Topics in Classical and Quantum Relativity

Torre教授将进行数学相对论和量子引力中的一些问题。 在数学相对论中，利用了真空爱因斯坦方程喷流空间结构的最新结果。 详细地说，现代理论的守恒定律，它提供了一个实用的框架计算守恒体积，表面和线积分相关联的一个系统的场方程，应用到爱因斯坦方程。 计算了有无Killing向量的爱因斯坦方程的守恒律。 渐近守恒律的一般理论，它描述了一个给定的渐近结构，如ADM能量的解决方案的表面积分守恒律，开发和使用给出一个形式主义计算这样的守恒律的一大类一般协变场方程和渐近结构。 这些结果被应用到的问题，给出了一个定义的动态黑洞熵。 最后，最近的结果偏微分方程的对称性约化的时空度规的一般协变场方程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和路径积分方法。 通过假设存在两个交换的Killing矢量场，得到了广义相对论的简化模型，讨论了正则量子引力中的泛函演化问题。 这些模型被严格量化，并与一般协方差一致。 利用代数量子化的方法讨论了在闵可夫斯基空间中实现参数化自由场理论的狄拉克量子化的可能性。 研究了使用Ashtekar变量的路径积分量子化。

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.