Giampieri, Marco
(2021)
System of Imprimitivity and Mackey's Theorem.
[Laurea], Università di Bologna, Corso di Studio in
Matematica [L-DM270]
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Abstract
The aim of this thesis is to study and classify the representations of the Poincaré group, whose elements are all the isometries of the Minkowski Spacetime. These representations are important in particle physics since
they can be considered the mathematical equivalent of the elementary particles of the Standard Model, hence it is possible to identify and classify all
such particles through the analysis of the Poincaré representations. In order to achieve this we introduce the concepts of group action, representation and semidirect product of groups and the structure of Lie group and Lie algebra. With those preliminary concepts we compute the representations of the key group SL(2,C) and introduce the fundamentals of the system of imprimitivity. Finally we obtain, through the Mackey’s Theorem, information on the representations of the semidirect product of two groups starting from the representations of the groups themselves. We apply those results to the Poincaré group, which is a semidirect product of translations and Lorentz transformations, and conclude with a brief classification of the Poincaré representations.
Abstract
The aim of this thesis is to study and classify the representations of the Poincaré group, whose elements are all the isometries of the Minkowski Spacetime. These representations are important in particle physics since
they can be considered the mathematical equivalent of the elementary particles of the Standard Model, hence it is possible to identify and classify all
such particles through the analysis of the Poincaré representations. In order to achieve this we introduce the concepts of group action, representation and semidirect product of groups and the structure of Lie group and Lie algebra. With those preliminary concepts we compute the representations of the key group SL(2,C) and introduce the fundamentals of the system of imprimitivity. Finally we obtain, through the Mackey’s Theorem, information on the representations of the semidirect product of two groups starting from the representations of the groups themselves. We apply those results to the Poincaré group, which is a semidirect product of translations and Lorentz transformations, and conclude with a brief classification of the Poincaré representations.
Tipologia del documento
Tesi di laurea
(Laurea)
Autore della tesi
Giampieri, Marco
Relatore della tesi
Scuola
Corso di studio
Ordinamento Cds
DM270
Parole chiave
semidirect product representation Mackey's theorem system of imprimitivity Poincaré group Lie algebra particle classification
Data di discussione della Tesi
28 Maggio 2021
URI
Altri metadati
Tipologia del documento
Tesi di laurea
(NON SPECIFICATO)
Autore della tesi
Giampieri, Marco
Relatore della tesi
Scuola
Corso di studio
Ordinamento Cds
DM270
Parole chiave
semidirect product representation Mackey's theorem system of imprimitivity Poincaré group Lie algebra particle classification
Data di discussione della Tesi
28 Maggio 2021
URI
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