theoretical physicist

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Here you can find many documents I’ve written over the years in order to learn a number of subjects. Some of them correspond to lecture notes I’ve written before teaching a course or a seminar, some were study notebooks written just for the sake of it and could be useful for someone, and some were the lecture notes I took (and typed) while watching a course. Some of these are in Portuguese and some are in English. I chose to leave the descriptions in agreement with the project’s language. Notice most titles are hyperlinks to the pdfs. The documents not currently available are absent due to some personal reason, but I might be willing to share some of them upon request.

Notes on Functional Analysis

This document is a personal study notebook about functional analysis. It is intended to guide my studies of the subject and work as a manner of registering my thoughts, doubts, and solutions. Hopefully, it will also be of use to other students. The book by Oliveira (2018) is the main reference in the sense I’ll follow its structure and order, but hopefully, my comments will be more authentic. Some other books on functional analysis that I might often check are those by Brezis (2011), Conway (2007), Kreyszig (1978), Reed and Simon (1980), and Rudin (1991), among others.

These notes are written from the point of view of a theoretical physicist interested in learning more mathematics. Hence, they might have a style somewhat different from usual mathematics texts. I’ll typically try to write to an audience of similar-minded people who already have some necessary prerequisites, but might like to be convinced that the definitions make sense. Well, at least that’s how I usually like to learn mathematics.

Talking about prerequisites„ I shall assume you to be familiar with topology, linear algebra, measure theory, and so on. These topics can be studied, for example, in the books by Axler (2015), Folland (1999), Geroch (1985), Munkres (2000), and Simon (2015).

Entropy Bounds and Holography in General Spacetimes

This notebook is an introduction to entropy bounds and holography in general spacetimes. Its main purpose is to gain practice with these themes. The themes and structure are greatly inspired by the lecture series by Bousso.

Statistical Mechanics

This is my study notebook for the course Statistical Mechanics, taught by Prof. Carlos E. Fiore at the University of São Paulo’s Institute of Physics (IFUSP) on the second semester of 2022. It was written as a way of keeping up with the course and correspond to my lecture notes with some additions from extra bibliography and sometimes slightly different notation. These notes are not endorsed by Prof. Fiore or IFUSP.

Flat Spacetimes

These are some examples of how some quite interesting results in Relativity can be found without ever leaving Minkowski spacetime. We consider some universes within Minkowski spacetime—viz. the Rindler and Milne universes—and use them to understand some properties of black holes and cosmology.

Applications of Topos Theory to the Mathematical Foundations of Quantum Theory

This is my study notebook for a minicourse given by Prof. Walter de Siqueira Pedra (DFMA-IFUSP) at the IV Jayme Tiomno School on Physics. The school was organized at the University of São Paulo’s Institute of Physics (IFUSP) by the Dead Physicists Society (DPS), a student-driven organization at IFUSP. These notes were written as a way of practicing with the theory and keeping up with the minicourse. I claim no originality on results or presentation. Notice though that these notes are, quite naturally, my interpretation of the lectures, which means I might as well have added some original mistakes to them. This text is not endorsed by Prof. Pedra, IFUSP, or DPS.

Clifford Algebras and Rotations

This is a study notebook on Clifford algebras, rotations, spin groups, and related topics. It was written as a way of keeping up with my studies on these subjects. The discussion begins with some properties of ordinary three-dimensional rotations and proceeds towards issues, such as gimbal lock, solves them by representing rotations in terms of quaternions, and uses this as motivation to define more general Clifford algebras, spin groups, and so on. Some familiarity with topology and Lie groups is assumed.

Técnicas de Demonstração

Este documento é uma breve introdução às principais técnicas de demonstrações matemáticas para estudantes de Física. É recomendada uma exposição prévia a alguns assuntos de Cálculo (como a definição formal de limite), mas o texto tem o objetivo de ser tão autocontido quanto possível. A princípio são estudadas, por meio de exemplos utilizando números inteiros, as técnicas de prova direta, por absurdo, por contrapositiva e por indução, além de alguns símbolos comuns em demonstrações matemáticas. A seguir, algumas destas técnicas são utilizadas para mostrar como os procedimentos usuais de cálculo de limites se relacionam com a definição formal.

Stress-Energy Tensor for a Scalar Field

This is a computation of the stress-energy-momentum tensor for a scalar field with non-minimal coupling to the background curved spacetime. The goal is mainly to keep the result ready at hand when I need it again in the future, so parts of the computation come from references instead of being redone entirely.

Vector Calculus from a Geometrical Perspective

This document is a brief exercise on Differential Geometry aimed at proving the usual expressions of vector calculus in curvilinear coordinates by employing Differential Geometry techniques. The current version only considers cylindrical coordinates (and does not include the vector Laplacian), but I might do spherical coordinates sometime and maybe some less usual coordinate system.

Geometric Methods of Theoretical Physics

Years ago I convinced myself that one of the main issues of an undergraduate course in Physics was the major risk that someone could graduate without being exposed to some Differential Geometry. As a relativist, I’m certainly have a few personal reasons to believe that, but I also believe Differential Geometry is a powerful tool in Theoretical Physics even if we pretend to forget about its role in General Relativity. For example, it can give us deep insights in Classical Mechanics, Thermodynamics, Electrodynamics, and certainly in even more cases. Eventually, I got the courage to start putting together a course in Differential Geometry for physicists. These notes are the current draft of what I’d like to put in such a course.

Penrose Diagrams

This document is a collection of computations of Penrose diagrams in a few spacetimes of interest (Minkowski, Schwarschild, and three different FLRW cosmologies). I wrote it to get acquainted with computing of Penrose diagrams in General Relativity. The only essential prerequisite should be a basic knowledge of General Relativity, but I tried to leave indications of where a few subjects are covered in Wald’s and Hawking & Ellis’ books.

Non-Perturbative Aspects of Quantum and Semiclassical Gravity

Not Currently Available

I’ve been writing these notes as a way to organize my literature review for my MSc project. While writing these, I’m interested in learning how to describe quantum fields in the presence of a classical gravitational field, which is a framework commonly known as Quantum Field Theory in Curved Spacetime. This is the main subject of these notes, which assume reasonable familiarity with both General Relativity and Quantum Field Theory in flat spacetime. It also covers the main ideas behind the Functional Renormalization Group, which is the main tool in my current project.

Hyperbolic Equations

Not Currently Available

These are some study notes I’ve developed throughout my undergraduate research project. The project’s goal was to study Hyperbolic Equations, but this requires a lot of previous knowledge and, as a consequence, most of the material covered in here involves not Hyperbolic Equations, but Topology, Functional Analysis and even some Differential Geometry.

Minicurso III Escola Jayme Tiomno: Métodos Algébricos da Física Teórica

Estas são notas de aula para um minicurso ministrado durante a III Escola Jayme Tiomno de Física Teórica, organizada entre 02 e 06 de agosto de 2021 pela Dead Physicists Society de modo online, embora voltada aos alunos de graduação do Instituto de Física da Universidade de São Paulo. O minicurso aborda uma introdução à teoria de grupos e representações para físicos com alguns exemplos e motivações vindos da Relatividade Geral e Física de Partículas.

Minicurso XV STO: Introdução ao LaTeX

Estas notas de aula foram desenvolvidas para o minicurso “Introdução ao LaTeX”, ofertado nos dias 10 e 12 de novembro de 2020 como parte da XV Semana Temática de Oceanografia. No curso, pretendeu-se apresentar a linguagem LaTeX para a escrita de documentos a partir de seus princípios básicos. Ao longo do minicurso, pretendeu-se mostrar o processo de escrita de um artigo científico em LaTeX, contando com exemplos de uso de pacotes e práticas úteis para profissionais das áreas das Ciências Exatas. Não foram assumidos conhecimentos prévios e o único requisito técnico é um computador com acesso a internet.

Minicurso DPS 2020: Mecânica Clássica

Estas são as notas de aula para um minicurso ministrado virtualmente aos estudantes de graduação do Instituto de Física da Universidade de São Paulo durante as férias de inverno do ano de 2020. Elas são uma introdução às formulações Lagrangeana e Hamiltoniana da Mecânica Clássica e incluem uma breve introdução à equação de Hamilton-Jacobi.

A Brief Introduction to Classical Field Theory

In this work, I present an introduction to classical field theory by exploring the limiting case of N coupled harmonic oscillators as N tends to infinity in order to obtain the equation of motion for a vibrating string. The Euler-Lagrange Equations for a collection of N three-dimensional fields are presented and, finally, Noether’s Theorem is proved, with the stress-energy tensor and the conservation of electric charge due to gauge invariance in QED being given as examples of application.

O Grupo de Lorentz

Relatividade Restrita aos olhos de um matemático

Este trabalho é a versão escrita de um seminário apresentado aos calouros do Instituto de Física da USP em fevereiro de 2020 com a intenção de divulgar o trabalho em linhas gerais da Física Matemática por meio do estudo da Teoria da Relatividade Restrita. Tomamos como hipótese a existência de intervalos do tipo tempo, luz e espaço e obtemos a forma do Grupo de Lorentz ao buscar um conjunto de transformações lineares que preservem causalidade no espaço de Minkowski. Obtém-se então as transformações de Lorentz por meio da estrutura algébrica do Grupo de Lorentz. Por fim, como exemplos de aplicação da Relatividade, mostra-se a contração dos comprimentos e a dilatação do tempo.

O seminário original foi apresentado aos calouros do IFUSP numa das aulas de recepção da Dead Physicists Society, em 07 de fevereiro de 2020.

Locally Convex Spaces and the Tempered Distributions

This text is a brief discussion about some elementary topics on the theory of locally convex spaces and tempered distributions as continuous linear functionals on a particular LCS: the Schwartz space of functions of rapid decrease. It is part of the evaluation of the course “Mathematical Physics III”, taught by Prof. Walter de S. Pedra at IFUSP on the first semester of 2020. Familiarity with general topology and linear topological spaces is assumed.

The High-Energy Cross Section for the e+e- -> mu+mu- Process in QED

In this work, elementary S-matrix theory is briefly explained and the expression for the differential cross-section of some process in terms of the S-matrix is obtained. With these in hand, the Feynman rules are used in order to compute the invariant amplitude for the e+e- -> mu+mu- process in QED in the high-energy limit, which is then utilised in order to calculate the high-energy cross-section for the e+e- -> mu+mu- process.

This text was written as an exercise for a course in High-Energy Physics.

A Not So Brief Introduction to Topology

This is the chapter on Topology from my study notes about Hyperbolic Equations.

Differential Geometry

This is the chapter on Differential Geometry from my study notes about Hyperbolic Equations, originally written as preparation for a seminar on manifolds and stuff for the Wigner Group Seminars at IFUSP.

An Introduction to Measure Theory

These are some study notes I’ve been developing while studying Measure Theory. Most of the material presented in here was strongly based upon Prof. Walter Pedra’s course on Mathematical Physics III presented at the Institute of Physics at the University of São Paulo in Fall 2020. Indeed, these notes were originally written as a way to keep up with the course.

As Time Goes By

Seeking a Solution to Maxwell’s Equations

The power of potential formulation of Electro and Magnetostatics is widely known by Physics students. It allows us to calculate the scalar potential in order to find the Electric Field and then use the mathematical similarities between Electro and Magnetostatics to find the vector potential and, then, the Magnetic Field. In this paper, aimed at undergraduate students, we explore the potential formulation of Electrodynamics, departing from static scalar and vector potentials and walking towards the potential expressions for electrodynamical fields. Using the Lorenz gauge, we find the retarded potentials and Jefimenko’s Equations, the latter being the solution to the microscopic version of Maxwell’s Equations.

Notas de Álgebra

Anotações em pdf para estudo pessoal de Álgebra I (atualmente chamada de Números Inteiros: Uma Introdução à Matemática). O trabalho ainda está em desenvolvimento e devido a isso há algumas anotações ao longo do texto sobre mudanças para realizar posteriormente. Além disso, as demonstrações ainda não foram revisadas com a atenção devida.