263
19.2. The generation of ‘photon mass’ in a superconductor
exactly cancel – perfectly screen – the applied flux density in the interior.
With n s ∼ 4 × 10
28 m
−3 (roughly one conduction electron per atom) we find
(
) 1/2
m e
λ =
≈ 10
−8 m,
(19.35)
n s e 2
which is the correct order of magnitude for the thickness of the surface layer
within which screening currents flow, and over which the applied field falls to
zero. As T → T c , n s → 0 and λ becomes arbitrarily large, so that flux is no
longer screened.
It is quite simple to interpret equation (19.31) in terms of an ‘effective
non-zero photon mass’. Consider the equation (19.8) for a free massive vector
field. Taking the divergence via ∂ ν leads to
M
2 ∂ ν X
ν = 0
(19.36)
(cf (19.11)), and so (19.8) can be written as
(❗ + M
2 )X
ν = 0,
(19.37)
which simply expresses the fact that each component of X
ν has mass M . Now
consider the static version of (19.37), in the rest frame of the X-particle in
which (see equation (19.13)) the ν = 0 component vanishes. Equation (19.37)
reduces to
∇
2
X = M
2
X
(19.38)
which is exactly the same in form as (19.31) (if X were the electromagnetic
field A, we could take the curl of (19.38) to obtain (19.31) via B = ∇ × A).
The connection is made precise by making the association
(
)
e
2 n s
1
M
2 =
=
.
(19.39)
m e
λ 2
Equation (19.39) shows very directly another way of understanding the ‘screening length ↔ photon mass’ connection: in our units ħ = c = 1, a mass has
the dimension of an inverse length, and so we naturally expect to be able to
interpret λ
−1 as an equivalent mass (for the photon, in this case).
The above treatment conveys much of the essential physics behind the
phenomenon of ‘photon mass generation’ in a superconductor. In particular,
it suggests rather strongly that a second field, in addition to the electromagnetic one, is an essential element in the story (here, it is the ψ field). This
provides a partial answer to the puzzle about the discontinuous change in
the number of spin degrees of freedom in going from a massless to a massive
gauge field: actually, some other field has to be supplied. Nevertheless, many
questions remain unanswered so far. For example, how is all the foregoing
related to what we learned in chapter 17 about spontaneous symmetry breaking? Where is the Goldstone mode? Is it really all gauge invariant? And
19.2. The generation of ‘photon mass’ in a superconductor
exactly cancel – perfectly screen – the applied flux density in the interior.
With n s ∼ 4 × 10
28 m
−3 (roughly one conduction electron per atom) we find
(
) 1/2
m e
λ =
≈ 10
−8 m,
(19.35)
n s e 2
which is the correct order of magnitude for the thickness of the surface layer
within which screening currents flow, and over which the applied field falls to
zero. As T → T c , n s → 0 and λ becomes arbitrarily large, so that flux is no
longer screened.
It is quite simple to interpret equation (19.31) in terms of an ‘effective
non-zero photon mass’. Consider the equation (19.8) for a free massive vector
field. Taking the divergence via ∂ ν leads to
M
2 ∂ ν X
ν = 0
(19.36)
(cf (19.11)), and so (19.8) can be written as
(❗ + M
2 )X
ν = 0,
(19.37)
which simply expresses the fact that each component of X
ν has mass M . Now
consider the static version of (19.37), in the rest frame of the X-particle in
which (see equation (19.13)) the ν = 0 component vanishes. Equation (19.37)
reduces to
∇
2
X = M
2
X
(19.38)
which is exactly the same in form as (19.31) (if X were the electromagnetic
field A, we could take the curl of (19.38) to obtain (19.31) via B = ∇ × A).
The connection is made precise by making the association
(
)
e
2 n s
1
M
2 =
=
.
(19.39)
m e
λ 2
Equation (19.39) shows very directly another way of understanding the ‘screening length ↔ photon mass’ connection: in our units ħ = c = 1, a mass has
the dimension of an inverse length, and so we naturally expect to be able to
interpret λ
−1 as an equivalent mass (for the photon, in this case).
The above treatment conveys much of the essential physics behind the
phenomenon of ‘photon mass generation’ in a superconductor. In particular,
it suggests rather strongly that a second field, in addition to the electromagnetic one, is an essential element in the story (here, it is the ψ field). This
provides a partial answer to the puzzle about the discontinuous change in
the number of spin degrees of freedom in going from a massless to a massive
gauge field: actually, some other field has to be supplied. Nevertheless, many
questions remain unanswered so far. For example, how is all the foregoing
related to what we learned in chapter 17 about spontaneous symmetry breaking? Where is the Goldstone mode? Is it really all gauge invariant? And