Representation theory of SU(2) and SU(3) with applications to spin and quark models

Abstract

In this thesis we provide an elementary introduction in finite dimensional representation theory of the Lie groups SU(2) and SU(3) for undergraduate students in physics and mathematics. We will also give two application of representation theory of these two groups in physics: the spin and quark models. We begin with first discussing representation theory for finite groups to create intuition for representations. We will explain notions such as intertwining maps and complete reducibility and we will mention some application of representation theory of finite groups inquantum mechanics. Hereafter, we begin with representation theory for Lie groups and Lie algebras, especially the groups SO(3) and SU(2), as these groups will play an important role in the description of spin. One of the main results is that SU(2) is the universal cover of SO(3). Furthermore, we give a description of spin by means of representation theory of SO(3) and its Lie algebra so(3). We will show that half integer representations of the Lie algebra so(3) cannot be exponentiated to representations of the Lie group SO(3), but it can be exponentiated to its universal cover SU(2). Moreover, we study the irreducible representations of SO(3) inside the Hilbert space L₂(R³). We will argue that one of the simplest quantum Hilbert spaces of a particle L₂(R³), can be modified to the completion of the tensor product L₂(R³) ⊗V, where, V is a finite dimensional Hilbert space that incorporates the internal degrees of freedom: spin. V carries an irreducible projective representation of SO(3). We will also discuss the addition of angular momentum of two particles in quantum mechanics. For this, we show how the tensor product of irreducible representations V and W of so(3) decomposes into SO(3) invariant subspaces of L2(R3). Hereafter, we will turn to representation theory of the Lie group SU(3) for setting up the mathematical framework for analysing the quark model. We will proof that there is a one-to-one correspondence between the irreducible representations of sl(3;C)and SU(3). We will also proof the theorem of the highest weight by which we can classify all the irreducible representations of SU(3) and sl(3;C) by their highest weight. We will also introduce the notion of the Weyl group and show that the Weyl group is a symmetry of weights of the finite dimensional representation of sl(3;C). Other properties of these representation, such as the dimension of the irreducible representations of sl(3;C) will be provided. Lastly, the quark model is discussed by means of representation theory of SU(3). We will show how this model can be used to classify two type of particles which also interact by means of the strong force: baryons and mesons. We show that we can classify the lightest mesons and baryons in so-called multiplets by the irreducible representations of SU(3). However, we will also introduce a modification of the strong force which further refines this model. A topic for further study would be how the symmetry group SU(3) describing Quantum Chromo Dynamics (QCD) can be used for the description of mesons and baryons.