18.1. The Nambu analogy
233
q ˆ(x) and q ˆ
† (x) gives the results (problem 18.1)
ˆ ˆ
ˆ
[Q i , Q j ] = i∈ ijk Q k
(18.24)
ˆ ˆ
ˆ
[Q i , Q j,5 ] = i∈ ijk Q k,5
(18.25)
ˆ
ˆ
ˆ
[Q i,5 , Q j,5 ] = i∈ ijk Q k .
(18.26)
Relation (18.24) has been seen before in (12.101), and simply says that the
ˆ
Q i ’s obey a SU(2) algebra. A simple trick reduces the rather complicated
algebra of (18.24)–(18.26) to something much simpler. Defining
1
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Q i,R = (Q i + Q i,5 )
Q i,L = (Q i Q
2
2
− i,5 )
(18.27)
we find (problem 18.2)
ˆ
ˆ
ˆ
[Q i,R , Q j,R ] = i∈ ijk Q k,R
(18.28)
ˆ
ˆ
ˆ
[Q i,L , Q j,L ] = i∈ ijk Q k,L
(18.29)
ˆ
ˆ
[Q i,R , Q j,L ] = 0.
(18.30)
ˆ
ˆ
The operators Q i,R , Q i,L therefore behave like two commuting (independent)
angular momentum operators, each obeying the algebra of SU(2). For this
reason, the symmetry group of the combined symmetries (ii) and (iv) is called
SU(2) f L × SU(2) f R .
The decoupling effected by (18.27) has a simple interpretation. Referring
to (18.22) and (18.23), we see that
∫
(
)
1 + γ
ˆ
5
τ i
Q
3
i,R = q ˆ
†
q ˆd x
(18.31)
2
2
ˆ
and similarly for Q i,L . But ((1 ± γ 5 )/2) are just the projection operators P R,L
introduced in section 12.3.2, which project out the chiral parts of any fermion
ˆ
field. Furthermore, it is easy to see that P
2
R = P R and P
2
L = P L , so that Q i,R
ˆ
and Q i,L can also be written as
∫
∫
τ
τ
ˆ
i
ˆ
i
Q i,R = q ˆ
†
q ˆ R d
3
R
x
Q i,L = q ˆ L
†
q ˆ L d
3
x,
(18.32)
2
2
where q ˆ R = ((1 + γ 5 )/2)q, ˆ q ˆ L = ((1 − γ 5 )/2)q ˆ. In a similar way, the currents
(18.20) and (18.21) can be written as
ˆ μ ˆ μ
ˆ μ
ˆ μ
ˆ μ
ˆ μ
j i = j i,R + j i,L
j i,5 = j i,R − j i,L ,
(18.33)
where
τ
ˆ
¯
i
μ
τ
¯
i
j
ˆ
i,
γ
μ
R = q ˆ R
q ˆ
j i,L = q ˆ L γ
μ
R
q ˆ L .
(18.34)
2
2
Thus the SU(2) L and SU(2) R refer to the two chiral components of the fermion
fields, which is why it is called chiral symmetry.
233
q ˆ(x) and q ˆ
† (x) gives the results (problem 18.1)
ˆ ˆ
ˆ
[Q i , Q j ] = i∈ ijk Q k
(18.24)
ˆ ˆ
ˆ
[Q i , Q j,5 ] = i∈ ijk Q k,5
(18.25)
ˆ
ˆ
ˆ
[Q i,5 , Q j,5 ] = i∈ ijk Q k .
(18.26)
Relation (18.24) has been seen before in (12.101), and simply says that the
ˆ
Q i ’s obey a SU(2) algebra. A simple trick reduces the rather complicated
algebra of (18.24)–(18.26) to something much simpler. Defining
1
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Q i,R = (Q i + Q i,5 )
Q i,L = (Q i Q
2
2
− i,5 )
(18.27)
we find (problem 18.2)
ˆ
ˆ
ˆ
[Q i,R , Q j,R ] = i∈ ijk Q k,R
(18.28)
ˆ
ˆ
ˆ
[Q i,L , Q j,L ] = i∈ ijk Q k,L
(18.29)
ˆ
ˆ
[Q i,R , Q j,L ] = 0.
(18.30)
ˆ
ˆ
The operators Q i,R , Q i,L therefore behave like two commuting (independent)
angular momentum operators, each obeying the algebra of SU(2). For this
reason, the symmetry group of the combined symmetries (ii) and (iv) is called
SU(2) f L × SU(2) f R .
The decoupling effected by (18.27) has a simple interpretation. Referring
to (18.22) and (18.23), we see that
∫
(
)
1 + γ
ˆ
5
τ i
Q
3
i,R = q ˆ
†
q ˆd x
(18.31)
2
2
ˆ
and similarly for Q i,L . But ((1 ± γ 5 )/2) are just the projection operators P R,L
introduced in section 12.3.2, which project out the chiral parts of any fermion
ˆ
field. Furthermore, it is easy to see that P
2
R = P R and P
2
L = P L , so that Q i,R
ˆ
and Q i,L can also be written as
∫
∫
τ
τ
ˆ
i
ˆ
i
Q i,R = q ˆ
†
q ˆ R d
3
R
x
Q i,L = q ˆ L
†
q ˆ L d
3
x,
(18.32)
2
2
where q ˆ R = ((1 + γ 5 )/2)q, ˆ q ˆ L = ((1 − γ 5 )/2)q ˆ. In a similar way, the currents
(18.20) and (18.21) can be written as
ˆ μ ˆ μ
ˆ μ
ˆ μ
ˆ μ
ˆ μ
j i = j i,R + j i,L
j i,5 = j i,R − j i,L ,
(18.33)
where
τ
ˆ
¯
i
μ
τ
¯
i
j
ˆ
i,
γ
μ
R = q ˆ R
q ˆ
j i,L = q ˆ L γ
μ
R
q ˆ L .
(18.34)
2
2
Thus the SU(2) L and SU(2) R refer to the two chiral components of the fermion
fields, which is why it is called chiral symmetry.