John C. Baez's Knots and quantum gravity PDF

By John C. Baez

Fresh paintings by means of mathematicians and physicists has exposed revelatory connections among knot conception and the matter of constructing a quantum idea of gravity. This e-book, the court cases of a workshop held to collect researchers in knot idea and quantum gravity, includes a variety of expository and learn papers that would relief considerably in last the distance among the 2 disciplines. it is going to function a advisor for mathematicians and physicists trying to comprehend this quickly constructing zone of study. The ebook represents a cutting-edge research of present learn and development. The editor is the writer of Gauge Fields, Knots, and Gravity (World Scientific), a graduate point textual content at the subject.

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Aref’eva, I. : The gauge field as chiral field on the path and its integrability, Lett. Math. Phys. 3 (1979) 241–247 2. Arms, J. , Marsden, J. : Symmetry and bifurcations of momentum mappings, Commun. Math. Phys. 78 (1981) 455–478 3. : Lectures on non-perturbative gravity, World Scientific, Singapore, 1991 4. : Mathematical problems of non-perturbative quantum general relativity, Les Houches lecture notes, preprint Syracuse SUGP-92/11-2 5. Ashtekar, A. and Isham, C. : Representations of the holonomy algebras of gravity and non-Abelian gauge theories, Classical & Quantum Gravity 9 (1992) 1433–1467 6.

To illustrate this point, in Appendix A we restrict ourselves to U (1) connections and, by exploiting the Abelian character of this group, show how one can obtain the main results of this paper using piecewise C 1 loops. Whether similar constructions are possible in the non-Abelian case is, however, an open question. Finally, in Appendix 26 Abhay Ashtekar and Jerzy Lewandowski B we consider another extension. In the main body of the paper, Σ is a 3-manifold and the gauge group G is taken to be SU (2).

Note that the affine space structure of A is lost in this projection; A/G is a genuinely non-linear space with a rather complicated topological structure. The next notion we need is that of closed loops in Σ. e. maps p : [0, s1 ] ∪ . . 2) which are continuous on the whole domain and C ω on the closed intervals [sk , sk+1 ]. Given two paths p1 : [0, 1] → Σ and p2 : [0, 1] → Σ such that p1 (1) = p2 (0), we denote by p2 ◦ p1 the natural composition: p2 ◦ p1 (s) = p1 (2s), for s ∈ [0, 12 ] p2 (2s − 1), for s ∈ [ 21 , 1].

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