197
7.3. The Maxwell field A
μ (x)
the Maxwell equation for the electromagnetic potential A
ν , namely (cf (2.22))
❗A
ν
− ∂
ν (∂ μ A
μ ) = j
ν .
(7.65)
em
The answer is (see problem 7.10)
1
F
μν
− j
ν
L em = − F μν
em A ν
(7.66)
4
where F μν = ∂ μ A ν −∂ ν A μ . So the pure A-field part is the Maxwell Lagrangian
L A = −
1
F
μν
F μν
.
(7.67)
4
Before proceeding to try to quantize (7.67), we need to understand some
important aspects of the free classical field A
ν (x).
When j em is set equal to zero, A
ν satisfies the equation
∂ μ F
μν = ❗A
ν
− ∂
ν (∂
μ A μ ) = 0.
(7.68)
As we have seen in section 2.3, these equations are left unchanged if we perform
the gauge transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(7.69)
We can use this freedom to choose the A
μ with which we work to satisfy the
condition
∂ μ A
μ = 0.
(7.70)
This is called the Lorentz condition. The process of choosing a particular
condition on A
μ so as to define it (ultimately) uniquely is called ‘choosing
a gauge’; actually the condition (7.70) does not yet define A
μ uniquely, as
we shall see shortly. The Lorentz condition is a very convenient one, since it
decouples the different components of A
μ in Maxwell’s equations (7.68) – in
a covariant way, moreover, leaving the very simple equation
❗A
μ = 0.
(7.71)
This has plane-wave solutions of the form
−ik·x
A
μ = N ∈
μ e
(7.72)
k
2
with k
2 = 0 (i.e. k
2 = ), where N is a normalization factor and ∈
μ is a
0
polarization vector for the wave. The gauge condition (7.70) now reduces to
a condition on ∈
μ :
k · ∈ = 0.
(7.73)
However, we have not yet exhausted all the gauge freedom. We are still free
to make another shift in the potential
A
μ
→ A
μ
− ∂
μ χ ˜
(7.74)
7.3. The Maxwell field A
μ (x)
the Maxwell equation for the electromagnetic potential A
ν , namely (cf (2.22))
❗A
ν
− ∂
ν (∂ μ A
μ ) = j
ν .
(7.65)
em
The answer is (see problem 7.10)
1
F
μν
− j
ν
L em = − F μν
em A ν
(7.66)
4
where F μν = ∂ μ A ν −∂ ν A μ . So the pure A-field part is the Maxwell Lagrangian
L A = −
1
F
μν
F μν
.
(7.67)
4
Before proceeding to try to quantize (7.67), we need to understand some
important aspects of the free classical field A
ν (x).
When j em is set equal to zero, A
ν satisfies the equation
∂ μ F
μν = ❗A
ν
− ∂
ν (∂
μ A μ ) = 0.
(7.68)
As we have seen in section 2.3, these equations are left unchanged if we perform
the gauge transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(7.69)
We can use this freedom to choose the A
μ with which we work to satisfy the
condition
∂ μ A
μ = 0.
(7.70)
This is called the Lorentz condition. The process of choosing a particular
condition on A
μ so as to define it (ultimately) uniquely is called ‘choosing
a gauge’; actually the condition (7.70) does not yet define A
μ uniquely, as
we shall see shortly. The Lorentz condition is a very convenient one, since it
decouples the different components of A
μ in Maxwell’s equations (7.68) – in
a covariant way, moreover, leaving the very simple equation
❗A
μ = 0.
(7.71)
This has plane-wave solutions of the form
−ik·x
A
μ = N ∈
μ e
(7.72)
k
2
with k
2 = 0 (i.e. k
2 = ), where N is a normalization factor and ∈
μ is a
0
polarization vector for the wave. The gauge condition (7.70) now reduces to
a condition on ∈
μ :
k · ∈ = 0.
(7.73)
However, we have not yet exhausted all the gauge freedom. We are still free
to make another shift in the potential
A
μ
→ A
μ
− ∂
μ χ ˜
(7.74)