Collaborative Research: Renormalization, Path Integrals, and Applications of Conformal Quantum Mechanics, Singular Potentials, and Quantum Field Theory

本研究将著重于奇异势的理论与应用，以及它们与量子场论的关系。特别强调的是那些系统表现出渐近紫外共形不变性。共形异常将在分子物理中应用的量子力学问题中进行分析，例如各向异性反平方势。路径积分方法将用于研究这一问题以及整个保形不变相互作用族。第二个课题是比较分析共形不变问题的两种方法，一种是重整化群分析，另一种是自伴随扩展方法。第三个项目是了解黑洞热力学是否可以用标量场的近视界共形对称来理解。第四个项目是理解奇异势在降维方案中的作用，其中低能物理被限制在一个膜中，而重力在一个额外的维度中传播。

The proposed research will focus on the theory and application of singular potentials and their relationship to quantum field theory. Particular emphasis will be given to those systems exhibiting asymptotic ultraviolet conformal invariance. The conformal anomaly will be analyzed in quantum mechanics problems that have applications in molecular physics such as the anisotropic inverse square potential. Path integral methods will be used to study this problem as well as a whole family of conformally invariant interactions. A second project will be to compare two methods of analyzing conformally invariant problems, one being a renormalization group analysis and the other being the method of self-adjoint extensions. The third project is to understand whether black hole thermodynamics can be understood in terms of the near-horizon conformal symmetry of a scalar field. The fourth project is to understand the role of singular potentials in dimensional reduction schemes where the low energy physics is confined to a brane but gravity propagates in one extra dimension.