By Lefschetz S.
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Extra resources for [Article] On the Real Folds of Abelian Varieties
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.