On elementary particles as representations of the Poincaré group

Abstract

This thesis is concerned with the definition of elementary particles as irreducible projective unitary representations of the Poincaré group. During the contents of this work, we will introduce the relevant prerequisites and results. Concerning differential geometry, we will discuss smooth manifolds, Lie groups and Lie algebras. About quantum mechanics, we will introduce Hilbert spaces and the basic structures of quantum mechanics, together with Wigner’s theorem on symmetries. With respect to special relativity, we will present the Minkowski spacetime as an affine space an derive its group of automorphisms, the Poincaré group. We will finally talk about representations of Lie groups and define an elementary particle to be an irreducible projective representation of the Poincaré group.