By Anton Alekseev (auth.), H. Gausterer, L. Pittner, Harald Grosse (eds.)

ISBN-10: 3540465529

ISBN-13: 9783540465522

ISBN-10: 3540671129

ISBN-13: 9783540671121

In smooth mathematical physics, classical including quantum, geometrical and practical analytic tools are used concurrently. Non-commutative geometry particularly is changing into a useful gizmo in quantum box theories. This publication, aimed toward complex scholars and researchers, offers an creation to those rules. Researchers will profit fairly from the huge survey articles on types with regards to quantum gravity, string conception, and non-commutative geometry, in addition to Connes' method of the traditional model.

**Read Online or Download Geometry and Quantum Physics: Proceeding of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999 PDF**

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**Additional resources for Geometry and Quantum Physics: Proceeding of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999**

**Sample text**

For each edge e of γ, an irreducible representation ρe of G, 3. for each vertex v of γ, an intertwiner ιv : ρe1 ⊗ · · · ⊗ ρen → ρe1 ⊗ · · · ⊗ ρem where e1 , . . , en are the edges incoming to v and e1 , . . , em are the edges outgoing from v. Here we say an edge is incoming to v if its target is v, and outgoing from v if its source is v. People have already begun formulating physical theories in which such abstract spin networks, not embedded in any manifold, describe the geometry of space. However it is still a bit difficult to relate such theories to more traditional physics.

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.

However, in dimensions 3 and 4, we can render it finite by adding an extra term to the Lagrangian of BF theory. In applications to gravity, this extra term corresponds to the presence of a cosmological constant. Finally, we discuss spin foam models of 4-dimensional quantum gravity. At present, work on spin foam models is spread throughout a large number of technical papers in various fields of mathematics and physics. This has the unfortunate effect of making the subject seem more complicated and less beautiful than it really is.

### Geometry and Quantum Physics: Proceeding of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999 by Anton Alekseev (auth.), H. Gausterer, L. Pittner, Harald Grosse (eds.)

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