270
19. Spontaneously Broken Local Symmetry
C is quantized:
∫
2πn
Φ = B · dS =
= nΦ 0
(19.61)
q
where Φ 0 = 2π/q is the flux equation (or 2πħ/q in ordinary units). It is not
entirely self-evident why ψ should be single-valued, but experiments do indeed
demonstrate the phenomenon of flux quantization, in units of Φ 0 with |q| = 2e
(which may be interpreted as the charge on a Cooper pair, as usual). The phenomenon is seen in non-simply connected specimens of type I superconductors
(i.e. ones with holes in them, such as a ring), and in the flux filaments of type
II materials; in the latter case each filament carries a single flux quantum Φ 0 .
It is interesting to consider now a situation – so far entirely hypothetical
– in which a magnetic monopole is placed in a superconductor. Dirac showed
(1931) that for consistency with quantum mechanics the monopole strength
g m had to satisfy the ‘Dirac quantization conduction’
qg m = n/2
(19.62)
where q is any electronic charge, and n is an integer. It follows from (19.62)
that the flux 4πg m out of any closed surface surrounding the monopole is
quantized in units of Φ 0 . Hence a flux filament in a superconductor can
originate from, or be terminated by, a Dirac monopole (with the appropriate
sign of g m ), as was first pointed out by Nambu (1974).
This is the basic model which, in one way or another, underlies many
theoretical attempts to understand confinement. The monopole–antimonopole
pair in a type II superconducting vacuum, joined by a quantized magnetic flux
filament, provides a model of a meson. As the distance between the pair –
the length of the filament – increases, so does the energy of the filament, at a
rate proportional to its length, since the flux cannot spread out in directions
transverse to the filament. This is exactly the kind of linearly rising potential
energy required by hadron spectroscopy (see equations (1.33) and (16.145)).
The configuration is stable, because there is no way for the flux to leak away;
it is a conserved quantized quantity.
For the eventual application to QCD, one will want (presumably) particles carrying non-zero values of the colour quantum numbers to be confined. These quantum numbers are the analogues of electric charge in the
U(1) case, rather than of magnetic charge. We imagine, therefore, interchanging the roles of magnetism and electricity in all of the foregoing. Indeed, the
Maxwell equations have such a symmetry when monopoles are present, as well
as charges. The essential feature of the superconducting ground state was that
it involved the coherent state formed by condensation of electrically charged
bosonic fermion pairs. A vacuum which confined filaments of E rather than B
may be formed as a coherent state of condensed magnetic monopoles (Mandelstam 1976, ’t Hooft 1976). These E filaments would then terminate on
electric charges. Now magnetic monopoles do not occur naturally as solutions
of QED: they would have to be introduced by hand. Remarkably enough,
19. Spontaneously Broken Local Symmetry
C is quantized:
∫
2πn
Φ = B · dS =
= nΦ 0
(19.61)
q
where Φ 0 = 2π/q is the flux equation (or 2πħ/q in ordinary units). It is not
entirely self-evident why ψ should be single-valued, but experiments do indeed
demonstrate the phenomenon of flux quantization, in units of Φ 0 with |q| = 2e
(which may be interpreted as the charge on a Cooper pair, as usual). The phenomenon is seen in non-simply connected specimens of type I superconductors
(i.e. ones with holes in them, such as a ring), and in the flux filaments of type
II materials; in the latter case each filament carries a single flux quantum Φ 0 .
It is interesting to consider now a situation – so far entirely hypothetical
– in which a magnetic monopole is placed in a superconductor. Dirac showed
(1931) that for consistency with quantum mechanics the monopole strength
g m had to satisfy the ‘Dirac quantization conduction’
qg m = n/2
(19.62)
where q is any electronic charge, and n is an integer. It follows from (19.62)
that the flux 4πg m out of any closed surface surrounding the monopole is
quantized in units of Φ 0 . Hence a flux filament in a superconductor can
originate from, or be terminated by, a Dirac monopole (with the appropriate
sign of g m ), as was first pointed out by Nambu (1974).
This is the basic model which, in one way or another, underlies many
theoretical attempts to understand confinement. The monopole–antimonopole
pair in a type II superconducting vacuum, joined by a quantized magnetic flux
filament, provides a model of a meson. As the distance between the pair –
the length of the filament – increases, so does the energy of the filament, at a
rate proportional to its length, since the flux cannot spread out in directions
transverse to the filament. This is exactly the kind of linearly rising potential
energy required by hadron spectroscopy (see equations (1.33) and (16.145)).
The configuration is stable, because there is no way for the flux to leak away;
it is a conserved quantized quantity.
For the eventual application to QCD, one will want (presumably) particles carrying non-zero values of the colour quantum numbers to be confined. These quantum numbers are the analogues of electric charge in the
U(1) case, rather than of magnetic charge. We imagine, therefore, interchanging the roles of magnetism and electricity in all of the foregoing. Indeed, the
Maxwell equations have such a symmetry when monopoles are present, as well
as charges. The essential feature of the superconducting ground state was that
it involved the coherent state formed by condensation of electrically charged
bosonic fermion pairs. A vacuum which confined filaments of E rather than B
may be formed as a coherent state of condensed magnetic monopoles (Mandelstam 1976, ’t Hooft 1976). These E filaments would then terminate on
electric charges. Now magnetic monopoles do not occur naturally as solutions
of QED: they would have to be introduced by hand. Remarkably enough,