144
5. Quantum Field Theory I: The Free Scalar Field
The ‘ˆ a
† a ˆ
† ’ term will give zero since <0|a ˆ
† = 0. For the other term we use the
commutation relation (5.130) to write it as
∫
ik
' x−iω
' t
N dk
e
<0|
√ [ˆ a
† (k
′ )ˆ a(k) + 2πδ(k − k
′ )]e
ikx−iωt
|0> = N √
(5.145)
2π 2ω
2ω ′
using the vacuum condition once again, and integrating over the δ function
using the property (5.116) which sets k = k
′ and hence ω = ω
′ . The vacuum
is normalized to unity, <0|0> = 1. The normalization constant N can be
adjusted according to the desired convention for the normalization of the
states and wavefunctions. The result is just the plane-wave wavefunction for
a particle in the state |k
′
>! Thus we discover that the vacuum to one-particle
matrix elements of the field operators are just the familiar wavefunctions of
single-particle quantum mechanics. In this connection we can explain some
common terminology. The path to quantum field theory that we have followed
is sometimes called ‘second quantization’ – ordinary single-particle quantum
mechanics being the first-quantized version of the theory.
5.3 Generalizations: four dimensions, relativity and mass
In the previous section we have shown how quantum mechanics may be married to field theory, but we have considered only one spatial dimension, for
simplicity. Now we must generalize to three and incorporate the demands of
relativity. This is very easy to do in the Lagrangian approach, for the scalar
field φ(x, t). ‘Scalar’ means that the field has only one independent component at each point (x, t) – unlike the electromagnetic field, for instance,
for which the analogous quantity has four components, making up a 4-vector
field A
μ (x, t) = (A 0 (x, t), A(x, t)) (see chapter 7). In the quantum case, a
one-component field (or wavefunction) is appropriate for spin-0 particles.
As we saw in (5.97), the three-dimensional Euler–Lagrange equations are
(
)
∂L
∂L
∂ ∂L
− ∇ ·
−
= 0
(5.146)
∂φ
∂(∇φ) ∂t ∂φ ˙
which may immediately be rewritten in relativistically invariant form
(
)
∂L
∂L
− ∂ μ
= 0
(5.147)
∂φ
∂(∂ μ φ)
where ∂ μ = ∂/∂x
μ . Similarly, the action
∫
∫
∫
S = dt d
3
x L = d
4 x L
(5.148)
5. Quantum Field Theory I: The Free Scalar Field
The ‘ˆ a
† a ˆ
† ’ term will give zero since <0|a ˆ
† = 0. For the other term we use the
commutation relation (5.130) to write it as
∫
ik
' x−iω
' t
N dk
e
<0|
√ [ˆ a
† (k
′ )ˆ a(k) + 2πδ(k − k
′ )]e
ikx−iωt
|0> = N √
(5.145)
2π 2ω
2ω ′
using the vacuum condition once again, and integrating over the δ function
using the property (5.116) which sets k = k
′ and hence ω = ω
′ . The vacuum
is normalized to unity, <0|0> = 1. The normalization constant N can be
adjusted according to the desired convention for the normalization of the
states and wavefunctions. The result is just the plane-wave wavefunction for
a particle in the state |k
′
>! Thus we discover that the vacuum to one-particle
matrix elements of the field operators are just the familiar wavefunctions of
single-particle quantum mechanics. In this connection we can explain some
common terminology. The path to quantum field theory that we have followed
is sometimes called ‘second quantization’ – ordinary single-particle quantum
mechanics being the first-quantized version of the theory.
5.3 Generalizations: four dimensions, relativity and mass
In the previous section we have shown how quantum mechanics may be married to field theory, but we have considered only one spatial dimension, for
simplicity. Now we must generalize to three and incorporate the demands of
relativity. This is very easy to do in the Lagrangian approach, for the scalar
field φ(x, t). ‘Scalar’ means that the field has only one independent component at each point (x, t) – unlike the electromagnetic field, for instance,
for which the analogous quantity has four components, making up a 4-vector
field A
μ (x, t) = (A 0 (x, t), A(x, t)) (see chapter 7). In the quantum case, a
one-component field (or wavefunction) is appropriate for spin-0 particles.
As we saw in (5.97), the three-dimensional Euler–Lagrange equations are
(
)
∂L
∂L
∂ ∂L
− ∇ ·
−
= 0
(5.146)
∂φ
∂(∇φ) ∂t ∂φ ˙
which may immediately be rewritten in relativistically invariant form
(
)
∂L
∂L
− ∂ μ
= 0
(5.147)
∂φ
∂(∂ μ φ)
where ∂ μ = ∂/∂x
μ . Similarly, the action
∫
∫
∫
S = dt d
3
x L = d
4 x L
(5.148)