Introdução à álgebra do espaço-tempo e à supersimetria
Data
2022-11
Autores
Ferreira, Danilo Cardoso [UNIFESP]
Orientadores
Senise Junior, Carlos Roberto [UNIFESP]
Tipo
Trabalho de conclusão de curso de especialização
Título da Revista
ISSN da Revista
Título de Volume
Resumo
As simetrias desempenham um papel fundamental no desenvolvimento, análise e construção de teorias na Física. Dentro deste campo de pesquisa, podemos destacar o grupo de Lorentz L, que é a base fundamental da teoria da relatividade restrita. O grupo de Poincaré P, que realiza as transformações reais no espaço de Minkowski, é um produto semidireto entre o grupo de Lorentz e grupo de translações do espaço-tempo quadridimensional. O grupo P possui a representação dos geradores de rotações e translações no espaço de Minkowski quadridimensional e foi firmemente estabelecido teórica e experimentalmente. É possível realizar uma extensão supersimétrica da álgebra de Poincaré a uma álgebra de Lie graduada, utilizando os espinores de Dirac e Majorana, além da teoria Weyl. Tal empreendimento nos conduz há um conjunto de relações de comutadores e anticomutadores, denominado álgebra de Super-Poincaré. A investigação dos operadores de Casimir da álgebra de Super-Poincaré nos conduz aos operadores de criação e aniquilação, e ao vácuo de Clifford. A representação irredutível da álgebra de Super-Poincaré nos conduz à supersimetria para N = 1 e a consequência de que o número de estados fermiônicos é igual ao número de estados bosônicos.
Symmetries play a fundamental role in the development, analysis and construction of theories in Physics. Within this field of research we can highlight the Lorentz group L which is the fundamental basis of the special theory of relativity. The Poincaré group P, which performs the real transformations in Minkowski space, is a semi-direct product between the Lorentz group and the four-dimensional space-time translation group. The group P has the representation of rotation and translation generators in four-dimensional Minkowski space and has been firmly established theoretically and experimentally. It is possible to perform a supersymmetric extension of the Poincaré algebra to a graded Lie algebra using Dirac and Majorana spinors in addition to Weyl theory. Such an undertaking leads us to a set of commutators and anticommutators relations called Super-Poincaré algebra. The investigation of the Casimir operators of the Super-Poincaré algebra leads us to the creation and annihilation operators, and the Clifford vacuum. The irreducible representation of the Super-Poincaré algebra leads us to supersymmetry for N = 1 and the consequence that the number of fermionic states is equal to the number of bosonic states.
Symmetries play a fundamental role in the development, analysis and construction of theories in Physics. Within this field of research we can highlight the Lorentz group L which is the fundamental basis of the special theory of relativity. The Poincaré group P, which performs the real transformations in Minkowski space, is a semi-direct product between the Lorentz group and the four-dimensional space-time translation group. The group P has the representation of rotation and translation generators in four-dimensional Minkowski space and has been firmly established theoretically and experimentally. It is possible to perform a supersymmetric extension of the Poincaré algebra to a graded Lie algebra using Dirac and Majorana spinors in addition to Weyl theory. Such an undertaking leads us to a set of commutators and anticommutators relations called Super-Poincaré algebra. The investigation of the Casimir operators of the Super-Poincaré algebra leads us to the creation and annihilation operators, and the Clifford vacuum. The irreducible representation of the Super-Poincaré algebra leads us to supersymmetry for N = 1 and the consequence that the number of fermionic states is equal to the number of bosonic states.