26
12. Global Non-Abelian Symmetries
and
ˆ
U
(
1
(
1 )
)
2
= exp(iα
1
· ˆ
T
2 )
(12.98)
ˆ
( )
1
where the T
2
ˆ
’s are Hermitian, so that U
( )
2
is unitary (cf (12.35)). It would
1
ˆ
( )
seem reasonable in this case too to regard the T
2 ’s as providing a field
theoretic representation of the generators of SU(2) f , an interpretation we shall
shortly confirm. In the infinitesimal case, (12.97) and (12.98) become
1
ˆ
( )
(
1
ˆ
)
(1 − i∊ · τ /2)q ˆ = (1 + i∊ · T
2 )ˆ q(1 − i∊ · T
2 ),
(12.99)
ˆ
(
1 )
using the Hermiticity of the T
2 ’s. Expanding the right-hand side of (12.99)
to first order in ∊, and equating coefficients of ∊ on both sides, (12.99) reduces
to (problem 12.9)
ˆ
(
1 )
[T
2 , q ˆ] = −(τ /2)q, ˆ
(12.100)
which is the analogue of (12.94). Equation (12.100) expresses a very specific
1
ˆ
( )
commutation property of the operators T
2 , which turns out to be satisfied
by the expression
∫
ˆ
(
1 )
T
2
= q ˆ
† (τ /2)q ˆd
3
x
(12.101)
as can be checked (problem 12.10) from the anticommutation relations of
the fermionic fields in q ˆ. We shall derive (12.101) from Noether’s theorem
(Noether 1918) in a little while. Note that if ‘τ /2’ is replaced by 1, (12.101)
reduces to the sum of the u and d number operators, as required for the oneparameter U(1) case. The ‘q ˆ
†
τ q ˆ’ combination is precisely the field-theoretic
version of the q
†
τ q coupling we discussed in section 12.1.3. It means that the
ˆ
(
1 )
three operators T
2
themselves belong to a T = 1 triplet of SU(2) f .
1
ˆ
( )
It is possible to verify that these T
2 ’s do indeed commute with the
ˆ
Hamiltonian H:
(
1 )
(
1
ˆ
dT
2 /dt = − ˆ
)
i[T
2
ˆ
, H] = 0
(12.102)
ˆ
(
1 )
so that their eigenvalues are conserved. That the T
2
are, as already suggested, a field theoretic representation of the generators of SU(2), appropriate
to the case T =
1 , follows from the fact that they obey the SU(2) algebra
2
(problem 12.11):
ˆ (
1 )
)
[T
2
ˆ (
1
(
1 )
, T
2
ˆ
i
j
] = i∈ ijk T
2
k .
(12.103)
For many purposes it is more useful to consider the raising and lowering
operators
ˆ (
1 )
ˆ (
1 )
T
2
= (T
2
1
± ˆ (
1 )
iT
2
2 ).
(12.104)
±
For example, we easily find
∫
ˆ (
1 )
T
2
+ = u ˆ
† ˆ
d d
3
x,
(12.105)
12. Global Non-Abelian Symmetries
and
ˆ
U
(
1
(
1 )
)
2
= exp(iα
1
· ˆ
T
2 )
(12.98)
ˆ
( )
1
where the T
2
ˆ
’s are Hermitian, so that U
( )
2
is unitary (cf (12.35)). It would
1
ˆ
( )
seem reasonable in this case too to regard the T
2 ’s as providing a field
theoretic representation of the generators of SU(2) f , an interpretation we shall
shortly confirm. In the infinitesimal case, (12.97) and (12.98) become
1
ˆ
( )
(
1
ˆ
)
(1 − i∊ · τ /2)q ˆ = (1 + i∊ · T
2 )ˆ q(1 − i∊ · T
2 ),
(12.99)
ˆ
(
1 )
using the Hermiticity of the T
2 ’s. Expanding the right-hand side of (12.99)
to first order in ∊, and equating coefficients of ∊ on both sides, (12.99) reduces
to (problem 12.9)
ˆ
(
1 )
[T
2 , q ˆ] = −(τ /2)q, ˆ
(12.100)
which is the analogue of (12.94). Equation (12.100) expresses a very specific
1
ˆ
( )
commutation property of the operators T
2 , which turns out to be satisfied
by the expression
∫
ˆ
(
1 )
T
2
= q ˆ
† (τ /2)q ˆd
3
x
(12.101)
as can be checked (problem 12.10) from the anticommutation relations of
the fermionic fields in q ˆ. We shall derive (12.101) from Noether’s theorem
(Noether 1918) in a little while. Note that if ‘τ /2’ is replaced by 1, (12.101)
reduces to the sum of the u and d number operators, as required for the oneparameter U(1) case. The ‘q ˆ
†
τ q ˆ’ combination is precisely the field-theoretic
version of the q
†
τ q coupling we discussed in section 12.1.3. It means that the
ˆ
(
1 )
three operators T
2
themselves belong to a T = 1 triplet of SU(2) f .
1
ˆ
( )
It is possible to verify that these T
2 ’s do indeed commute with the
ˆ
Hamiltonian H:
(
1 )
(
1
ˆ
dT
2 /dt = − ˆ
)
i[T
2
ˆ
, H] = 0
(12.102)
ˆ
(
1 )
so that their eigenvalues are conserved. That the T
2
are, as already suggested, a field theoretic representation of the generators of SU(2), appropriate
to the case T =
1 , follows from the fact that they obey the SU(2) algebra
2
(problem 12.11):
ˆ (
1 )
)
[T
2
ˆ (
1
(
1 )
, T
2
ˆ
i
j
] = i∈ ijk T
2
k .
(12.103)
For many purposes it is more useful to consider the raising and lowering
operators
ˆ (
1 )
ˆ (
1 )
T
2
= (T
2
1
± ˆ (
1 )
iT
2
2 ).
(12.104)
±
For example, we easily find
∫
ˆ (
1 )
T
2
+ = u ˆ
† ˆ
d d
3
x,
(12.105)