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

拟开展的研究将集中在奇异势的理论和应用以及它们与量子场论的关系上。我们将特别强调那些表现出渐近紫外共形不变性的系统。共形反常将在量子力学问题中进行分析，这些问题在分子物理中有应用，如各向异性反平方势。路径积分方法将被用来研究这个问题以及整个共形不变相互作用家族。第二个项目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方法，另一种是自伴延拓方法。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性来理解。第四个项目是理解奇异势在降维方案中的作用，在降维方案中，低能物理被限制在薄膜中，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.