Effective Descriptions of Quantum Gravity and Cosmology

这个项目的主要目的是用规范语言完成对量子系统的有效描述。尤其是正则量子重力场(包括圈量子引力)需要比标准有效作用更一般的设置，例如允许放大的规范结构、一般协方差和相关的改变时间变量的自由度，以及作为展开基础的非高斯态或非真空态。与现有的公式相比，该项目的工作提供的主要扩展将导致有效量子场论的规范版本。应用将包括分析正则量子引力中的重整化，与现有有效量子重力结果的比较，以检查环状量子重力的低能量极限，以及早期宇宙中的结构形成和对初始态的可能限制。量子重力预计在高能量密度下很重要，例如在大爆炸期间或之后的早期宇宙中。在这样的体系中，时空本身，而不仅仅是时空中的物质，都是由一个受量子不确定性和涨落影响的波函数来描述的。由于几个概念和技术上的原因，很难为潜在的物理观测评估这种波函数，但就像在粒子物理学中一样，它们的含义通常可以使用有效的方程来近似和推导。粒子物理的标准方法不能直接适用于量子引力，因为量子引力不再适用于经典的时空结构。这个项目的目标是完成适用于量子引力的有效方程的建立，在这个过程中，还将涉及几个数学感兴趣的问题(如泊松几何和代数结构)。它将为早期宇宙中包括量子引力效应的结构形成提供新的场景和方程，从而为物理宇宙学做出贡献。

The main aim of this project is to complete the formulation of effective descriptions of quantum systems in canonical language. Especially the field of canonical quantum gravity (including loop quantum gravity) requires a more general setting than standard effective actions, allowing for instance for an enlarged gauge structure, general covariance and the associated freedom to change the time variable, and a non-Gaussian or non-vacuum state as expansion basis. Compared to existing formulations, the major extension provided by work in the project will lead to a canonical version of effective quantum field theory. Applications will include an analysis of renormalization in canonical quantum gravity, comparisons with existing results in effective quantum gravity to check the low-energy limit of loop quantum gravity, as well as structure formation in the early universe and possible restrictions on initial states.Quantum gravity is expected to be important at high energy density, for instance in the early universe during or just after the big-bang. In such regimes, space-time itself, and not just matter in space-time, is described by a wave function subject to quantum uncertainty and fluctuations. For several conceptual and technical reasons, it is difficult to evaluate such wave functions for potential physical observations, but as in particle physics, their implications can often be approximated and derived using effective equations. Standard methods of particle physics are not directly applicable to quantum gravity, in which the classical structure of space-time is no longer available. This project aims to complete the setting of effective equations suitable for quantum gravity, a process in which several questions of mathematical interest (such as Poisson geometry and algebraic structures) will be touched as well. It will contribute to physical cosmology by providing new scenarios and equations for structure formation in the early universe that include quantum-gravity effects.