SymmetriesinPhysicsIsospinand.PDFVIP

Chapter 12 Symmetries in Physics: Isospin and the Eightfold Way by Melih Sener and Klaus Schulten Symmetries and their consequences are central to physics. In this chapter we will discuss a particular set of symmetries that have played a seminal role in the development of elementary particle and nuclear physics. These are the isospin symmetry of nuclear interactions and its natural extension, the so-called Eightfold Way. The organization of this chapter is as follows: In the next section we will discuss the relation between symmetries of a quantum mechanical system and the degeneracies between its energy levels. We will particularly use the example of spherically symmetric potentials. In the following section we will introduce the concept of isospin as an approximate SU(2) symmetry, which identifies the proton and the neutron as different states of the same particle. We will also introduce the quark model as a natural framework to represent the observed symmetries. We will apply these concepts to an analysis of nucleon-nucleon and nucleon-meson scattering. In the final section, we will discuss the SU(3) symmetry of three quark flavors. The algebraic structure and the representations of SU(3) will be discussed in parallel to SU(2) and particle families will be identified in terms of representations of the underlying symmetry group. 12.1 Symmetry and Degeneracies The degeneracies of energy levels of a quantum mechanical system are related to its symmetries. Let us assume a continuous symmetry obeyed by a quantum mechanical system. The action of the symmetry operations on quantum mechanical states are given by elements of a corresponding Lie group, i.e., O = exp (∑ k αkSk ) . (12.1) Then the generators, Sk, will commute with the Hamiltonian of the system, [H,Sk] = 0. (12.2) 373 374 Isospin and the Eightfold Way The action of any symmetry generator, Sk, on an energy eigenstate, ψE,λ1,...,λn , leaves the energy of the state invariant H exp(iαkSk) ψE,λ1,...,λn = E exp(iαk