Megascopic Quantum Phenomena
379
Hamiltonian. We have put forward five sets of well-known equations as limiting
cases of the resulting equations of our procedure; a test that our solutions must pass
and exactly agree with each other
(1) Pople’s equations for ab initio calculations of vibrational frequencies [104]
(2) Fröhlich’s expression for the correction of the ground state energy [102, 105]
(3) Fröhlich’s effective two-electron interaction [102]
(4) Lee-Low-Pines’ polarons and their self-energy [106]
(5) Born-Huang’s ansatz [7].
Starting from the field electron-vibrational (electron-phonon) Hamiltonian [101,
103], only the first four sets pass the test successfully. For instance, in small molecules
such as H 2 one obtains only a fragment of the whole contribution from the BornHuang ansatz, and this is a major failure that unfortunately is not well-known in
quantum chemistry and solid state physics. In quantum chemistry field methods
are not often used, and usually only on the electronic level [99, 100]. However, in
solid state physics electron-phonon field mechanisms are frequently employed, and
in most cases the contribution of the Born-Huang ansatz is negligible to such an
extent that many physicists do not even realize any differences between the BornOppenheimer (B-O) and the adiabatic approximations and mistakenly even regard
them as synonyms.
Hence the concept of the electron-vibrational field theory is clearly insufficient,
and it must be replaced by a more rigorous field theory comprising two major
upgrades [107, 109]:
(a) Together with the phonons two new types of quasiparticles appear: rotons and
translons descending from rotational and translational degrees of freedom. The
contribution from the roton and the translon quanta occurs even if the molecule
as whole does not rotate nor move.
(b) In analogy with Lorentz covariance, binding together space and time coordinates, a new covariance is necessary, binding together internal and external degrees of freedom, without the centre-of-mass (COM) separation, which
normally applies in both classical and quantum mechanics.
Let us compare these results with the content of Goldstone’s theorem [91]: “The
spontaneous breaking of a continuous symmetry can be associated with a massless
and spinless particle”. According to this theorem there are three types of broken symmetries in molecules and crystals, composed of N nuclei (3N degrees of freedom):
the Galilean associated with 3N − 6 phonons (or 3N − 5 for diatomic molecules),
translational associated with 3 translons, and rotational associated with 3 (or 2)
rotons. Unfortunately the scientific community does not seem to be aware of this
fundamental fact with its origin in the Goldstone theorem, cf. the picture reflected
in the (2018) Wikipedia phrase “Goldstone boson“ that states the following incongruity: “In solids, the situation is more complicated; the Goldstone bosons are the
longitudinal and transverse phonons and they happen to be the Goldstone bosons
of spontaneously broken Galilean, translational, and rotational symmetry with no
379
Hamiltonian. We have put forward five sets of well-known equations as limiting
cases of the resulting equations of our procedure; a test that our solutions must pass
and exactly agree with each other
(1) Pople’s equations for ab initio calculations of vibrational frequencies [104]
(2) Fröhlich’s expression for the correction of the ground state energy [102, 105]
(3) Fröhlich’s effective two-electron interaction [102]
(4) Lee-Low-Pines’ polarons and their self-energy [106]
(5) Born-Huang’s ansatz [7].
Starting from the field electron-vibrational (electron-phonon) Hamiltonian [101,
103], only the first four sets pass the test successfully. For instance, in small molecules
such as H 2 one obtains only a fragment of the whole contribution from the BornHuang ansatz, and this is a major failure that unfortunately is not well-known in
quantum chemistry and solid state physics. In quantum chemistry field methods
are not often used, and usually only on the electronic level [99, 100]. However, in
solid state physics electron-phonon field mechanisms are frequently employed, and
in most cases the contribution of the Born-Huang ansatz is negligible to such an
extent that many physicists do not even realize any differences between the BornOppenheimer (B-O) and the adiabatic approximations and mistakenly even regard
them as synonyms.
Hence the concept of the electron-vibrational field theory is clearly insufficient,
and it must be replaced by a more rigorous field theory comprising two major
upgrades [107, 109]:
(a) Together with the phonons two new types of quasiparticles appear: rotons and
translons descending from rotational and translational degrees of freedom. The
contribution from the roton and the translon quanta occurs even if the molecule
as whole does not rotate nor move.
(b) In analogy with Lorentz covariance, binding together space and time coordinates, a new covariance is necessary, binding together internal and external degrees of freedom, without the centre-of-mass (COM) separation, which
normally applies in both classical and quantum mechanics.
Let us compare these results with the content of Goldstone’s theorem [91]: “The
spontaneous breaking of a continuous symmetry can be associated with a massless
and spinless particle”. According to this theorem there are three types of broken symmetries in molecules and crystals, composed of N nuclei (3N degrees of freedom):
the Galilean associated with 3N − 6 phonons (or 3N − 5 for diatomic molecules),
translational associated with 3 translons, and rotational associated with 3 (or 2)
rotons. Unfortunately the scientific community does not seem to be aware of this
fundamental fact with its origin in the Goldstone theorem, cf. the picture reflected
in the (2018) Wikipedia phrase “Goldstone boson“ that states the following incongruity: “In solids, the situation is more complicated; the Goldstone bosons are the
longitudinal and transverse phonons and they happen to be the Goldstone bosons
of spontaneously broken Galilean, translational, and rotational symmetry with no