12
12. Global Non-Abelian Symmetries
In concluding this section we remark that, in this two-dimensional n-p
space, the electromagnetric charge operator is represented by the matrix
(
)
1 0
1
Q em =
= (1 2 + τ 3 ).
(12.44)
0 0
2
It is clear that although Q em commutes with τ 3 , it does not commute with
either τ 1 or τ 2 . Thus, as we would expect, electromagnetic corrections to the
strong interaction Hamiltonian will violate SU(2) symmetry.
12.2.2 Larger (higher-dimensional) multiplets of SU(2) in
nuclear physics
For the single nucleon states considered so far, the foregoing is really nothing
more than the general quantum mechanics of a two-state system, phrased in
‘spin-1/2’ language. The real power of the isospin (SU(2)) symmetry concept
becomes more apparent when we consider states of several nucleons. For
A nucleons in the nucleus, we introduce three ‘total isospin operators’ T =
(T 1 , T 2 , T 3 ) via
1
1
1
T =
τ (2) + . . . + τ (A) ,
(12.45)
τ (1) +
2
2
2
which are Hermitian. Here τ (n) is the τ -matrix for the nth nucleon. The
Hamiltonian H describing the strong interactions of this system is presumed
to be invariant under the transformation (12.40) for all the nucleons independently. It then follows that
[H, T] = 0.
(12.46)
Thus the eigenvalues of the T operators are constants of the motion. Further,
since the isospin operators for different nucleons commute with each other
(they are quite independent), the commutation relations (12.28) for each of
the individual τ ’s imply (see problem 12.4) that the components of T defined
by (12.45) satisfy the commutation relations
[T i , T j ] = i∈ ijk T k
(12.47)
for i, j, k = 1, 2, 3, which are simply the standard angular momentum commutation relations, once more. Thus the energy levels of nuclei ought to be
characterized – after allowance for electromagnetic effects, and correcting for
the slight neutron-proton mass difference – by the eigenvalues of T
2 and T 3 ,
say, which can be simultaneously diagonalized along with H. These eigenvalues should then be, to a good approximation, ‘good quantum numbers’ for
nuclei, if the assumed isospin invariance is true.
What are the possible eigenvalues? We know that the T’s are Hermitian
and satisfy exactly the same commutation relations (12.47) as the angular
momentum operators. These conditions are all that are needed to show that
1
the eigenvalues of T
2 are of the form T (T + 1), where T = 0, 2 , 1, . . ., and that
12. Global Non-Abelian Symmetries
In concluding this section we remark that, in this two-dimensional n-p
space, the electromagnetric charge operator is represented by the matrix
(
)
1 0
1
Q em =
= (1 2 + τ 3 ).
(12.44)
0 0
2
It is clear that although Q em commutes with τ 3 , it does not commute with
either τ 1 or τ 2 . Thus, as we would expect, electromagnetic corrections to the
strong interaction Hamiltonian will violate SU(2) symmetry.
12.2.2 Larger (higher-dimensional) multiplets of SU(2) in
nuclear physics
For the single nucleon states considered so far, the foregoing is really nothing
more than the general quantum mechanics of a two-state system, phrased in
‘spin-1/2’ language. The real power of the isospin (SU(2)) symmetry concept
becomes more apparent when we consider states of several nucleons. For
A nucleons in the nucleus, we introduce three ‘total isospin operators’ T =
(T 1 , T 2 , T 3 ) via
1
1
1
T =
τ (2) + . . . + τ (A) ,
(12.45)
τ (1) +
2
2
2
which are Hermitian. Here τ (n) is the τ -matrix for the nth nucleon. The
Hamiltonian H describing the strong interactions of this system is presumed
to be invariant under the transformation (12.40) for all the nucleons independently. It then follows that
[H, T] = 0.
(12.46)
Thus the eigenvalues of the T operators are constants of the motion. Further,
since the isospin operators for different nucleons commute with each other
(they are quite independent), the commutation relations (12.28) for each of
the individual τ ’s imply (see problem 12.4) that the components of T defined
by (12.45) satisfy the commutation relations
[T i , T j ] = i∈ ijk T k
(12.47)
for i, j, k = 1, 2, 3, which are simply the standard angular momentum commutation relations, once more. Thus the energy levels of nuclei ought to be
characterized – after allowance for electromagnetic effects, and correcting for
the slight neutron-proton mass difference – by the eigenvalues of T
2 and T 3 ,
say, which can be simultaneously diagonalized along with H. These eigenvalues should then be, to a good approximation, ‘good quantum numbers’ for
nuclei, if the assumed isospin invariance is true.
What are the possible eigenvalues? We know that the T’s are Hermitian
and satisfy exactly the same commutation relations (12.47) as the angular
momentum operators. These conditions are all that are needed to show that
1
the eigenvalues of T
2 are of the form T (T + 1), where T = 0, 2 , 1, . . ., and that