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M. Svrˇ cek
simple one-to-one correspondence between the Goldstone modes and the broken
symmetries.”
First of all the Goldstone theorem is a complement to the first Noether theorem
where a one-to-one correspondence between the conservation laws and associated
symmetries always exists. Therefore the unique and unambiguous one-to-one correspondence between the Goldstone modes and spontaneously broken symmetries
in solids must necessarily exist as well. It means that vibrational modes/phonons in
molecules/solids do not represent the full set of Goldstone bosons. The errors caused
by disregarding the rest of the Goldstone modes are usually negligible in adiabatic
cases, but in non-adiabatic situations they might lead to fatal consequences, i.e. a
deficient view of spontaneously broken systems on the quantum level.
We have demonstrated that a field theory based only on phonons is not sufficient for
the description of Born-Oppenheimer (B-O) degenerate states, such as Jahn-Teller
(J-T) systems and superconductors, where a general field COM covariant theory
incorporating all Goldstone bosons, i.e. phonons, rotons and translons is unavoidable.
We have derived the formulae for the ground state energy and the excitation spectra of
B-O degenerate quantum chemical and solid state systems together with a complete
set of Goldstone bosons.
Within this background the Born-Huang ansatz plays the same role for the field
COM covariance as Maxwell’s equations for Lorentz covariance. Even though we
have here only presented a special proof of the Goldstone theorem for interacting
systems of electrons and nuclei, based on the field COM covariance resulting from
the Born-Huang ansatz, we have nonetheless arrived at the correct set of Goldstone
bosons. However, the general Goldstone-Salam-Weinberg proof [92] of Goldstone’s
theorem, based on the conserved currents of the first Noether theorem, appears not
transparent enough in the case of a correct identification of the Goldstone bosons.
The physicists inaccurately identify them only with phonons.
In fact this mistake has led to unfortunate consequences, i.e. an inadequate BCS
theory [78] using only the electron-phonon interaction which has never been revisited. Moreover, Anderson in his article [95] added to this misunderstanding claiming
that “the solid crystal violates translational and rotational invariance, and possesses
phonons”. This paper soon became a fateful marker, leading to the ‘celebrated’ Higgs
mechanism, that was incorporated in the Standard model attempting to explain, how
particles in the electro-weak interaction gain the mass. Higgs named his Nobel lecture “Evading the Goldstone theorem” [97]. This was alas no ‘evading’, but rather
a misrepresentation of the theorem, since Goldstone bosons cannot after all be suppressed. We have summarized Comay’s work [88], where he, after a careful analysis
of Higgs’ equations, did come to the conclusion that they do violate Bohr’s correspondence principle, and therefore cannot be approved as valid equations of quantum
physics.
Goldstone bosons, arising as the result of symmetry violations, i.e. rotons and
translons, produce singular behaviour in the original symmetrical positions. The
system is forced to avoid them by removing the degeneracy, and after apt oneparticle transformations, finding itself in new asymmetric positions. This principle
unifies the formation of ground states of J-T molecules and superconductors. It should
M. Svrˇ cek
simple one-to-one correspondence between the Goldstone modes and the broken
symmetries.”
First of all the Goldstone theorem is a complement to the first Noether theorem
where a one-to-one correspondence between the conservation laws and associated
symmetries always exists. Therefore the unique and unambiguous one-to-one correspondence between the Goldstone modes and spontaneously broken symmetries
in solids must necessarily exist as well. It means that vibrational modes/phonons in
molecules/solids do not represent the full set of Goldstone bosons. The errors caused
by disregarding the rest of the Goldstone modes are usually negligible in adiabatic
cases, but in non-adiabatic situations they might lead to fatal consequences, i.e. a
deficient view of spontaneously broken systems on the quantum level.
We have demonstrated that a field theory based only on phonons is not sufficient for
the description of Born-Oppenheimer (B-O) degenerate states, such as Jahn-Teller
(J-T) systems and superconductors, where a general field COM covariant theory
incorporating all Goldstone bosons, i.e. phonons, rotons and translons is unavoidable.
We have derived the formulae for the ground state energy and the excitation spectra of
B-O degenerate quantum chemical and solid state systems together with a complete
set of Goldstone bosons.
Within this background the Born-Huang ansatz plays the same role for the field
COM covariance as Maxwell’s equations for Lorentz covariance. Even though we
have here only presented a special proof of the Goldstone theorem for interacting
systems of electrons and nuclei, based on the field COM covariance resulting from
the Born-Huang ansatz, we have nonetheless arrived at the correct set of Goldstone
bosons. However, the general Goldstone-Salam-Weinberg proof [92] of Goldstone’s
theorem, based on the conserved currents of the first Noether theorem, appears not
transparent enough in the case of a correct identification of the Goldstone bosons.
The physicists inaccurately identify them only with phonons.
In fact this mistake has led to unfortunate consequences, i.e. an inadequate BCS
theory [78] using only the electron-phonon interaction which has never been revisited. Moreover, Anderson in his article [95] added to this misunderstanding claiming
that “the solid crystal violates translational and rotational invariance, and possesses
phonons”. This paper soon became a fateful marker, leading to the ‘celebrated’ Higgs
mechanism, that was incorporated in the Standard model attempting to explain, how
particles in the electro-weak interaction gain the mass. Higgs named his Nobel lecture “Evading the Goldstone theorem” [97]. This was alas no ‘evading’, but rather
a misrepresentation of the theorem, since Goldstone bosons cannot after all be suppressed. We have summarized Comay’s work [88], where he, after a careful analysis
of Higgs’ equations, did come to the conclusion that they do violate Bohr’s correspondence principle, and therefore cannot be approved as valid equations of quantum
physics.
Goldstone bosons, arising as the result of symmetry violations, i.e. rotons and
translons, produce singular behaviour in the original symmetrical positions. The
system is forced to avoid them by removing the degeneracy, and after apt oneparticle transformations, finding itself in new asymmetric positions. This principle
unifies the formation of ground states of J-T molecules and superconductors. It should