16
12. Global Non-Abelian Symmetries
represent the generators of SU(2) f in the 2* representation (i.e. the complex
conjugate of the original two-dimensional representation, which we will now
call 2). Referring to (12.25), we see that τ 1
∗ = τ 1 , τ 2
∗ = −τ 2 and τ 3
∗ = τ 3 . It is
then easy to check that the three matrices −τ 1 /2, +τ 2 /2 and −τ 3 /2 do indeed
satisfy the required commutation relations (12.28), and thus provide a valid
matrix representation of the SU(2) generators. Also, since the third component of isospin is here represented by −τ 3
∗ /2 = −τ 3 /2, the desired reversal in
sign of the additively conserved eigenvalue does occur.
¯
Although the quark doublet (u, d) and antiquark doublet (u ¯, d) do transform differently under SU(2) f transformations, there is nevertheless a sense
in which the 2* and 2 representations are somehow the ‘same’: after all, the
quantum numbers T =
1 , T 3 =
1 describe them both. In fact, the two
2
± 2
representations are ‘unitarily equivalent’, in that we can find a unitary matrix
U C such that
U C exp(−iα · τ
∗ /2)U
−1
C = exp(iα · τ /2).
(12.53)
This requirement is easier to disentangle if we consider infinitesimal transformations, for which (12.53) becomes
U C (−τ
∗ )U
−1
C = τ ,
(12.54)
or
U C τ 1 U
C τ
1
C
−1 = −τ 1 , U 2 U
−1
C = τ 2 , U C τ 3 U C
− = −τ 3 .
(12.55)
Bearing the commutation relations (12.28) in mind, and the fact that τ i
−1 = τ i ,
it is clear that we can choose U C proportional to τ 2 , and set
(
)
0
1
U C = iτ 2 =
(12.56)
−1 0
to obtain a convenient unitary form. From (12.52) and (12.53) we obtain
(U C q
∗′ ) = V(U C q
∗ ), which implies that the doublet
(
) (
)
¯
u ¯
d
U C
=
(12.57)
¯
d
−u ¯
transforms in exactly the same way as (u, d). This result is useful, because
it means that we can use the familiar tables of (Clebsch-Gordan) angular
momentum coupling coefficients for combining quark and antiquark states
¯
together, provided we include the relative minus sign between the d and u ¯
¯
components which has appeared in (12.57). Note that, as expected, the d is
in the T 3 = +
1 position, and the u ¯ is in the T 3 = −
1 position.
2
2
As an application of these results, let us compare the T = 0 combination
of the p and n states to form the (isoscalar) deuteron, and the combination
¯
of (u, d) and (u, ¯ d) states to form the isoscalar ω-meson. In the first, the
isospin part of the wavefunction is √ 1 (ψ p ψ n
2
− ψ n ψ p ), corresponding to the
S = 0 combination of two spin1 particles in quantum mechanics given by
2
12. Global Non-Abelian Symmetries
represent the generators of SU(2) f in the 2* representation (i.e. the complex
conjugate of the original two-dimensional representation, which we will now
call 2). Referring to (12.25), we see that τ 1
∗ = τ 1 , τ 2
∗ = −τ 2 and τ 3
∗ = τ 3 . It is
then easy to check that the three matrices −τ 1 /2, +τ 2 /2 and −τ 3 /2 do indeed
satisfy the required commutation relations (12.28), and thus provide a valid
matrix representation of the SU(2) generators. Also, since the third component of isospin is here represented by −τ 3
∗ /2 = −τ 3 /2, the desired reversal in
sign of the additively conserved eigenvalue does occur.
¯
Although the quark doublet (u, d) and antiquark doublet (u ¯, d) do transform differently under SU(2) f transformations, there is nevertheless a sense
in which the 2* and 2 representations are somehow the ‘same’: after all, the
quantum numbers T =
1 , T 3 =
1 describe them both. In fact, the two
2
± 2
representations are ‘unitarily equivalent’, in that we can find a unitary matrix
U C such that
U C exp(−iα · τ
∗ /2)U
−1
C = exp(iα · τ /2).
(12.53)
This requirement is easier to disentangle if we consider infinitesimal transformations, for which (12.53) becomes
U C (−τ
∗ )U
−1
C = τ ,
(12.54)
or
U C τ 1 U
C τ
1
C
−1 = −τ 1 , U 2 U
−1
C = τ 2 , U C τ 3 U C
− = −τ 3 .
(12.55)
Bearing the commutation relations (12.28) in mind, and the fact that τ i
−1 = τ i ,
it is clear that we can choose U C proportional to τ 2 , and set
(
)
0
1
U C = iτ 2 =
(12.56)
−1 0
to obtain a convenient unitary form. From (12.52) and (12.53) we obtain
(U C q
∗′ ) = V(U C q
∗ ), which implies that the doublet
(
) (
)
¯
u ¯
d
U C
=
(12.57)
¯
d
−u ¯
transforms in exactly the same way as (u, d). This result is useful, because
it means that we can use the familiar tables of (Clebsch-Gordan) angular
momentum coupling coefficients for combining quark and antiquark states
¯
together, provided we include the relative minus sign between the d and u ¯
¯
components which has appeared in (12.57). Note that, as expected, the d is
in the T 3 = +
1 position, and the u ¯ is in the T 3 = −
1 position.
2
2
As an application of these results, let us compare the T = 0 combination
of the p and n states to form the (isoscalar) deuteron, and the combination
¯
of (u, d) and (u, ¯ d) states to form the isoscalar ω-meson. In the first, the
isospin part of the wavefunction is √ 1 (ψ p ψ n
2
− ψ n ψ p ), corresponding to the
S = 0 combination of two spin1 particles in quantum mechanics given by
2