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= O[ oo/(x, t) [0) =
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= O[ o/lf'(x, t) [0) = 0
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(11-12)
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The last equality follows from the fact that V j(x, t) = i[P, j(x, t)J. Assuming the matrix element of j(x, t) to be meaningful the matrix element of its threedivergence must vanish between zero momentum states. Finally O/lj/l(x) being a Lorentz scalar, Eq. (11-12) generalizes to arbitrary matrix elements between the vacuum and any other state. Hence o/lj/l annihilates the vacuum. One may show that this is only possible for a local operator if it vanishes. We conclude that (11-13) The symmetry is exact and unitarily implemented by operators of the form U = e iroQ
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11-2 MASS SPECTRUM, MULTIPLETS, AND GOLDSTONE BOSONS 11-2-1 The Octet Model of GeII-Mann and Ne'eman
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When the internal symmetries form a compact Lie group, we may choose the phases of the representative operators to obtain a unitary representation of its covering group. This representation splits into irreducible components, acting on
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subspaces of the original Hilbert space. The latter is therefore generated by multiplets of states. Here we shall briefly describe the SU(3) symmetry or octet model valid with some degree of approximation at the level of strong interactions. It generalizes the familiar isospin in variance of nuclear forces introduced by Heisenberg in the 1930s. Larger flavor groups, as they are called nowadays, manifest themselves at higher energies. There is already some evidence for an SU(4) symmetry through the study of the new multi-GeV narrow resonances, and it may well be that this is not the end of the story. The members ofthe elementary isodoublet, proton and neutron, are classified according to the eigenvalues ofthe third component of isospin T3 with eigenvalues i- in such a way that (11-14) Here N is the baryonic charge. In the lagrangian framework, the corresponding fields will form an isospinor ljJ and, neglecting mass differences, the kinetic term
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ff' 0 = ljJ(i
+ M)ljJ
(11-15)
is invariant under isospin rotations
with U a 2 x 2 unimodular unitary matrix. In the simplest case we couple this field to the pion isovector 1t in an invariant way (see Chap. 5). For the pion, a relation analogous to (11-14) holds with N = and T3 taking the values + 1,0, or -1. Using the pseudo scalar character of 1t the only (renormalizable) invariant interaction is
(11-16) The boldface characters used for the Pauli matrices or the pion field refer to isospace. Equation (11-16) expresses in a compact form a set of relations on the various couplings (11-17) asserting the dynamical content of the symmetry. These relations were obtained by noting that the usual fields n(+), n(-) = (n(+ )t, and n(O) are related to the cartesian coordinates through
i-(ni
+ n~ + n~) =
n(-)n(+)
+ i-n(W
(11-18)
SYMMETRIES
)( )(
1f)( )(
..1.-
..1. 0
..1.+
..1.++
Isopin
4(nucleon)
Figure 11-1 Weight diagrams for
SU(2).
Isopin I (pion) Isopin ~ (..1. resonance)
Note that n(-) creates a positive pion and annihilates a negative one. When discussing pion-nucleon scattering we mentioned some dynamical consequences of this symmetry such as triangular inequalities on cross sections. In the early 1960s, Gell-Mann and Ne'eman succeeded in obtaining a generalization of isospin symmetry to a larger group SU(3). More precisely, all known multiplets correspond to representations of the factor group SU(3)jZ3 where Z3 is the abelian center of SU(3) generated by the cubic roots of the identity. The latter therefore acts as the identity on all hadronic states which are called of triality zero. We recall that the representations are conveniently described in terms of the Lie algebra of the infinitesimal hermitian generators of the group. The procedure is familiar in quantum mechanics. We diagonalize a maximal set of commuting generators (Cartan subalgebra). Basis states are represented by weight vectors with components equal to the eigenvalues of these operators in a space of dimension eqwil to the rank of the Lie algebra, i.e., the dimension of its Cartan subalgebra. For the familiar example of SU(2), of rank one, weight diagrams are one-dimensional, the abscissa being the eigenvalue of T3 (Fig. 11-1). A particular role is played by the adjoint representation acting on the Lie algebra itself through the linear action of commutation. In the case of SU(2) it corresponds to isospin 1 with
T = Tl [T3' T ] =