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Symmetries in Physics: Isospin and the Eightfold Way

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Symmetries in Physics: Isospin and the Eightfold Way 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 αkSk . (12.1) k 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(iαkSk) ψE,λ1,...,λn . (12.3) If the newly obtained state is linearly independent of the original one, this implies a degeneracy in the spectrum. We will investigate this shortly in more detail in the case of systems with spherical symmetry, where the symmetry generators, Sk, can be identified with the angular momentum operators, Jk, as studied in chapter 7. Lie groups play an essential role in the discussion of mass degeneracies in particle physics. In order to illustrate this, we first consider a particular example of the implications of symmetry, namely motion in a spherically symmetric potential described by the group SO(3) (or its double covering SU(2) as discussed in section 5.12). In chapter 7 the dynamics of a particle moving in three dimensions under the influence of a spher- ically symmetric potential, V (r), has been discussed. The spherical symmetry implies the commu- tation of the Hamiltonian with angular momentum operators (7.8) [H,ˆ Jk]=0,k=1, 2, 3 . (12.4) The stationary Schr¨odinger equation, (7.18), can then be reduced to a one-dimensional radial equation, (7.24), which yields a set of eigenstates of the form ψE,,m(r)=vE,,m(r) Ym(θ, φ), (12.5) with m = −l, . ···,l and the corresponding energy levels are independent of m. Therefore, each energy level is (2l + 1)-fold degenerate. This degeneracy follows from the fact that any rotation, as represented by an element of SO(3), (7.39), generates a state which has the same energy as the original one. Presence of additional symmetries may further increase the degeneracy of the system. As an example of this we will consider the Coulomb problem with the Hamiltonian p2 k H = − . (12.6) 2m r From elementary quantum mechanics we know the spectrum of the hydrogen atom. The energy levels are mk2 n − E = 2 2 , (12.7) 2~ n where the orbital angular momentum, l, is allowed to take values in 0,...,n− 1. The energy levels are totally independent of l. For example, the states 3s,3p and 3d all have the same energy. We want to understand this extra degeneracy in terms of the extra symmetry of the hydrogen atom given by an additional set of symmetry generators introduced below. Classically, the additional symmetry generators of the Coulomb problem are the three components of the so-called eccentricity vector discovered by Hamilton 1 r = p × J − k . (12.8) m r 12.1: Symmetry and Degeneracies 375 The vector points along the symmetry axis of the elliptical orbit and its length equals the eccen- tricity of the orbit. The vector in (12.8) is not a hermitian operator. The corresponding quantum mechanical hermitian operator can be defined by 1 x = (p J − p J + J p − J p ) − k 1 , (12.9) 1 2m 2 3 3 2 3 2 2 3 r 1 − x1 = (J p − J p + i ~p ) k , (12.10) 1 2m 3 2 2 3 1 r 1 − x2 = (J p − J p + i ~p ) k , (12.11) 2 2m 1 3 3 1 2 r 1 − x3 = (J p − J p + i ~p ) k , (12.12) 3 2m 2 1 1 2 3 r (12.13) It can be verified explicitly that its components commute with the Hamiltonian. In order to understand the aforementioned extra degeneracy, we will compute the hydrogen spec- trum using the additional symmetry. For this purpose we first note that J · =0. (12.14) This follows from a · (a ×b)=(a ×b) · a = 0, which is valid even when a and b do not commute. We will also need the following identity [4] 2 2H 2 2 2 = J + ~ + k , (12.15) m which can be proved after very considerable algebra. In the following we consider the bound states, which have a negative energy E. Therefore, in the subspace of the Hilbert space corresponding to a certain energy we can replace H by E.Nowwe scale the eccentricity vector as follows m K = − . (12.16) 2E Through some algebra [4] the following commutation relations can be verified [Ki,Jj]=i ~ijkKk, (12.17) [Ki,Kj]=i ~ijkJk, (12.18) which complement the familiar angular momentum algebra of section 5.3. We introduce the following new operators 1 A = (J + K ), (12.19) 2 1 B = (J − K ), (12.20) 2 which can be shown to satisfy [Ai,Aj]=i ~ijkAk, (12.21) 376 Isospin and the Eightfold Way [Bi,Bj]=i ~ijkBk, (12.22) [Ai,Bj]=0, (12.23) A, H =0, (12.24) B,H =0. (12.25) So far we have shown that the symmetry generators form an algebra, which is identical to the the direct sum of the Lie algebra of of two SO(3) (or SU(2)) algebras. By comparing to the rotation algebra introduced in chapter 5, we can read off the eigenvalues of A2 and B 2 from (12.21) and (12.22): 2 2 1 A = a(a +1)~ ,a=0, , 1,... , (12.26) 2 2 2 1 B = b(b +1)~ ,b=0, , 1,... (12.27) 2 Following (12.14) we note that A2 − B 2 = J · =0. (12.28) This implies that a = b. In order to arrive at the spectrum a final bit algebra is needed A2 + B 2 = J2 + K 2 (12.29) m = J2 − 2 (12.30) 2E 2 mk − 1 2 = − ~ , (12.31) 4E 2 where we have used (12.15). Using this equation the energy eigenvalues can be written in terms of the eigenvalues of A2 and B 2 operators. Noticing that A2 and B 2 have the same eigenvalues because of (12.28), the energy eigenvalues are found to be 2 − mk 1 E = 2 2 ,a=0, , 1,... (12.32) 2~ (2a +1) 2 A comparison with (12.7) tells us that (2a +1)=n. Furthermore the bound on the orbital angular momentum, l, can be seen to follow from the triangle inequality as applied to J = A + B ,namely that J > A − B = 0 (12.33) J < A + B =2A (12.34) It follows that l has to have values in {0=|a − b|, 1,...,a+ b = n − 1}. This illustrates the effect of additional symmetries to the degeneracy structure of a quantum mechanical system. In contrast to the discussion above about extra symmetries, a lack of symmetry implies a lack of degeneracy in the energy levels of a quantum mechanical system. The most extreme case of this is the quantum analogue of a classically chaotic system. Chaos is described classically as exponential 12.2: Isospin and the SU(2) flavor symmetry 377 sensitivity to initial conditions, in the sense that nearby trajectories in the phase space diverge from each other over time. However, another manifestation of chaos is the lack of independent operators commuting with the Hamiltonian. A typical example of this so called quantum chaos is the quantum billiard problem, which is a particle in box problem in two dimensions with a boundary which can be chosen arbitrarily. If the chosen boundary is ‘irregular’ in a suitably defined sense, the classical trajectories will diverge from each other after successive bounces from the boundary. For a more detailed discussion of quantum chaos in billiard systems we refer the reader to [7] and the references therein. In the case of billiards and other examples of quantum chaos one common observation is the almost nonexistence of degeneracies and the fact that the energy levels are more evenly spaced. This is known as level repulsion. In the next section we will proceed with the discussion of a symmetry, which was discovered by observing degeneracies in the particle spectrum.
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