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| loop quantum gravity | |
| mathematical physics | |
| quantum field theory | |
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IN THE CONTEXT OF LOOP QUANTUM GRAVITY One of the key results of loop quantum gravity is quantization of areas: according to several related derivations based on loop quantum gravity, the operator of the area of a two-dimensional surface should have a discrete Spectrum . Every spin network is an Eigenstate of each such operator, and the area eigenvalue equals : where the sum goes over all intersections of with the spin network. In this formula,
Similar quantization applies to the volume operators but mathematics behind these derivations is less convincing. MORE GENERAL GAUGE THEORIES (Outside the context of LQG, the name ''spin'' networks is a bit of a misnomer...) As mentioned, it was noticed that analogous constructions can be made for general gauge theories with a compact Lie group G and a Connection Form . This is actually an exact Duality over a lattice. Over a Manifold however, assumptions like Diffeomorphism Invariance are needed to make the duality exact (smearing Wilson Loop s is tricky). Later, it was generalized by Robert Oeckl to representations of Quantum Group s in 2 and 3 dimensions using the Tannaka-Krein Duality . Michael Levin and Xiao-Gang Wen have also defined another generalization of spin networks which they call String-net s using Tensor Categories . String-net Condensation produces Topologically Ordered states in condensed matter. PUBLICATIONS Some random early papers (none of them actually called them spin networks; that is Penrose 's name for them):
Modern papers:
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