12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 25
(cf (12.1)) which is associated with baryon number conservation. It is also
invariant under global SU(2) f transformations acting in the flavour u-d space
(cf (12.32)):
′
q ˆ = exp(−iα · τ /2)ˆ q
(12.89)
(for the change in sign with respect to (12.51), compare section 7.1 and section
7.2 in the U(1) case). In (12.89), the three parameters α are independent of
x.
What are the conserved quantities associated with the invariance of L ˆ
under (12.89) ? Let us recall the discussion of the simpler U(1) cases studied
in sections 7.1 and 7.2. Considering the complex scalar field of section 7.1, the
−iα ˆ
analogue of (12.89) was just φ ˆ → φ ˆ ′ = e φ, and the conserved quantity was
the Hermitian operator N ˆ φ which appeared in the exponent of the unitary
operator U ˆ that effected the transformation φ ˆ → φ ˆ′ via
φ ˆ ′ = U ˆ φ ˆ U ˆ † ,
(12.90)
with
U ˆ = exp(iαN ˆ φ ).
(12.91)
For an infinitesimal α, we have
φ ˆ ′
φ, U ˆ ≈ 1 + i∈N ˆ φ ,
≈ (1 − i∈) ˆ
(12.92)
so that (12.90) becomes
(1 − i∈)φ ˆ = (1 + i∈N ˆ φ )φ ˆ (1 − i∈N ˆ φ ) ≈ φ ˆ + i∈[N ˆ φ , φ ˆ ];
(12.93)
hence we require
[N ˆ φ , φ ˆ ] = −φ ˆ
(12.94)
for consistency. Insofar as N ˆ φ determines the form of an infinitesimal version
of the unitary transformation operator U ˆ , it seems reasonable to call it the
generator of these global U(1) transformations (compare the discussion after
(12.27) and (12.35), but note that here N ˆ φ is a quantum field operator, not a
matrix).
Consider now the SU(2) f transformation (12.89), in the infinitesimal case:
′
q ˆ = (1 − i∊ · τ /2)ˆ q.
(12.95)
Since the single U(1) parameter ∈ is now replaced by the three parameters
∊ = (∈ 1 , ∈ 2 , ∈ 3 ), we shall need three analogues of N ˆ φ , which we call
1
2
1
2
1
2
1
2
ˆ
T
( )
= (T ˆ
1
)
(
ˆ
T
)
(
ˆ
T
) ),
(12.96)
,
,
2
3
corresponding to the three independent infinitesimal SU(2) transformations.
The generalizations of (12.90) and (12.91) are then
1
2
1
2
′
U ˆ (
=
) q ˆU ˆ ( )†
(12.97)
q ˆ
(cf (12.1)) which is associated with baryon number conservation. It is also
invariant under global SU(2) f transformations acting in the flavour u-d space
(cf (12.32)):
′
q ˆ = exp(−iα · τ /2)ˆ q
(12.89)
(for the change in sign with respect to (12.51), compare section 7.1 and section
7.2 in the U(1) case). In (12.89), the three parameters α are independent of
x.
What are the conserved quantities associated with the invariance of L ˆ
under (12.89) ? Let us recall the discussion of the simpler U(1) cases studied
in sections 7.1 and 7.2. Considering the complex scalar field of section 7.1, the
−iα ˆ
analogue of (12.89) was just φ ˆ → φ ˆ ′ = e φ, and the conserved quantity was
the Hermitian operator N ˆ φ which appeared in the exponent of the unitary
operator U ˆ that effected the transformation φ ˆ → φ ˆ′ via
φ ˆ ′ = U ˆ φ ˆ U ˆ † ,
(12.90)
with
U ˆ = exp(iαN ˆ φ ).
(12.91)
For an infinitesimal α, we have
φ ˆ ′
φ, U ˆ ≈ 1 + i∈N ˆ φ ,
≈ (1 − i∈) ˆ
(12.92)
so that (12.90) becomes
(1 − i∈)φ ˆ = (1 + i∈N ˆ φ )φ ˆ (1 − i∈N ˆ φ ) ≈ φ ˆ + i∈[N ˆ φ , φ ˆ ];
(12.93)
hence we require
[N ˆ φ , φ ˆ ] = −φ ˆ
(12.94)
for consistency. Insofar as N ˆ φ determines the form of an infinitesimal version
of the unitary transformation operator U ˆ , it seems reasonable to call it the
generator of these global U(1) transformations (compare the discussion after
(12.27) and (12.35), but note that here N ˆ φ is a quantum field operator, not a
matrix).
Consider now the SU(2) f transformation (12.89), in the infinitesimal case:
′
q ˆ = (1 − i∊ · τ /2)ˆ q.
(12.95)
Since the single U(1) parameter ∈ is now replaced by the three parameters
∊ = (∈ 1 , ∈ 2 , ∈ 3 ), we shall need three analogues of N ˆ φ , which we call
1
2
1
2
1
2
1
2
ˆ
T
( )
= (T ˆ
1
)
(
ˆ
T
)
(
ˆ
T
) ),
(12.96)
,
,
2
3
corresponding to the three independent infinitesimal SU(2) transformations.
The generalizations of (12.90) and (12.91) are then
1
2
1
2
′
U ˆ (
=
) q ˆU ˆ ( )†
(12.97)
q ˆ