199
7.3. The Maxwell field A
μ (x)
(linear combinations of (7.81) and (7.82)), which correspond to circularly polarized radiation. The phase convention in (7.84) and (7.85) is the standard
one in quantum mechanics for states of definite spin projection (‘helicity’)
λ = ±1 along the direction of motion (the z-axis here). We may easily check
that
∈
∗ (λ) · ∈(λ
′ ) = δ λλ '
(7.86)
or, in terms of the corresponding 4-vectors ∈
μ = (0, ∈),
∈
∗ (λ) · ∈(λ
′ ) = −δ λλ ' .
(7.87)
We have therefore arrived at the result, familiar in classical electromagnetic
theory, that the free electromagnetic fields are purely transverse. Though they
are described in this formalism by a vector potential with apparently four
independent components (V, A), the condition (7.70) reduces this number by
one, and the further gauge freedom exploited in (7.74)–(7.76) reduces it by
one more.
A crucial point to note is that the reduction to only two independent field
components (polarization states) can be traced back to the fact that the free
photon is massless: see the remark after (7.76). By contrast, for massive spin1 bosons, such as the W
± and Z
0 , all three expected polarization states are
indeed present. However, weak interactions are described by a gauge theory,
and the W
± and Z
0 particles are gauge-field quanta, analogous to the photon.
How gauge invariance can be reconciled with the existence of massive gauge
quanta with three polarization states will be explained in volume 2.
We may therefore write the plane-wave mode expansion for the classical
A
μ (x) field in the form
∫
d
3
k
∑
A
μ (x) =
√
[∈
μ (k, λ)α(k, λ)e
−ik·x + ∈
μ∗ (k, λ)α
∗ (k, λ)e
ik·x ]
(2π) 3 2ω λ
(7.88)
where the sum is over the two possible polarization states λ, for given k, as
described by the suitable polarization vector ∈
μ (k, λ) and ω = |k|.
It would seem that all we have to do now, in order to ‘quantize’ (7.88), is
to promote α and α
∗ to operators ˆ
α and ˆ
α
† , as usual. However, things are
actually not nearly so simple.
7.3.2 Quantizing A μ (x)
Readers familiar with Lagrangian mechanics may already suspect that quantizing A
ν is not going to be straightforward. The problem is that, clearly,
A
ν (x) has four (Lorentz) components – but, equally clearly in view of the
previous section, they are not all independent field components or field degrees of freedom. In fact, there are only two independent degrees of freedom,
both transverse. Thus there are constraints on the four fields, for instance the
gauge condition (7.70). Constrained systems are often awkward to handle in
7.3. The Maxwell field A
μ (x)
(linear combinations of (7.81) and (7.82)), which correspond to circularly polarized radiation. The phase convention in (7.84) and (7.85) is the standard
one in quantum mechanics for states of definite spin projection (‘helicity’)
λ = ±1 along the direction of motion (the z-axis here). We may easily check
that
∈
∗ (λ) · ∈(λ
′ ) = δ λλ '
(7.86)
or, in terms of the corresponding 4-vectors ∈
μ = (0, ∈),
∈
∗ (λ) · ∈(λ
′ ) = −δ λλ ' .
(7.87)
We have therefore arrived at the result, familiar in classical electromagnetic
theory, that the free electromagnetic fields are purely transverse. Though they
are described in this formalism by a vector potential with apparently four
independent components (V, A), the condition (7.70) reduces this number by
one, and the further gauge freedom exploited in (7.74)–(7.76) reduces it by
one more.
A crucial point to note is that the reduction to only two independent field
components (polarization states) can be traced back to the fact that the free
photon is massless: see the remark after (7.76). By contrast, for massive spin1 bosons, such as the W
± and Z
0 , all three expected polarization states are
indeed present. However, weak interactions are described by a gauge theory,
and the W
± and Z
0 particles are gauge-field quanta, analogous to the photon.
How gauge invariance can be reconciled with the existence of massive gauge
quanta with three polarization states will be explained in volume 2.
We may therefore write the plane-wave mode expansion for the classical
A
μ (x) field in the form
∫
d
3
k
∑
A
μ (x) =
√
[∈
μ (k, λ)α(k, λ)e
−ik·x + ∈
μ∗ (k, λ)α
∗ (k, λ)e
ik·x ]
(2π) 3 2ω λ
(7.88)
where the sum is over the two possible polarization states λ, for given k, as
described by the suitable polarization vector ∈
μ (k, λ) and ω = |k|.
It would seem that all we have to do now, in order to ‘quantize’ (7.88), is
to promote α and α
∗ to operators ˆ
α and ˆ
α
† , as usual. However, things are
actually not nearly so simple.
7.3.2 Quantizing A μ (x)
Readers familiar with Lagrangian mechanics may already suspect that quantizing A
ν is not going to be straightforward. The problem is that, clearly,
A
ν (x) has four (Lorentz) components – but, equally clearly in view of the
previous section, they are not all independent field components or field degrees of freedom. In fact, there are only two independent degrees of freedom,
both transverse. Thus there are constraints on the four fields, for instance the
gauge condition (7.70). Constrained systems are often awkward to handle in