Isospin rotations

It should be obvious that if A vanishes, the phase degree of freedom has to become redundant, as seen later. It would be worth mentioning that similar configuration has been studied in other contexts [21-23], Note that the configuration in (44) breaks rotational invariance as well as translational invariance, but the latter invariance is recovered by an isospin rotation [26]. 

A similar mechanism can be invoked to give mass to the quarks [though there may be other sources of mass for the quarks (see Section 20.3)], but this would leave the up-type quarks, the analogues of the neutrinos, massless. In order to construct an interaction invariant under weak SU(2) and weak U 1) which will give mass to the quarks with = 1/2, it turns out, on the basis of Table 9.2, that we need a Higgs doublet with Yw = —1-Recall that = has Y y = 1, and is a weak isospin doublet. Recall also, that for ordinary isospin, the nucleon doublet N = ( ) and the antinucleon doublet N = (jp) (note the minus sign ) transform identically under isospin rotations. By analogy... 

As an example consider the group of isospin rotations, 517(2). In this case the iJ j = 1,2,3) are the isospin matrices for example, for an isospin-half field they are r/2 where the r are the Pauli matrices, and the diagonal charge Qa is Ta, the operator whose eigenvalues are the third components of isospin. 

According to the general relationship (5.9), rotations in isospin space transform the electron creation operators by the D-matrix of rank 1/2. If we go over from these operators to the one-electron wave functions they produce, then we shall have the unitary transformation of radial orbitals... 

Under rotations in isospin space the wave functions (18.27) are transformed as follows ... 

Thus, in the limiting case, where the expansion of the electrostatic interaction operator in terms of the multipoles (see (19.6)) includes only the central-symmetric part (i.e. only the terms with k = 0), dependent on the term in (18.52) is only the summand with the operator T2.1116 eigenvalues of the operator T2, according to (18.28), are equal to T(T + 1), i.e. in this approximation we obtain the spectrum of energy levels rotational with respect to isospin. 

When the isospin quantum number is good , the states characterized by a larger value of the isospin T have a higher energy, which is mainly conditioned by the presence of the rotational term T(T + l)Go(nil,n2l) in diagonal (in isospin basis) matrix elements of the electrostatic energy operator. 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

The spin structure is irrelevant here, so let us simply denote by Jj a set of three Hermitian currents (j = 1,2,3) which transform like an isospin triplet under isotopic spin rotations. Thus from (1.2.18), since for SU 2) the are just the antisymmetric Cjki, we have... 

A more striking result can be achieved, if, for the type of isospin averaged data just discussed, one has also data for the antineutrino cross-sections. At the leptonic vertex, going from a i/ reaction to a P one has the effect of changing 75 to —75 as explained in Section 1.3. At the hadronic vertex —> h- and vice versa. But with our isotopically neutral initial and final states we can make a rotation of ir/2 about the 1 axis in isospace without affecting the states, and thereby change h back to h+, provided we are in a situation where it is a good approximation to take h hi i/i2. So the hadronic expression in the P cross-section is the same as it was in the v reaction. 

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