Introduction to the Mathematical Methods of Theoretical Physics

• Grant number: 25/23674-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: January 01, 2026
• End date: December 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Physics - Elementary Particle Physics and Fields
• Principal Investigator: João Carlos Alves Barata
• Grantee: Thiago Rondon Bassul Campos
• Host Institution: Instituto de Física (IF). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In broad terms, the project aims to prepare the student in themes related to Functional Analysis and Operator Algebras, aiming for a more in-depth future research in Quantum Field Theories. The main work towards which the student is to dedicate himself is the study of the Reeh-Schlieder Theorem and one of its main applications to Quantum Field Theories: a manifestation of the Tomita-Takesaki Theorem (the central theme of this project) in the description of movements under constant acceleration (Lorentz boosts) in Minkowski spacetime, as in the work of J. Bisognano and E. H. Wichmann: "On the duality condition for a Hermitian scalar field". That work presents a mathematical explanation for the Unruh effect, whose analysis can be extended to spacetimes with a bipartite horizon, further allowing an analysis of the Hawking effect associated with radiation emission by black holes.The initial phase of the project, in which the student is already involved, includes an introductory study of Topology, especially of metric spaces, and Analysis, with basic notions from the theory of measure and integration. It also includes the study of basic notions of Banach and Hilbert spaces and of the theory of continuous (bounded) operators acting on those spaces. Some of the themes to be explored by the student in this phase include: * Basic notions of Hilbert and Banach spaces.* Linear operators in Normed Vector Spaces.* Banach Spaces of Operators.* The Hahn-Banach Theorem and some of its Consequences.* The Banach-Steinhaus Theorem or the Uniform Boundedness Principle.* The Open Mapping Theorem and the Closed Graph Theorem.* Bounded Operators in Hilbert Spaces.* Self-adjoint, Normal and Unitary Operators, Orthogonal Projectors and Partial Isometries in Hilbert Spaces.* Outlines of the Theory of Banach and C*-Algebras.* The Inverse of Bounded Operators.* Spectral Theory.* The Gelfand Homomorphism in C*-Algebras.* States and Representations in C*-Algebras. GNS Representation.* Pure and Mixed States, and the Irreducibility of GNS Representations. * Von Neumann Algebras. A Minimum.* The Bicommutant Theorem in von Neumann Algebras.At a later stage, the student will be introduced to Classical Field Theory, exploring the Lagrangian and Hamiltonian formalisms in the context of scalar, Dirac and classical electromagnetic fields and studying important themes like Noether's Theorem and the relationship between symmetry principles and conservation laws. Then, the student will be presented with the theory of quantized free fields, the canonical commutation relations in the bosonic and fermionic cases and the quantization problems in Electromagnetism. He will also be introduced to perturbation theory, approaching themes like the Wick Theorem and the basic problem behind regularization processes in that theory.By that point, the student is to turn himself toward the study of more mathematically precise formulations of Quantum Field Theory, with particular emphasis given to the study of analytic properties of Wightman distributions, intimately related to locality and Lorentz-invariance properties and of importance to the discussion of the so called PCT Theorem. Such study combines itself perfectly with the project's aforementioned central theme: the Tomita-Takesaki Theorem.At last, the student will study the Tomita-Takesaki Modular Theory and its uses in physics. The main focus in this final step is the Reeh-Schlieder Theorem and one of its important consequences to the algebraic formulation of Quantum Field Theories: the description, due to Bisognano and Wichmann, of the relationship between Lorentz boosts in Minkowski spacetime and the modular group. Studying the work of those scientists, the student will learn a mathematical explanation for the aforementioned Unruh effect and, basing himself off the work of Geoffrey L. Sewell, he will also get acquainted with the extension of that analysis to the Hawking effect. (AU)