Topics in Classical and Quantum Relativity

托瑞教授将解决一些数学相对论和量子引力方面的问题。在数学相对论中，利用了关于真空爱因斯坦方程喷流空间结构的最新结果。详细地说，现代守恒定律理论为计算与场方程组相关的守恒体积分、面积分和线积分提供了一个实用的框架，它被应用于爱因斯坦方程。计算了有、无Kill矢量的爱因斯坦方程的守恒律。发展了一个描述具有给定渐近结构(如ADM能量)的解的表面积分守恒律的渐近守恒律的一般理论，并用它给出了一大类一般协变场方程组和渐近结构的此类守恒律的计算公式。这些结果被应用于给出“动态黑洞熵”的定义问题。最后，利用偏微分方程对称约化的最新结果，给出了一般协变场方程在时空度规下的对称性约化的一般定理。在量子引力中，工作涉及引力场量子化的正则方法和路径积分方法。通过假设存在两个对易Kill矢量场而得到的对称性约化广义相对论模型，研究了正则量子引力中的泛函演化问题。关于一般协方差，这些模型被严格且一致地量化。利用代数量子化技术，讨论了在Minkowski空间中实现参数自由场理论狄拉克量子化的可能性。利用Ashtekar变量研究了路径积分量子化。

A number of problems in mathematical relativity and quantum gravity will be carried out by Prof. Torre. In mathematical relativity advantage is taken of recent results on the jet space structure of the vacuum Einstein equations. In detail, the modern theory of conservation laws, which provides a practical framework for computing conserved volume, surface and line integrals associated to a system of field equations, is applied to the Einstein equations. The conservation laws of the Einstein equations with and without Killing vectors are computed. A general theory of asymptotic conservation laws, which describes surface integral conservation laws of solutions with a given asymptotic structure, such as the ADM energy, is developed and used to give a formalism for computing such conservation laws for a large class of generally covariant field equations and asymptotic structures. These results are applied to the problem of giving a definition of `dynamical black hole entropy`. Finally, recent results on symmetry reductions of partial differential equations are used to give general theorems on symmetry reductions of generally covariant field equations for a spacetime metric. In quantum gravity, work involves both canonical and path integral methods of quantization of the gravitational field. The problem of functional evolution in canonical quantum gravity is addressed via symmetry-reduced models of general relativity obtained by assuming the existence of two commuting Killing vector fields. These models are quantized rigorously and consistently with respect to general covariance. The possibility of accomplishing a Dirac quantization of a parametrized free field theory in Minkowski space is addressed using techniques from algebraic quantization. Path integral quantization is studied using Ashtekar variables.