338
M. Svrˇ cek
One small question remains: Is the Born-Huang ansatz experimentally credible
to the same extent as the Maxwell equations? If so we can authorize the formulation of a new type of field covariance. It was the merit of Handy [110] and many
others, that thousands of Born-Huang ansatz calculations and their comparison with
experimental data were performed, since for a long time a reasonable doubt prevailed
regarding the equivalency of these results, and those of the exact quantum mechanical COM separation and the Born-Huang ansatz as the first correction to the B-O
approximation. Kutzelnigg therefore renamed the Born-Huang ansatz as the BornHandy ansatz [8]: “Handy and co-workers have never claimed to have invented the
ansatz referred to here as the “Born-Handy ansatz”, but they certainly convinced a
large audience that this ansatz is of enormous practical value, even if it has not been
completely obvious why it leads to correct results. Handy and co-workers realized
that the difficulties with the traditional approach come from the separation of the
COM motion (and the need to define internal coordinates after this separation has
been made). They therefore decided to renounce the separation.”
Once the Born-Huang (Born-Handy) ansatz was experimentally confirmed, we
have subsequently experimental evidence of the lost Goldstone bosons—rotons and
translons—as well. Their role in quantum systems is quite curious. Their contribution to the corrections of the ground state energy is already significant in small
molecules like hydrogen molecule with only one vibrational mode and the five lost
Goldstone boson modes, and is even several times greater than the contribution of
the pure vibrations. The bigger the molecule, the lesser effect have these rotons and
translons. In most of solids, with a huge amount of phonons, such as conductors,
semiconductors or insulators, the effect of the six rotons/translons becomes negligible and the electron-phonon field theory is sufficient. However, in B-O degenerate
systems suddenly a surprise appears. We have arrived at the same formula for the
ground states of finite as well as infinite systems, both molecules and crystals, in
the case of a B-O electronic degeneracy. However, Goldstone bosons arising from
the violation of rotational and translational symmetries, rotons and translons, produce singularities in the original symmetrical positions, and the system is forced to
avoid them, removing the degeneracy in an effectively one-particle manner at new
asymmetric positions. This principle unifies the formation of the ground states of J-T
molecules and superconductors, showing the way how quantum field formulations
deal with virtual degeneracies originating from approximative B-O solutions. They
are profoundly different from real degeneracies, where the principle of superposition
takes place after the removal by some external perturbation (Stark or Zeeman effect).
Quantum field theories simply do not share the centre of mass defined in quantum
mechanics but solves the centre of mass problem “on its own”.
If the lost Goldstone bosons—rotons and translons—are responsible for the
mechanism of the formation of quantum states with spontaneously broken symmetries, it means, that the spontaneous symmetry breaking (SSB) was only phenomenologically described by classical physics, but it was completely misunderstood on the
quantum level. We will continue with the analysis of the SSB in the next section.
M. Svrˇ cek
One small question remains: Is the Born-Huang ansatz experimentally credible
to the same extent as the Maxwell equations? If so we can authorize the formulation of a new type of field covariance. It was the merit of Handy [110] and many
others, that thousands of Born-Huang ansatz calculations and their comparison with
experimental data were performed, since for a long time a reasonable doubt prevailed
regarding the equivalency of these results, and those of the exact quantum mechanical COM separation and the Born-Huang ansatz as the first correction to the B-O
approximation. Kutzelnigg therefore renamed the Born-Huang ansatz as the BornHandy ansatz [8]: “Handy and co-workers have never claimed to have invented the
ansatz referred to here as the “Born-Handy ansatz”, but they certainly convinced a
large audience that this ansatz is of enormous practical value, even if it has not been
completely obvious why it leads to correct results. Handy and co-workers realized
that the difficulties with the traditional approach come from the separation of the
COM motion (and the need to define internal coordinates after this separation has
been made). They therefore decided to renounce the separation.”
Once the Born-Huang (Born-Handy) ansatz was experimentally confirmed, we
have subsequently experimental evidence of the lost Goldstone bosons—rotons and
translons—as well. Their role in quantum systems is quite curious. Their contribution to the corrections of the ground state energy is already significant in small
molecules like hydrogen molecule with only one vibrational mode and the five lost
Goldstone boson modes, and is even several times greater than the contribution of
the pure vibrations. The bigger the molecule, the lesser effect have these rotons and
translons. In most of solids, with a huge amount of phonons, such as conductors,
semiconductors or insulators, the effect of the six rotons/translons becomes negligible and the electron-phonon field theory is sufficient. However, in B-O degenerate
systems suddenly a surprise appears. We have arrived at the same formula for the
ground states of finite as well as infinite systems, both molecules and crystals, in
the case of a B-O electronic degeneracy. However, Goldstone bosons arising from
the violation of rotational and translational symmetries, rotons and translons, produce singularities in the original symmetrical positions, and the system is forced to
avoid them, removing the degeneracy in an effectively one-particle manner at new
asymmetric positions. This principle unifies the formation of the ground states of J-T
molecules and superconductors, showing the way how quantum field formulations
deal with virtual degeneracies originating from approximative B-O solutions. They
are profoundly different from real degeneracies, where the principle of superposition
takes place after the removal by some external perturbation (Stark or Zeeman effect).
Quantum field theories simply do not share the centre of mass defined in quantum
mechanics but solves the centre of mass problem “on its own”.
If the lost Goldstone bosons—rotons and translons—are responsible for the
mechanism of the formation of quantum states with spontaneously broken symmetries, it means, that the spontaneous symmetry breaking (SSB) was only phenomenologically described by classical physics, but it was completely misunderstood on the
quantum level. We will continue with the analysis of the SSB in the next section.