圈量子引力和量子宇宙学中的路径积分

Loop quantum gravity is a program to quantize gravity by directly combining the principles of general relativity and quantum mechanics. This project is planed to contribute the following aspects: (1) to investigate the quantum dynamics, semi-classical ananlysis and effective theory of loop quantum cosmology, especially to focus on the physical prediction of path-integral formalism and the corresponding effective action; furthermore, to explore the intrinsic relation between the Moyal-star product appearing in effective hamiltonian and non-commutative geometry based on our work about the coherent state functional integrals and effective theory of quantum cosmology; (2) to investigate the semiclssical behavior of the Spin-Foam models in the low-energy regime, in particular, to calculate the Feynman propagator,n-point function and to give the asymptotic behavior of the Spin-Foam amplitude; in order to test or/and improve the path-integral formalism of loop quantum gravity via comparing with the classical theory and quantum field theory.

圈量子引力是一种直接将广义相对论和量子力学的基本原理相结合的引力量子化方案。本项目的研究涉及以下两方面的内容：（1）考察圈量子宇宙学的量子动力学、半经典分析和有效理论，特别关注其路径积分形式和相应的有效作用量所给出的物理预言；在我们关于量子宇宙学的相干态路径积分及有效理论工作的基础上，进一步探索有效哈密顿量中的Moyal-星乘积与非对易几何的内在联系；（2）考察Spin-Foam模型的半经典性质，包括费曼传播子和n-点函数的计算以及Spin-Foam振幅的渐近行为；通过与经典理论以及量子场论的对比来检验Spin-Foam模型的正确性，为提出新的Spin-Foam模型进而完善量子引力的路径积分表述提供重要启示。

本项目的工作主要分为两方面的内容：（1）用有效运动方程的方法全面考察量子宇宙学的约束代数、动力学和修正的弗里德曼方程，得到含 Moyal-星乘积的有效哈密顿量，验证了圈量子宇宙学避免大爆炸奇点的结果，表明了 Moyal-星乘积的出现在约束系统量子化有效理论中的普适性；（2）考察四维Chern-Simons 作用量在空间均匀各项同性宇宙学中的应用，成功构造了引力部分的哈密顿约束算符及其在半经典态下的有效哈密顿量，拓展了圈量子宇宙学的应用范围。