348
M. Svrˇ cek
for molecular structure [132]. Since one knows that physical quantities, such as mass,
charge, energy, momentum etc., need exact definitions, how is it then possible that
molecular and crystalline structures are defined by an approximation. The latter must
either be erroneous or alternatively not principally an approximation at all, hiding
a precise and fundamental meaning. One reason for accepting the second choice,
would be the argument by Sutcliffe and Woolley, see Sect. 2, regarding the simplest case of broken symmetries. For instance the Monkhorst-Cafiero-Adamowitz
approach, dealing with isolated molecules, does not recognize any isomerism. They
appear only on the B-O level, that deals exclusively with individual molecules. The
conclusion is clear, no approximation will lead to new phenomena such as symmetry
violations or even to making an ontological shift from isolated to individual order of
the systems under investigation. Quoting again the challenge by Sutcliffe and Woolley, see Sect. 2: “The interesting question is how to get from the quantum theory
of an Isolated Molecule to a quantum theory of an individual molecule by rational
mathematics.”
Pioneering quantum mechanics, describing a system of nuclei and electrons, cannot answer the above mentioned dilemma. We need a field theoretical description of
the fermions as renormalized electrons and the nuclei replaced by Goldstone bosons
as vibrational, rotational and translational modes. As was shown in Sect. 11, the field
Hamiltonian (11.33) satisfies this request finally resulting in the clamped-nuclei concept, the first step of the B-O approximation, yielding all the equations known from
this approximation, such as those of Pople for the ab initio calculation of vibrational frequencies (11.40), (11.41) and the Born-Huang ansatz (11.49). However, the
clamped-nuclei concept is a contradictio in adjecto. For instance either one keeps the
nuclei in mind, which can never be “clamped” since the nuclear positions do not commute with the total Hamiltonian, or one becomes fixed on the adjective “clamped”,
emphasizing instead of nuclei rather to speak about some traces or footprints, such as
e.g. traces of electrons on the screen or traces of photons on a photographic plate. In
a previous paper [109] I did introduce the notion of property-object dualism as follows: In quantum mechanics, nuclei and electrons represent objects to be described
on an equal footing, and the vibrational modes are their common property. On the
other hand, in quantum field theories the objects are represented by electrons and the
Goldstone bosons, and “nuclear“ positions or “clamped nuclei” are properties of the
pertinent field equations.
As far as adiabatic systems are concerned, we have shown a proof of the equivalency between the field equations and the mechanical equations based on the B-O
concept. What happens for nonadiabatic systems? While the practice of the B-O
model is limited to adiabatic systems, there are no such limitations for the field
equations. They are valid on the whole scale from adiabatic to nonadiabatic. For
example we proved in Sect. 11 how the true field equations, based on Goldstone’s
theorem, do lead to symmetry broken ground states with the degeneracy removed
on the one-particle level in both J-T systems and superconductors. In contrast the
B-O approach to nonadiabatic systems leads to metaphysical solutions that are based
on linear combinations of degenerate states. For the case of an infinite number of
degenerate states the B-O concept gives rise to Mexican hats and the Berry-phase
M. Svrˇ cek
for molecular structure [132]. Since one knows that physical quantities, such as mass,
charge, energy, momentum etc., need exact definitions, how is it then possible that
molecular and crystalline structures are defined by an approximation. The latter must
either be erroneous or alternatively not principally an approximation at all, hiding
a precise and fundamental meaning. One reason for accepting the second choice,
would be the argument by Sutcliffe and Woolley, see Sect. 2, regarding the simplest case of broken symmetries. For instance the Monkhorst-Cafiero-Adamowitz
approach, dealing with isolated molecules, does not recognize any isomerism. They
appear only on the B-O level, that deals exclusively with individual molecules. The
conclusion is clear, no approximation will lead to new phenomena such as symmetry
violations or even to making an ontological shift from isolated to individual order of
the systems under investigation. Quoting again the challenge by Sutcliffe and Woolley, see Sect. 2: “The interesting question is how to get from the quantum theory
of an Isolated Molecule to a quantum theory of an individual molecule by rational
mathematics.”
Pioneering quantum mechanics, describing a system of nuclei and electrons, cannot answer the above mentioned dilemma. We need a field theoretical description of
the fermions as renormalized electrons and the nuclei replaced by Goldstone bosons
as vibrational, rotational and translational modes. As was shown in Sect. 11, the field
Hamiltonian (11.33) satisfies this request finally resulting in the clamped-nuclei concept, the first step of the B-O approximation, yielding all the equations known from
this approximation, such as those of Pople for the ab initio calculation of vibrational frequencies (11.40), (11.41) and the Born-Huang ansatz (11.49). However, the
clamped-nuclei concept is a contradictio in adjecto. For instance either one keeps the
nuclei in mind, which can never be “clamped” since the nuclear positions do not commute with the total Hamiltonian, or one becomes fixed on the adjective “clamped”,
emphasizing instead of nuclei rather to speak about some traces or footprints, such as
e.g. traces of electrons on the screen or traces of photons on a photographic plate. In
a previous paper [109] I did introduce the notion of property-object dualism as follows: In quantum mechanics, nuclei and electrons represent objects to be described
on an equal footing, and the vibrational modes are their common property. On the
other hand, in quantum field theories the objects are represented by electrons and the
Goldstone bosons, and “nuclear“ positions or “clamped nuclei” are properties of the
pertinent field equations.
As far as adiabatic systems are concerned, we have shown a proof of the equivalency between the field equations and the mechanical equations based on the B-O
concept. What happens for nonadiabatic systems? While the practice of the B-O
model is limited to adiabatic systems, there are no such limitations for the field
equations. They are valid on the whole scale from adiabatic to nonadiabatic. For
example we proved in Sect. 11 how the true field equations, based on Goldstone’s
theorem, do lead to symmetry broken ground states with the degeneracy removed
on the one-particle level in both J-T systems and superconductors. In contrast the
B-O approach to nonadiabatic systems leads to metaphysical solutions that are based
on linear combinations of degenerate states. For the case of an infinite number of
degenerate states the B-O concept gives rise to Mexican hats and the Berry-phase