203
7.3. The Maxwell field A
μ (x)
the ∈’s in section 7.3.1, and choose two transverse polarization vectors as (cf
(7.81), (7.82))
∈
μ (k, λ = 1) = (0, 1, 0, 0)
‘transverse polarizations’.
(7.105)
∈
μ (k, λ = 2) = (0, 0, 1, 0)
The other two ∈’s are
∈
μ (k, λ = 0) = (1, 0, 0, 0)
‘time-like polarization’
(7.106)
and
∈
μ (k, λ = 3) = (0, 0, 0, 1)
‘longitudinal polarization’.
(7.107)
Making (7.104) consistent with (7.103) then requires
†
[ˆ α λ (k), α ˆ λ ' (k
′ )] = −g λλ ' (2π)
3 δ
3 (k − k
′ ).
(7.108)
This is where the wrong sign in (7.103) has come back to haunt us: we have
the wrong sign in (7.108) for the case λ = λ
′ = 0 (time-like modes).
What is the consequence of this? It seems natural to assume that the
vacuum is defined by
α ˆ λ (k)|0> = 0
for all λ = 0, 1, 2, 3.
(7.109)
But suppose we use (7.108) and (7.109) to calculate the normalization overlap
of a ‘one time-like photon’ state; this is
†
′ )|0>
0
= −(2π)
3 δ
3 (k − k
′ )
(7.110)
k
′
and the state effectively has a negative norm (the k =
infinity is the standard plane-wave artefact). Such states would threaten fundamental properties
such as the conservation of total probability if they contributed, uncancelled,
in physical processes.
At this point we would do well to recall the condition ‘∂ μ A ˆ μ = 0’, which
still needs to be taken into account, somehow, and it does indeed save us.
Gupta (1950) and Bleuler (1950) proposed that, rather than trying (unsuccessfully) to impose it as an operator condition, one should replace it by the
weaker condition
A ˆ μ(+) (x)|Ψ>
∂ μ
= 0
(7.111)
where the (+) signifies the positive frequency part of A ˆ , i.e. the part involving
annihilation operators, and |Ψ> is any physical state (including |0>). From
(7.111) and its Hermitian conjugate
<Ψ|∂ μ A ˆ μ(−) (x) = 0
(7.112)
7.3. The Maxwell field A
μ (x)
the ∈’s in section 7.3.1, and choose two transverse polarization vectors as (cf
(7.81), (7.82))
∈
μ (k, λ = 1) = (0, 1, 0, 0)
‘transverse polarizations’.
(7.105)
∈
μ (k, λ = 2) = (0, 0, 1, 0)
The other two ∈’s are
∈
μ (k, λ = 0) = (1, 0, 0, 0)
‘time-like polarization’
(7.106)
and
∈
μ (k, λ = 3) = (0, 0, 0, 1)
‘longitudinal polarization’.
(7.107)
Making (7.104) consistent with (7.103) then requires
†
[ˆ α λ (k), α ˆ λ ' (k
′ )] = −g λλ ' (2π)
3 δ
3 (k − k
′ ).
(7.108)
This is where the wrong sign in (7.103) has come back to haunt us: we have
the wrong sign in (7.108) for the case λ = λ
′ = 0 (time-like modes).
What is the consequence of this? It seems natural to assume that the
vacuum is defined by
α ˆ λ (k)|0> = 0
for all λ = 0, 1, 2, 3.
(7.109)
But suppose we use (7.108) and (7.109) to calculate the normalization overlap
of a ‘one time-like photon’ state; this is
†
′ )|0>
0
= −(2π)
3 δ
3 (k − k
′ )
(7.110)
k
′
and the state effectively has a negative norm (the k =
infinity is the standard plane-wave artefact). Such states would threaten fundamental properties
such as the conservation of total probability if they contributed, uncancelled,
in physical processes.
At this point we would do well to recall the condition ‘∂ μ A ˆ μ = 0’, which
still needs to be taken into account, somehow, and it does indeed save us.
Gupta (1950) and Bleuler (1950) proposed that, rather than trying (unsuccessfully) to impose it as an operator condition, one should replace it by the
weaker condition
A ˆ μ(+) (x)|Ψ>
∂ μ
= 0
(7.111)
where the (+) signifies the positive frequency part of A ˆ , i.e. the part involving
annihilation operators, and |Ψ> is any physical state (including |0>). From
(7.111) and its Hermitian conjugate
<Ψ|∂ μ A ˆ μ(−) (x) = 0
(7.112)