By Gausterer H., Grosse H., Pittner L. (eds.)

In glossy mathematical physics, classical including quantum, geometrical and sensible analytic equipment are used concurrently. Non-commutative geometry specifically is changing into a great tool in quantum box theories. This ebook, aimed toward complicated scholars and researchers, presents an creation to those rules. Researchers will profit quite from the large survey articles on versions on the subject of quantum gravity, string thought, and non-commutative geometry, in addition to Connes' method of the traditional version.

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From now on we shall follow this custom. Thus instead of the above skein relation, we write: ρ = dim(ρ) An Introduction to Spin Foam Models 37 Moving on to something a bit more complicated, let us consider spin networks with trivalent vertices. Given any pair of irreducible representations ρ1 , ρ2 of G, their tensor product can be written as a direct sum of irreducible representations. Picking one of these and calling it ρ3 , the projection from ρ1 ⊗ ρ2 to ρ3 is an intertwiner that we can use to label a trivalent vertex.

In 3 dimensional spacetime, the vacuum Einstein equations simply say that the metric is flat. Of course, many different A and E fields correspond to the same metric, but they all differ by gauge transformations. So in 3 dimensions, BF theory with gauge group SO(2, 1) is really just an alternate formulation of Lorentzian general relativity without matter fields — at least when E is one-to-one. When E is not one-to-one, the metric g defined above will be degenerate, but the field equations of BF theory still make perfect sense.

So in 3 dimensions, BF theory with gauge group SO(2, 1) is really just an alternate formulation of Lorentzian general relativity without matter fields — at least when E is one-to-one. When E is not one-to-one, the metric g defined above will be degenerate, but the field equations of BF theory still make perfect sense. Thus 3d BF theory with gauge group SO(2, 1) may be thought of as an extension of the vacuum Einstein equations to the case of degenerate metrics. If instead we take G = SO(3), all these remarks still hold except that the metric g is Riemannian rather than Lorentzian when E is one-to-one.