266
19. Spontaneously Broken Local Symmetry
But (19.50) is nothing but the equation (19.8) for a free massive vector field,
with mass M = vq! This fundamental observation was first made, in the
relativistic context, by Englert and Brout (1964), Higgs (1964), and Guralnik
et al. (1964); for a full account, see Higgs (1966).
The foregoing analysis shows us two things. First, the current (19.46) is
indeed a relativistic analogue of (19.26), in that it provides a ‘screening’ (mass
generation) effect on the gauge field. Second, equation (19.48) shows how the
phase degree of freedom of the Higgs field φ ˆ has been incorporated into a new
gauge field A ˆ
′ ν , which is massive, and therefore has ‘three’ spin degrees of
freedom. In fact, we can go further. If we imagine plane wave solutions for
ˆ
′ ν , ˆ
θ, we see that the ∂
ν ˆ
A A
ν and ˆ
θ/vq part of (19.48) will contribute something
ν
proportional to k
ν /M to the polarization vector of A
′
(recall M = vq). But
this is exactly the (large k) behaviour of the longitudinal polarization vector of
a massive vector particle. We may therefore say that the massless gauge field
A ˆ ν has ‘swallowed’ the Goldstone field θ ˆ via (19.48) to make the massive vector
field A ˆ
′ ν . The Goldstone field disappears as a massless degree of freedom, and
reappears, via its gradient, as the longitudinal part of the massive vector field.
In this way the four degrees of freedom are all now safely accounted for: three
are in the massive vector field A ˆ
′ ν , and one is in the real scalar field h ˆ (to
which we shall return shortly).
In this (relativistic) case, we know from Lorentz covariance that all the
components (transverse and longitudinal) of the vector field must have the
same mass, and this has of course emerged automatically from our covariant
treatment. But the transverse and longitudinal degrees of freedom respond
differently in the non-relativistic (superconductor) case. There, the longitudinal part of A couples strongly to longitudinal excitations of the electrons:
primarily, as Bardeen (1957) first recognized, to the collective density fluctuation mode of the electron system – that is, to plasma oscillations. This is
a high frequency mode, and is essentially the one discussed in section 17.3.2,
after equation (17.46). When this aspect of the dynamics of the electrons is
included, a fully gauge invariant description of the electromagnetic properties of superconductors, within the BCS theory, is obtained (Schreiffer 1964,
chapter 8).
We return to equations (19.48)–(19.50). Taking the divergence of (19.50)
leads, as we have seen, to the condition
ˆ
′ μ
∂ μ A = 0
(19.51)
on A ˆ
′ μ . It follows that in order to interpret the relation (19.48) as a gauge
transformation on A ˆ ν we must, to be consistent with (19.51), regard A ˆ μ as
being in a gauge specified by
∂ μ A ˆ μ =
1 ❗θ ˆ =
1 ❗θ. ˆ
(19.52)
vq
M
μ
In going from the situation described by A ˆ μ and θ ˆ to one described by A ˆ
′
19. Spontaneously Broken Local Symmetry
But (19.50) is nothing but the equation (19.8) for a free massive vector field,
with mass M = vq! This fundamental observation was first made, in the
relativistic context, by Englert and Brout (1964), Higgs (1964), and Guralnik
et al. (1964); for a full account, see Higgs (1966).
The foregoing analysis shows us two things. First, the current (19.46) is
indeed a relativistic analogue of (19.26), in that it provides a ‘screening’ (mass
generation) effect on the gauge field. Second, equation (19.48) shows how the
phase degree of freedom of the Higgs field φ ˆ has been incorporated into a new
gauge field A ˆ
′ ν , which is massive, and therefore has ‘three’ spin degrees of
freedom. In fact, we can go further. If we imagine plane wave solutions for
ˆ
′ ν , ˆ
θ, we see that the ∂
ν ˆ
A A
ν and ˆ
θ/vq part of (19.48) will contribute something
ν
proportional to k
ν /M to the polarization vector of A
′
(recall M = vq). But
this is exactly the (large k) behaviour of the longitudinal polarization vector of
a massive vector particle. We may therefore say that the massless gauge field
A ˆ ν has ‘swallowed’ the Goldstone field θ ˆ via (19.48) to make the massive vector
field A ˆ
′ ν . The Goldstone field disappears as a massless degree of freedom, and
reappears, via its gradient, as the longitudinal part of the massive vector field.
In this way the four degrees of freedom are all now safely accounted for: three
are in the massive vector field A ˆ
′ ν , and one is in the real scalar field h ˆ (to
which we shall return shortly).
In this (relativistic) case, we know from Lorentz covariance that all the
components (transverse and longitudinal) of the vector field must have the
same mass, and this has of course emerged automatically from our covariant
treatment. But the transverse and longitudinal degrees of freedom respond
differently in the non-relativistic (superconductor) case. There, the longitudinal part of A couples strongly to longitudinal excitations of the electrons:
primarily, as Bardeen (1957) first recognized, to the collective density fluctuation mode of the electron system – that is, to plasma oscillations. This is
a high frequency mode, and is essentially the one discussed in section 17.3.2,
after equation (17.46). When this aspect of the dynamics of the electrons is
included, a fully gauge invariant description of the electromagnetic properties of superconductors, within the BCS theory, is obtained (Schreiffer 1964,
chapter 8).
We return to equations (19.48)–(19.50). Taking the divergence of (19.50)
leads, as we have seen, to the condition
ˆ
′ μ
∂ μ A = 0
(19.51)
on A ˆ
′ μ . It follows that in order to interpret the relation (19.48) as a gauge
transformation on A ˆ ν we must, to be consistent with (19.51), regard A ˆ μ as
being in a gauge specified by
∂ μ A ˆ μ =
1 ❗θ ˆ =
1 ❗θ. ˆ
(19.52)
vq
M
μ
In going from the situation described by A ˆ μ and θ ˆ to one described by A ˆ
′