Mathematical Sciences: Study of Feynman-Dyson Algebras

本课题是泛函分析中的数学研究 适用于理论物理学。 费曼-戴森的想法 代数，主要的研究对象，是它允许提升， 两个非交换算子到一个时间序交换算子的 状态，在这种状态下，可以执行分析操作，然后 回到原来的非交换状态。 吉尔教授 我将扩展早期的工作，包括线性的一般理论， 和非线性演化，具有最弱的可能约束。 第二个目标是扩展他的 方法和路径积分通过寻找方法来构建 转换内核直接从进化生成器。 最终 目标是完善整合理论， 事实上，许多与物理学相关的路径积分 概率论不需要在一个广义 维纳空间。

This project is mathematical research in functional analysis applicable to theoretical physics. The idea of a Feynman - Dyson algebra, the main object of study, is that it permits the lifting of two non-commuting operators to a time-ordered commutative state, in which one can perform analytic operations and then return to the original non-commutative situation. Professor Gill will extend earlier work to include the general theory of linear and non-linear evolutions, with the weakest possible constraints. A second objective is to extend the connection between his approach and path integrals by finding methods to construct the transition kernel directly from the evolution generator. A final objective is to refine the integration theory to allow for the fact that many path integrals associated with physics and probability theory need not generate a measure on a generalized Wiener space.