International Research Fellowship Program: Moving Forward in Loop Quantum Gravity Via Self-Duality

[601844 .]国际研究奖学金计划使美国科学家和工程师能够在国外进行9至24个月的研究。该计划的奖励为联合研究提供了机会，并利用独特或互补的设施、专业知识和国外的实验条件。该奖项将支持Jonathan S. Engle博士与Carlo Rovelli博士在法国马赛体质理论中心开展为期24个月的研究。在过去的十年里，环量子引力已经取得了重大进展。然而，问题依然存在。其中一个突出的问题是，动态的实现在概念上有些不自然，而且在技术上相当复杂。这样做的一个直接后果是无法将环量子引力与自旋泡沫完全联系起来，自旋泡沫是环量子引力的路径积分版本。解决这个问题的一种方法是使用广义相对论的自对偶公式。经典地，在自对偶公式中，动力学是尽可能简单的。然而，在过去，有一个障碍阻碍了基于自对偶公式的量子理论的严格定义：人们不知道如何构建一个可接受的内积，即状态之间的“重叠”概念，这对于从理论中提取预测至关重要。该项目的目标是找到自双环量子引力的内积。许多内积是可用的，包括文献中已经提到的规范和微分同构不变的内积，以及PI引入的一类内积。关键的问题是，这些是否实现了自对偶理论的正确现实条件——也就是说，这些是否使得经典水平的复共轭延续到量子水平的厄米共轭。如果PI成功地找到可接受的内积，PI将尝试在新创建的自对偶框架中再现量子几何，然后尝试构建简化的哈密顿约束。然后，PI将观察简化的哈密顿约束是否允许环量子引力更直接地与自旋泡沫模型相关。

0601844EngleThe International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.This award will support a twenty-four-month research fellowship by Dr. Jonathan S. Engle to work with Dr. Carlo Rovelli at Centre de Physique Theorique, in Marseille, France.Significant progress has been made over the past decade in loop quantum gravity. However, issues remain. One prominent among them is that the implementation of dynamics is conceptually somewhat unnatural and technically quite complex. A direct consequence of this is an inability to fully relate loop quantum gravity to spin-foams, the path integral version of loop quantum gravity. One way of resolving this problem is to use a self-dual formulation of general relativity. Classically, in the self-dual formulation, the dynamics are as simple as they possibly could be. However, in the past, there was an obstacle preventing the rigorous definition of a quantum theory based on the self-dual formulation: One did not know how to construct an acceptable inner-product a notion of "overlap" between states, which is essential in order to extract predictions from the theory. The goal of the project is to find an inner-product for self-dual loop quantum gravity. A number of inner products are available, including gauge and diffeomorphism-invariant ones already mentioned in the literature, as well as a class introduced by the PI. The key question is whether or not any of these implement the correct reality conditions of the self-dual theory --- that is, whether or not any of these are such that complex conjugation at the classical level carries over to Hermitian conjugation at the quantum level. If the PI is successful in finding an acceptable inner product, the PI will try to reproduce quantum geometry in the newly created self-dual framework and then attempt to construct a simplified Hamiltonian constraint. The PI will then see whether the simplified Hamiltonian constraint allows loop quantum gravity to be more directly related to a spin-foam model.