Subnanosecond photodissociation atomic iodine laser

Abstract

In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only a finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise-smooth finite paths and loops. In particular, we (a) characterize the spectrum of the Ashtekar-Isham configuration space, (b) introduce spin-web states, a generalization of the spinnetwork states, (c) extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism-invariant states and finally (d) extend the 3-geometry operators and the Hamiltonian operator. PACS number: 0460D J Lewandowski and T Thiemannconcerning diffeomorphism-invariant states of connections and associated diffeomorphisminvariant operators. Finally, in section 6 we apply the machinery of the previous sections to the operators that have proved to be elementary building blocks for the quantization of the gravitational field. The algebra of cylindrical functions on the space of connectionsThis section is divided into three parts. In the first part we recall the basic notions of cylindrical functions on the space of distributional connections and inductive limit structures, and review the results that one obtains if one labels cylindrical functions by piecewise analytic, finitely generated graphs. In the second part we review the construction, due to Baez and Sawin, of a new label set, called 'webs', which are appropriate for the case where the loops defining the Wilson functions are only piecewise smooth and may intersect each other an infinite number of times. Finally, in the third part we prove the master theorem on which the rest of this paper is based. PreliminariesIn the loop quantization of gravity as initiated by Ashtekar [1] and Jacobson, Rovelli and Smolin [2] and Gambini et al (see, e.g., [3] and references therein) through embedding general relativity into a Yang-Mills theory phase space and making use of Wilson loop variables as well as in a general framework for quantization of a diffeomorphism-invariant theory of connections with local degrees of freedom [4]