260
19. Spontaneously Broken Local Symmetry
metry. From his discussion, it is clear that Anderson had his doubts about
the hadronic application, precisely because, as he remarked, gauge bosons can
only acquire a mass if the symmetry is spontaneously broken. This has the
consequence, as we saw in chapter 17, that the multiplet structure ordinarily
associated with a non-Abelian symmetry would be lost. But we know that
flavour symmetry, even if admittedly not exact, certainly leads to identifiable
multiplets, which are at least approximately degenerate in mass. It was Weinberg (1967) and Salam (1968) who made the correct application of these ideas,
to the generation of mass for the gauge quanta associated with the weak force.
There is, however, nothing specifically relativistic about the basic mechanism
involved, nor need we start with the non-Abelian case. In fact, the physics
is well illustrated by the non-relativistic Abelian (i.e. electromagnetic) case
– which is nothing but the physics of superconductivity. Our presentation is
influenced by that of Anderson (1963).
19.2 The generation of ‘photon mass’ in a superconductor: Ginzburg–Landau theory and
the Meissner effect
In chapter 17, section 17.7, we gave a brief introduction to some aspects
of the BCS theory of superconductivity. We were concerned mainly with
the nature of the BCS ground state, and with the non-perturbative origin
of the energy gap for elementary excitations. In particular, as noted after
(17.128), we omitted completely all electromagnetic couplings of the electrons
in the ‘microscopic’ Hamiltonian. It is certainly possible to complete the BCS
theory in this way, so as to include within the same formalism a treatment of
electromagnetic effects (e.g. the Meissner effect) in a superconductor. We refer
interested readers to the book by Schrieffer (1964), chapter 8. Instead, we shall
follow a less ‘microscopic’ and somewhat more ‘phenomenological’ approach,
which has a long history in theoretical studies of superconductivity, and is in
some ways actually closer (at least formally) to our eventual application in
particle physics.
In section 17.3.1 we introduced the concept of an ‘order parameter’, a
quantity which was a measure of the ‘degree of ordering’ of a system below
some transition temperature. In the case of superconductivity, the order parameter (in this sense) is taken to be a complex scalar field ψ, as originally
proposed by Ginzburg and Landau (1950), well before the appearance of BCS
theory. Subsequently, Gorkov (1959) and others showed how the Ginzburg–
Landau description could be derived from BCS theory, in certain domains of
temperature and magnetic field. This work all relates to static phenomena.
More recently, an analogous ‘effective theory’ for time-dependent phenomena
19. Spontaneously Broken Local Symmetry
metry. From his discussion, it is clear that Anderson had his doubts about
the hadronic application, precisely because, as he remarked, gauge bosons can
only acquire a mass if the symmetry is spontaneously broken. This has the
consequence, as we saw in chapter 17, that the multiplet structure ordinarily
associated with a non-Abelian symmetry would be lost. But we know that
flavour symmetry, even if admittedly not exact, certainly leads to identifiable
multiplets, which are at least approximately degenerate in mass. It was Weinberg (1967) and Salam (1968) who made the correct application of these ideas,
to the generation of mass for the gauge quanta associated with the weak force.
There is, however, nothing specifically relativistic about the basic mechanism
involved, nor need we start with the non-Abelian case. In fact, the physics
is well illustrated by the non-relativistic Abelian (i.e. electromagnetic) case
– which is nothing but the physics of superconductivity. Our presentation is
influenced by that of Anderson (1963).
19.2 The generation of ‘photon mass’ in a superconductor: Ginzburg–Landau theory and
the Meissner effect
In chapter 17, section 17.7, we gave a brief introduction to some aspects
of the BCS theory of superconductivity. We were concerned mainly with
the nature of the BCS ground state, and with the non-perturbative origin
of the energy gap for elementary excitations. In particular, as noted after
(17.128), we omitted completely all electromagnetic couplings of the electrons
in the ‘microscopic’ Hamiltonian. It is certainly possible to complete the BCS
theory in this way, so as to include within the same formalism a treatment of
electromagnetic effects (e.g. the Meissner effect) in a superconductor. We refer
interested readers to the book by Schrieffer (1964), chapter 8. Instead, we shall
follow a less ‘microscopic’ and somewhat more ‘phenomenological’ approach,
which has a long history in theoretical studies of superconductivity, and is in
some ways actually closer (at least formally) to our eventual application in
particle physics.
In section 17.3.1 we introduced the concept of an ‘order parameter’, a
quantity which was a measure of the ‘degree of ordering’ of a system below
some transition temperature. In the case of superconductivity, the order parameter (in this sense) is taken to be a complex scalar field ψ, as originally
proposed by Ginzburg and Landau (1950), well before the appearance of BCS
theory. Subsequently, Gorkov (1959) and others showed how the Ginzburg–
Landau description could be derived from BCS theory, in certain domains of
temperature and magnetic field. This work all relates to static phenomena.
More recently, an analogous ‘effective theory’ for time-dependent phenomena