System of Imprimitivity and Mackey's Theorem

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.