Megascopic Quantum Phenomena
381
be noted how the quantum field formulation deals with the virtual degeneracies
originating from approximative B-O solutions. They are profoundly different from
the real degeneracies, where the quantum principle of superposition takes place,
after being removed by some external perturbation, such as the Stark or the Zeeman
effect. The quantum field simply does not share the centre of mass, defined in standard
quantum mechanics, rather it solves the problem of the centre of gravity “in its own
way”.
We have also offered a reformulated version of the J-T theorem as it follows
directly from the Goldstone theorem. Molecular and crystalline entities in a particular geometry of an electronically degenerate ground state are unstable except for
the case when all matrix elements of electron-rotational and electron-translational
interactions equal zero. This is a striking result, as it shows how a true quantum field,
respecting fully the Goldstone theorem, copes with B-O degenerate systems, such
as J-T molecules and superconductors. The superposition principle for the removal
of the degeneracy is simply bypassed, since the rotons and the translons are actually responsible for the violation of symmetry. For instance in a superconductor, the
symmetry breaking produces several geometrically different symmetry broken states,
splitting the original half-occupied conducting band of the conductor into two bands,
one fully occupied valence band and one empty conducting band, in such a way that
the original conductor under the critical temperature becomes a multi-ground-state
insulator.
The results discussed above clearly disqualify all theories based on B-O degeneracies, such as the usual solutions of the J-T effect [11, 12] and the BCS theory
of superconductivity [78]. Their validity was never justified from simple reasons,
i.e. one query of extraordinary importance was never answered, cf. the question of
Sutcliffe and Woolley [4]: “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.” We have further concluded that quantum mechanics describing a system of nuclei and electrons can never resolve this task. The quantum theory
of an Isolated Molecule is quantum mechanics with a solution avoiding the B-O
concept, and this is the method of Monkhorst [5, 6] that “takes a very atomic view of
a molecule: instead of fixing the nuclei as in the B-O approximation, the electrons
and nuclei are both described quantum-dynamically within a centrosymmetric shell
structure.” Unfortunately this approach leads neither to any B-O degenerate states
nor to any symmetry breaking.
A valid quantum theory of an individual molecule should be a true quantum field
theory of fermions represented by renormalized electrons and Goldstone bosons
standing for vibrational, rotational and translational modes, i.e. a system where nuclei
from the mechanical description are replaced by the Goldstone bosons in the field
description. This means, that true quantum mechanics deals solely with isolated
systems, while true quantum field theory works with individual systems. Only a
quantum field formulation can produce symmetry broken states and, unlike the B-O
model, removing degeneracies bypassing the superposition principle.
Returning to the question of teleological principles in science, one might first
deliberate over how classical physics copes with the spontaneous symmetry breaking
381
be noted how the quantum field formulation deals with the virtual degeneracies
originating from approximative B-O solutions. They are profoundly different from
the real degeneracies, where the quantum principle of superposition takes place,
after being removed by some external perturbation, such as the Stark or the Zeeman
effect. The quantum field simply does not share the centre of mass, defined in standard
quantum mechanics, rather it solves the problem of the centre of gravity “in its own
way”.
We have also offered a reformulated version of the J-T theorem as it follows
directly from the Goldstone theorem. Molecular and crystalline entities in a particular geometry of an electronically degenerate ground state are unstable except for
the case when all matrix elements of electron-rotational and electron-translational
interactions equal zero. This is a striking result, as it shows how a true quantum field,
respecting fully the Goldstone theorem, copes with B-O degenerate systems, such
as J-T molecules and superconductors. The superposition principle for the removal
of the degeneracy is simply bypassed, since the rotons and the translons are actually responsible for the violation of symmetry. For instance in a superconductor, the
symmetry breaking produces several geometrically different symmetry broken states,
splitting the original half-occupied conducting band of the conductor into two bands,
one fully occupied valence band and one empty conducting band, in such a way that
the original conductor under the critical temperature becomes a multi-ground-state
insulator.
The results discussed above clearly disqualify all theories based on B-O degeneracies, such as the usual solutions of the J-T effect [11, 12] and the BCS theory
of superconductivity [78]. Their validity was never justified from simple reasons,
i.e. one query of extraordinary importance was never answered, cf. the question of
Sutcliffe and Woolley [4]: “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.” We have further concluded that quantum mechanics describing a system of nuclei and electrons can never resolve this task. The quantum theory
of an Isolated Molecule is quantum mechanics with a solution avoiding the B-O
concept, and this is the method of Monkhorst [5, 6] that “takes a very atomic view of
a molecule: instead of fixing the nuclei as in the B-O approximation, the electrons
and nuclei are both described quantum-dynamically within a centrosymmetric shell
structure.” Unfortunately this approach leads neither to any B-O degenerate states
nor to any symmetry breaking.
A valid quantum theory of an individual molecule should be a true quantum field
theory of fermions represented by renormalized electrons and Goldstone bosons
standing for vibrational, rotational and translational modes, i.e. a system where nuclei
from the mechanical description are replaced by the Goldstone bosons in the field
description. This means, that true quantum mechanics deals solely with isolated
systems, while true quantum field theory works with individual systems. Only a
quantum field formulation can produce symmetry broken states and, unlike the B-O
model, removing degeneracies bypassing the superposition principle.
Returning to the question of teleological principles in science, one might first
deliberate over how classical physics copes with the spontaneous symmetry breaking