Introduction to the Mathematical Methods of Theoretical Physics

• Grant number: 25/00964-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: March 01, 2025
• End date: February 28, 2026
• Field of knowledge: Physical Sciences and Mathematics - Physics
• Principal Investigator: João Carlos Alves Barata
• Grantee: Pedro de Oliveira Freitas
• Host Institution: Instituto de Física (IF). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The emphasis of the project is on improving the student's education in physics and mathematics, with a view to preparing them for their future involvement in research projects in the areas of mathematical physics and theoretical physics in general. In general terms, we will try to make progress towards preparing the student in topics related to Functional Analysis and Operator Algebras, with a view to a more in-depth study of Quantum Field Theories in the future. The first steps, which are already underway, will focus on an introductory study of Topology, especially metric spaces, and Analysis, with basic notions of measure theory and integration. In a second stage, the student will focus on studying the basics of Banach and Hilbert spaces and then devote himself to studying the theory of continuous (bounded) operators acting on such spaces. This important stage is preparatory to the more important stage involving the study of the so-called C* and von Neumann algebras. The development of operator algebras is of paramount importance for a precise mathematical formalization of many-body systems, as is the case in Statistical Mechanics and Quantum Field Theory. The project will also explore the CAR and CCR algebras, which are proving to be fundamental in the description of quantum many-body systems, as well as the Modular Theory of Tomita and Takesaki, which has shown increasing relevance in Quantum Physics.