Introduction to the Mathematical Methods in Theoretical Physics

• Grant number: 24/16165-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: October 01, 2024
• End date: September 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Physics - Elementary Particle Physics and Fields
• Principal Investigator: João Carlos Alves Barata
• Grantee: Louis Bergamo Radial
• Host Institution: Instituto de Física (IF). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The research project aims to enhance the education of Louis Radial, a student at the Institute of Physics of the University of São Paulo (IFUSP), in Mathematical Physics and Theoretical Physics, preparing him for future studies in advanced areas such as Functional Analysis and Operator Algebras. The project, which is expected to last two years, is already underway and involves a progressive approach, starting with introductory topics in Topology, Measure Theory, and Integration, followed by an in-depth study of Banach and Hilbert spaces, and culminating in the study of the Spectral Theorem and C¿ and von Neumann algebras.The development of operator algebras is fundamental for a precise mathematical formulation of many-body systems, as occurs in Statistical Mechanics and Quantum Field Theory. The project will also explore CAR and CCR algebras, which are important in the description of quantum many-body systems, and the Tomita-Takesaki Modular Theory, which has increasing relevance in Quantum Physics.