New methods for analyzing quantum field theories

我们开发并应用了几种分析量子场论的新方法。特别地，我们考虑了量子引力和弦理论，并为研究这些系统构建了有用的非微扰框架。首先，我们发展了2+ 2维量子引力的2 -展开格式。利用该方案，我们研究了高维量子引力的标度特性。接下来我们考虑R^2拉格朗日量可以在红外和紫外范围内精确求解。此外，我们还发明了一种可以应用于时空量子涨落的转移矩阵形式。利用该方法，我们准确地解决了二维量子引力问题，并证明了时空具有分形结构。这种方法可能是分析弦理论的非微扰效应的有力工具。这方面的一些尝试目前正在进行中。

We developed and applied several new methods for analyzing quantum field theories. In particular, we considered quantum gravity and string theories, and constructed useful nonperturbative frameworks for investigating those systems. First of all, we developed and epsilon - expansion scheme for 2+ epsilon dimensional quantum gravity. Using this scheme, we studied scaling properties of higher dimensional quantum gravity. We next considered R^2 Lagrangian can be solved exactly both in the infrared and ultraviolet limits. Furtheremore, we invented a transfer metrix formalism that can be applied in the presence of quantum fluctuations of the space-time. Using this method, we exactly solved the problem of two-dimensional quantum gravity, and proved that the space-time has fractal structures. This method may be a powerful tool for analyzing nonperturbative effects of string theories. Some attempts in this direction are now in progress.