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

In glossy mathematical physics, classical including quantum, geometrical and practical analytic equipment are used concurrently. Non-commutative geometry specifically is changing into a great tool in quantum box theories. This ebook, geared toward complicated scholars and researchers, offers an creation to those principles. Researchers will gain really from the huge survey articles on types on the subject of quantum gravity, string concept, and non-commutative geometry, in addition to Connes' method of the normal version.

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**Example text**

This is not true in the smooth context: for example, two smoothly embedded paths can intersect in a Cantor set. One can generalize the construction of L2 (A) and L2 (A/G) to the smooth context, but one needs a generalization of graphs known as ‘webs’. The smooth and real-analytic categories are related as nicely as one could hope: a paracompact smooth manifold of any dimension admits a real-analytic structure, and this structure is unique up to a smooth diffeomorphism. 2. There are various ways to modify a spin network in S without changing the state it defines: – We can reparametrize an edge by any orientation-preserving diffeomorphism of the unit interval.

J2 t ..... ...... j4 50 John C. Baez Remarks 1. For a deeper understanding of BF theory with gauge group SU(2), it is helpful to start with a classical phase space describing tetrahedron geometries and apply geometric quantization to obtain a Hilbert space of quantum states. We can describe a tetrahedron in R3 by specifying vectors E1 , . . , E4 normal to its faces, with lengths equal to the faces’ areas. We can think of these vectors as elements of so(3)∗ , which has a Poisson structure familiar from the quantum mechanics of angular momentum: {J a , J b } = abc J c.

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. We can then write any intertwiner labeling v as a linear combination of intertwiners of the following special form: ιv : ρuv ⊗ ρdv → ρhv where ρuv (resp. ρdv , ρhv ) is an irreducible summand of the tensor product of all the representations labeling upwards (resp. downwards, horizontal) edges. This lets us write any spin network state with γ as its underlying graph as a finite linear combination of spin network states with intertwiners of his special form.