Models 43 two classes according to which side of Σ they lie on; using the orientation of Σ we call these classes ‘upwards’ and ‘downwards’. Reversing orientations of edges if necessary, we may assume all the upwards and downwards edges are incoming to v while the horizontal ones are outgoing.

Thus spin network observables give a way to measure correlations among the holonomies of A around a collection of loops. When G = U(1) it is also easy to construct gauge-invariant functions of E. We simply take any compact oriented (n − 2)-dimensional submanifold Σ in S, possibly with boundary, and do the integral Σ E. An Introduction to Spin Foam Models 41 This measures the flux of the electric field through Σ. Unfortunately, this integral is not gauge-invariant when G is nonabelian, so we need to modify the construction slightly to handle the nonabelian case.

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