Megascopic Quantum Phenomena
339
12 The Paradox of Spontaneous Symmetry Breaking
Goldstone’s theorem implies a quantum field reformulation of the J-T effect which is
also valid for superconductors [107]: “Molecular and crystallin entities in a geometry
of electronically degenerate ground states are unstable at this geometry except for the
case when all matrix elements of the electron-rotational and electron-translational
interactions equal zero.” When I published this definition [107], I did not, at first,
realize that it was a direct consequence of the Goldstone theorem, and secondly I did
look upon it as an alternative to the official quantum mechanical definition. Later I
saw that only the field J-T definition should be correct, since it leads to a truly broken
symmetry for the ground state.
The official mechanical definition starts with the clamped-nuclei concept of the
B-O approximation (see Sect. 2), whereas the exact mechanical solution, based on
the Monkhorst-Cafiero-Adamowitz shell method for nuclei, leads to no symmetry
breaking since the nuclei can be delocalized. On the other hand, the BCS theory
of superconductivity is a quantum field theory where a Bloch-type field is used
that does not respect the Goldstone theorem, and therefore the BCS theory cannot
be correct. Yet the Bloch field was successfully applied to insulators, conductors,
and semiconductors, but from the perspective of materials with B-O degenerate
ground states and broken symmetries, such a field will be crippled and not usable. In
such a case it permits linear superpositions of degenerate states, as it was originally
described in the BCS paper [78]: “The normal phase is described by the Bloch
individual-particle model. The ground state of a superconductor, formed from a linear
combination of normal state configurations in which electrons are virtually excited
in pairs of opposite spin and momentum, is lower in energy than the normal state by
amount that is proportional to the average (èω)
2 , which is consistent with the isotope
effect. A mutually orthogonal set of excited states, in one-to-one correspondence with
those of the normal phase, obtains by specifying the occupation of certain Bloch states
and by using the rest to form a linear combination of virtual pair configurations.”
It is interesting to observe that, the BCS paper [78] does not mention any symmetry breakings. Four decades later, Weinberg wondered how Bardeen, Cooper, and
Schrieffer could not see it [111]: “A superconductor is simply a material in which
electromagnetic gauge invariance is spontaneously broken. This is not the way that
most experts have historically thought about superconductivity. Early phenomenological theories were known to violate electromagnetic gauge invariance, but this was
regarded as more annoying than enlightening. Broken symmetry is never mentioned
in the seminal paper by Bardeen et al. [78] that first gave us a microscopic theory of superconductivity. Anderson [95] subsequently stressed the important role of
broken symmetry in superconductors, but even today most textbooks explain superconductivity in terms of detailed dynamical models, with broken symmetry rarely
mentioned.”
Looking closely at the structure of the BCS theory, there is no asymmetry
input/output, and hence there was absolutely no reason for Bardeen, Cooper, and
Schrieffer to discuss any violations of symmetry. But we have already seen above
339
12 The Paradox of Spontaneous Symmetry Breaking
Goldstone’s theorem implies a quantum field reformulation of the J-T effect which is
also valid for superconductors [107]: “Molecular and crystallin entities in a geometry
of electronically degenerate ground states are unstable at this geometry except for the
case when all matrix elements of the electron-rotational and electron-translational
interactions equal zero.” When I published this definition [107], I did not, at first,
realize that it was a direct consequence of the Goldstone theorem, and secondly I did
look upon it as an alternative to the official quantum mechanical definition. Later I
saw that only the field J-T definition should be correct, since it leads to a truly broken
symmetry for the ground state.
The official mechanical definition starts with the clamped-nuclei concept of the
B-O approximation (see Sect. 2), whereas the exact mechanical solution, based on
the Monkhorst-Cafiero-Adamowitz shell method for nuclei, leads to no symmetry
breaking since the nuclei can be delocalized. On the other hand, the BCS theory
of superconductivity is a quantum field theory where a Bloch-type field is used
that does not respect the Goldstone theorem, and therefore the BCS theory cannot
be correct. Yet the Bloch field was successfully applied to insulators, conductors,
and semiconductors, but from the perspective of materials with B-O degenerate
ground states and broken symmetries, such a field will be crippled and not usable. In
such a case it permits linear superpositions of degenerate states, as it was originally
described in the BCS paper [78]: “The normal phase is described by the Bloch
individual-particle model. The ground state of a superconductor, formed from a linear
combination of normal state configurations in which electrons are virtually excited
in pairs of opposite spin and momentum, is lower in energy than the normal state by
amount that is proportional to the average (èω)
2 , which is consistent with the isotope
effect. A mutually orthogonal set of excited states, in one-to-one correspondence with
those of the normal phase, obtains by specifying the occupation of certain Bloch states
and by using the rest to form a linear combination of virtual pair configurations.”
It is interesting to observe that, the BCS paper [78] does not mention any symmetry breakings. Four decades later, Weinberg wondered how Bardeen, Cooper, and
Schrieffer could not see it [111]: “A superconductor is simply a material in which
electromagnetic gauge invariance is spontaneously broken. This is not the way that
most experts have historically thought about superconductivity. Early phenomenological theories were known to violate electromagnetic gauge invariance, but this was
regarded as more annoying than enlightening. Broken symmetry is never mentioned
in the seminal paper by Bardeen et al. [78] that first gave us a microscopic theory of superconductivity. Anderson [95] subsequently stressed the important role of
broken symmetry in superconductors, but even today most textbooks explain superconductivity in terms of detailed dynamical models, with broken symmetry rarely
mentioned.”
Looking closely at the structure of the BCS theory, there is no asymmetry
input/output, and hence there was absolutely no reason for Bardeen, Cooper, and
Schrieffer to discuss any violations of symmetry. But we have already seen above