186
7. Quantum Field Theory III
is conserved:
μ
∂ μ N ˆ
φ = 0.
(7.11)
Such conserved 4-vector operators are called symmetry currents, often denoted
generically by J ˆ μ . There is a general theorem (due to Noether (1918) in the
classical field case) to the effect that if a Lagrangian is invariant under a
continuous transformation, then there will be an associated symmetry current.
We shall consider Noether’s theorem again in volume 2.
What does all this have to do with symmetry operators? Written out in
full, (7.11) is
∂N ˆ
φ
0 /∂t + ∇ · N ˆ φ = 0.
(7.12)
Integrating this equation over all space, we obtain
∫
∫
d
N ˆ
φ
0 d
3
ˆ
x +
N φ · dS = 0
(7.13)
dt V →∞
S→∞
where we have used the divergence theorem in the second term. Normally the
fields may be assumed to die off sufficiently fast at infinity that the surface
integral vanishes (by using wave packets, for example), and we can therefore
deduce that the quantity N ˆ φ is constant in time, where
∫
N ˆ φ = N ˆ
φ
0 d
3
x
(7.14)
that is, the volume integral of the μ = 0 component of a symmetry current is
a symmetry operator.
In order to see how N ˆ φ serves to distinguish ‘particle’ from ‘antiparticle’
in the simple example we are considering, it turns out to be convenient to
regard φ ˆ 1 and φ ˆ 2 as components of a single complex field
1
φ ˆ = √ (φ ˆ 1 − iφ ˆ 2 )
2
(7.15)
1
φ ˆ† = √ (φ ˆ 1 + iφ ˆ 2 ).
2
The plane-wave expansions of the form (5.155) for φ ˆ 1 and φ ˆ 2 imply that φ ˆ has
the expansion
∫
d
3
k
φ ˆ =
√ [ˆ a(k)e
−ik·x + ˆ b
† (k)e
ik·x ]
(7.16)
(2π) 3 2ω
where
1
a ˆ(k) = √ (ˆ a 1 − iˆ a 2 )
2
1
†
†
(7.17)
ˆ b
† (k) = √ (ˆ a − iˆ a )
1
2
2
†
and ω = (M
2 + k
2 )
1/2 . The operators ˆ
a, ˆ
a , ˆ b, ˆ b
† obey the commutation
relations
[ˆ a(k), a ˆ
† (k
′ )] = (2π)
3 δ
3 (k − k
′ )
(7.18)
[ ˆ b(k), ˆ b
† (k
′ )] = (2π)
3 δ
3 (k − k
′ )
7. Quantum Field Theory III
is conserved:
μ
∂ μ N ˆ
φ = 0.
(7.11)
Such conserved 4-vector operators are called symmetry currents, often denoted
generically by J ˆ μ . There is a general theorem (due to Noether (1918) in the
classical field case) to the effect that if a Lagrangian is invariant under a
continuous transformation, then there will be an associated symmetry current.
We shall consider Noether’s theorem again in volume 2.
What does all this have to do with symmetry operators? Written out in
full, (7.11) is
∂N ˆ
φ
0 /∂t + ∇ · N ˆ φ = 0.
(7.12)
Integrating this equation over all space, we obtain
∫
∫
d
N ˆ
φ
0 d
3
ˆ
x +
N φ · dS = 0
(7.13)
dt V →∞
S→∞
where we have used the divergence theorem in the second term. Normally the
fields may be assumed to die off sufficiently fast at infinity that the surface
integral vanishes (by using wave packets, for example), and we can therefore
deduce that the quantity N ˆ φ is constant in time, where
∫
N ˆ φ = N ˆ
φ
0 d
3
x
(7.14)
that is, the volume integral of the μ = 0 component of a symmetry current is
a symmetry operator.
In order to see how N ˆ φ serves to distinguish ‘particle’ from ‘antiparticle’
in the simple example we are considering, it turns out to be convenient to
regard φ ˆ 1 and φ ˆ 2 as components of a single complex field
1
φ ˆ = √ (φ ˆ 1 − iφ ˆ 2 )
2
(7.15)
1
φ ˆ† = √ (φ ˆ 1 + iφ ˆ 2 ).
2
The plane-wave expansions of the form (5.155) for φ ˆ 1 and φ ˆ 2 imply that φ ˆ has
the expansion
∫
d
3
k
φ ˆ =
√ [ˆ a(k)e
−ik·x + ˆ b
† (k)e
ik·x ]
(7.16)
(2π) 3 2ω
where
1
a ˆ(k) = √ (ˆ a 1 − iˆ a 2 )
2
1
†
†
(7.17)
ˆ b
† (k) = √ (ˆ a − iˆ a )
1
2
2
†
and ω = (M
2 + k
2 )
1/2 . The operators ˆ
a, ˆ
a , ˆ b, ˆ b
† obey the commutation
relations
[ˆ a(k), a ˆ
† (k
′ )] = (2π)
3 δ
3 (k − k
′ )
(7.18)
[ ˆ b(k), ˆ b
† (k
′ )] = (2π)
3 δ
3 (k − k
′ )