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。Feynman-Dyson代数的主要研究对象是，它允许将两个非对易算子提升到时序对易状态，在这种状态下，一个人可以执行分析运算，然后返回到原始的非对易状态。吉尔教授将扩展早期的工作，包括线性和非线性演化的一般理论，以及最弱的可能约束。第二个目标是通过寻找直接从进化生成器构造转移核的方法来扩展他的方法和路径积分之间的联系。最后一个目标是改进积分理论，以允许与物理学和概率论相关的许多路径积分不需要在广义Wiener空间上生成度量的事实。

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.