Quantum gravity and non-linear sigma models

我的研究重点是构建一个描述场和粒子与引力场相互作用的量子理论。这仍然是基础理论物理学中尚未解决的问题。我在这一领域的具体研究计划涉及构建玩具量子引力模型；也就是说，这些理论解决了基本问题，但不太受与此问题相关的深刻技术困难的束缚。通常，这些玩具模型所涉及的物理空间不到三个维度。量子膨胀引力就是这样一类理论，它只有一个空间维度。这些模型可以推广到很远的地方，其中可以解决量子引力中的一些难题--例如，时间问题、奇点的出现和黑洞的衰变。我目前在这一领域的工作源于一种构建量子理论的新程序，并预测了一种离散的、类似格子的空间结构。我研究的另一个问题与非线性西格玛模型有关。弦理论是一种非线性西格玛模型，它的背景时空几何是扁平的，就像狭义相对论中的那样。如果背景不是平坦的，那么理论的自洽要求某些方程，称为RG流，有不动点，而这些不动点是允许的背景。我正在研究RG流，并回答有关这些方程与几何和其他场的动态变化(即，时间变化)之间的关系的问题。最后，我来看看推广共形场理论的理论。后者是二维时空上的场论，其特征是无限维对称代数，称为Kac-Moody代数。我正在研究的是四维时空中的场论的推广，其特征是Kac-Moody代数的某些推广，称为2环李代数。

My research focuses on the problem of constructing a quantum theory which describes the interaction of fields and particles with the gravitational field. This remains THE unsolved problem in fundamental theoretical physics. My particular research program in this area involves constructing toy quantum gravity models; that is, theories which address the basic issue but are less encumbered by the profound technical difficulties associated with this problem. Typically, these toy models involve physics in which`space' has less than three dimensions. One such class of theories, quantum dilaton gravity, has only one spatial dimension. These models can be pushed quite far, and in them one can address some of the difficult issues in quantum gravity- for instance, the problem of time, the occurrence of singularities and the decay of black holes. My current work in this area proceeds from a novel procedure for constructing quantum theories, and predicts a discrete, lattice-like structure for space. Another problem that I work on concerns non-linear sigma models. String theories are a type of non-linear sigma model in which the background spacetime geometry is flat, as in the special theory of relativity. If the background is not flat, then the self-consistency of the theory requires that certain equations, called the RG flow, have fixed points and these fixed points are the permissible backgrounds. I am looking at RG flows and addressing questions concerning the relationship between these equations and the dynamical changes (that is, changes in time) of geometries and other fields. Finally, I am looking at theories which generalize conformal field theories. The latter are field theories over a two dimensional spacetime, characterized by an infinite dimensional symmetry algebra, called the Kac-Moody algebra. I am looking at generalizations which are field theories in a four dimensional spacetime, and are characterized by a certain generalizations of the Kac-Moody algebras, called 2-Toroidal Lie algebras.