12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 35
Equation (12.158) then implies that P ˆ Q ˆ 5 P ˆ −1 = −Q ˆ 5 , following the normal
rule for operator transformations in quantum mechanics. Consider now the
state Q ˆ 5 |+›. We have
(
)
P ˆ Q ˆ 5 |+› =
P ˆ Q ˆ 5 P ˆ −1 P ˆ |+›
= −Q ˆ 5 |+›
(12.160)
showing that Q ˆ 5 |+› is an eigenstate of P ˆ with the opposite eigenvalue, -1.
A very important physical consequence now follows from the fact that (in
this simple m = 0 model) Q ˆ 5 is a symmetry operator commuting with the
Hamiltonian H ˆ . We have
H ˆ Q ˆ 5 |ψ› = Q ˆ 5 H ˆ |ψ› = EQ ˆ 5 |ψ›.
(12.161)
Hence for every state |ψ› with energy eigenvalue E, there should exist a state
ˆ
Q 5 |ψ› with the same eigenvalue E and the opposite parity: that is, chiral
symmetry apparently implies the existence of ‘parity doublets’.
Of course, it may reasonably be objected that all of the above refers not
only to the massless, but also the non-interacting case. However, this is just
where the analysis begins to get interesting. Suppose we allow the fermion
field ψ ˆ to interact with a U(1)-gauge field A ˆ μ via the standard electromagnetic
coupling
ˆ
¯ ˆ
L int = qψγ
μ ψ ˆ A ˆ μ .
(12.162)
Remarkably enough, L ˆ int is also invariant under the chiral transformation
(12.152), for the simple reason that the ‘Dirac’ structure of (12.162) is exactly
¯
the same as that of the free kinetic term ψ ˆ / ∂ψ ˆ : the ‘covariant derivative’
prescription ∂
μ
→ D
μ = ∂
μ + iqA ˆ μ automatically means that any ‘Dirac’ (e.g.
γ 5 ) symmetry of the kinetic part will be preserved when the gauge interaction
is included. Thus chirality remains a ‘good symmetry’ in the presence of a
U(1) gauge interaction.
The generalization of this to the more physical m u ≈ m d ≈ 0 case is quite
straightforward. The Lagrangian (12.87) becomes
L ˆ = ¯ q ˆi / ∂q ˆ
(12.163)
as m → 0, which is invariant under the γ 5 -version of (12.89),
6 namely
′
q ˆ = exp(−iβ · τ /2γ 5 )ˆ q.
(12.164)
There are three associated Noether currents (compare (12.109))
(
1
T ˆ 2 ) μ
= ¯ qγ ˆ
μ γ 5
τ q ˆ
(12.165)
5
2
6 L ˆ 0 is also invariant under ˆ
q ′ = e −iθγ 5 q ˆ which is an ‘axial’ version of the global U(1)
associated with quark number conservation. We shall discuss this additional U(1)-symmetry
in section 18.1.1.
Equation (12.158) then implies that P ˆ Q ˆ 5 P ˆ −1 = −Q ˆ 5 , following the normal
rule for operator transformations in quantum mechanics. Consider now the
state Q ˆ 5 |+›. We have
(
)
P ˆ Q ˆ 5 |+› =
P ˆ Q ˆ 5 P ˆ −1 P ˆ |+›
= −Q ˆ 5 |+›
(12.160)
showing that Q ˆ 5 |+› is an eigenstate of P ˆ with the opposite eigenvalue, -1.
A very important physical consequence now follows from the fact that (in
this simple m = 0 model) Q ˆ 5 is a symmetry operator commuting with the
Hamiltonian H ˆ . We have
H ˆ Q ˆ 5 |ψ› = Q ˆ 5 H ˆ |ψ› = EQ ˆ 5 |ψ›.
(12.161)
Hence for every state |ψ› with energy eigenvalue E, there should exist a state
ˆ
Q 5 |ψ› with the same eigenvalue E and the opposite parity: that is, chiral
symmetry apparently implies the existence of ‘parity doublets’.
Of course, it may reasonably be objected that all of the above refers not
only to the massless, but also the non-interacting case. However, this is just
where the analysis begins to get interesting. Suppose we allow the fermion
field ψ ˆ to interact with a U(1)-gauge field A ˆ μ via the standard electromagnetic
coupling
ˆ
¯ ˆ
L int = qψγ
μ ψ ˆ A ˆ μ .
(12.162)
Remarkably enough, L ˆ int is also invariant under the chiral transformation
(12.152), for the simple reason that the ‘Dirac’ structure of (12.162) is exactly
¯
the same as that of the free kinetic term ψ ˆ / ∂ψ ˆ : the ‘covariant derivative’
prescription ∂
μ
→ D
μ = ∂
μ + iqA ˆ μ automatically means that any ‘Dirac’ (e.g.
γ 5 ) symmetry of the kinetic part will be preserved when the gauge interaction
is included. Thus chirality remains a ‘good symmetry’ in the presence of a
U(1) gauge interaction.
The generalization of this to the more physical m u ≈ m d ≈ 0 case is quite
straightforward. The Lagrangian (12.87) becomes
L ˆ = ¯ q ˆi / ∂q ˆ
(12.163)
as m → 0, which is invariant under the γ 5 -version of (12.89),
6 namely
′
q ˆ = exp(−iβ · τ /2γ 5 )ˆ q.
(12.164)
There are three associated Noether currents (compare (12.109))
(
1
T ˆ 2 ) μ
= ¯ qγ ˆ
μ γ 5
τ q ˆ
(12.165)
5
2
6 L ˆ 0 is also invariant under ˆ
q ′ = e −iθγ 5 q ˆ which is an ‘axial’ version of the global U(1)
associated with quark number conservation. We shall discuss this additional U(1)-symmetry
in section 18.1.1.