Extending loop quantum gravity to supergravity.

摘要

Loop quantum gravity in Ashtekar's variables has during the past fifteen years been studied as a candidate for a quantum theory of general relativity underlying a non-perturbative and background independent approach. The purpose of this thesis is to extend non-perturbative techniques in loop quantum gravity to supergravity, and explore the connections among different approaches to quantum gravity. First we construct the first order formalism of N = 1, N = 2 supergravity in four dimensions, as well as eleven dimensional supergravity, as constrained topological field theories. Their canonical formulations are presented as well. Next we investigate the quantum theory of supergravity in the framework of loop quantum gravity. We study the construction of supersymmetric spin networks based on the representation theory of super Lie algebras. In particular, the Osp(1|2 n) super spin networks are explored in a systematic way. As a direct application, the spectrum of the area operator in simple supergravity is derived by acting on the Hilbert space in the basis of super spin network states. Furthermore, the holographic formulation of quantum supergravity with cosmological constant is obtained by setting appropriate boundary conditions, and the Bekenstein bound is tested at a quantum mechanical level.