350
M. Svrˇ cek
quantum mechanical approach, fully ignoring the B-O concept, one concludes that
there is no isomerism, no ferromagnetism, no J-T effect, and no superconductivity.
Ferromagnets, in calculations similar to those of Cafiero and Adamowitz, would look
like giant molecules with full spherical symmetry and therefore without any SSB.
J-T molecules would look like any other molecules with a non-degenerate electronic
spectrum and superconductors looking like insulators. Only after the introduction
of the B-O approximation, here a “dirty trick”, the isomerism appears, and we can
construct the simplest Heisenberg model of ferromagnetism. But this simplification
makes only sense for adiabatic systems. However, in cases such as J-T molecules
and superconductors, it yields ontologically incorrect results and here field methods
based on the full set of Goldstone bosons must be applied.
In agreement with Weinberg’s statement that every quantum system can be
described by two profoundly different descriptions, either as a mechanical system
or as a field system, and that only the second one recognizes SSB, one may ask if
these two descriptions are still equivalently valid or whether only one of them is
fundamentally correct. There is also the temptation to believe that the field approach
supersedes the mechanical one, i.e. that the quantum field description should be
seen as a generalization of quantum mechanics in the same way as general relativity becomes a generalization of special relativity. But listen to what Einstein said
about this topic [19]: “Newton’s theory deserves the name of a classical theory. It
has nevertheless been abandoned since Maxwell and Hertz have shown that the idea
of forces at a distance has to be relinquished and that one cannot manage without
the idea of continuous “fields.” The opinion that continuous fields are to be viewed
as the only acceptable basic concepts, which must also be assumed to underlie the
theory of the material particles, soon won out. Now this conception became, so to
speak, “classical”; but a proper, and in principle complete, theory has not grown out
of it. Maxwell’s theory of the electric field remained a torso, because it was unable to
set up laws for the behaviour of electric density, without which there can, of course,
be no such thing as an electro-magnetic field. Analogously the general theory of relativity furnished then a field theory of gravitation, but no theory of the field-creating
masses.”
Einstein’s quote above indicates that we must accept both the mechanical- and
the field description as equally valid. However, the mechanical description pictures
every system as isolated with no SSB, while the field formulation allows SSB at the
same time being responsible for the individual character of the system. The question becomes whether there is any mathematical transformation between them, as
inquired by Sutcliffe and Woolley. This reminds on the first stage of the development
of quantum physics, when it turned out that elementary entities had both mechanical
properties (particles) and field properties (waves). Even if Bohr had introduced the
concept of complementarity into physics, some physicists developed a negative attitude towards it and attempted to find direct transformations between particles and
waves without the need for any complementarity, such as Einstein or de Broglie. In
analogy with just how a century ago the different aspects of the mechanical and the
field descriptions of elementary entities led to Bohr’s concept of complementarity,
we are now in a similar situation with respect to the descriptions of the many-body
M. Svrˇ cek
quantum mechanical approach, fully ignoring the B-O concept, one concludes that
there is no isomerism, no ferromagnetism, no J-T effect, and no superconductivity.
Ferromagnets, in calculations similar to those of Cafiero and Adamowitz, would look
like giant molecules with full spherical symmetry and therefore without any SSB.
J-T molecules would look like any other molecules with a non-degenerate electronic
spectrum and superconductors looking like insulators. Only after the introduction
of the B-O approximation, here a “dirty trick”, the isomerism appears, and we can
construct the simplest Heisenberg model of ferromagnetism. But this simplification
makes only sense for adiabatic systems. However, in cases such as J-T molecules
and superconductors, it yields ontologically incorrect results and here field methods
based on the full set of Goldstone bosons must be applied.
In agreement with Weinberg’s statement that every quantum system can be
described by two profoundly different descriptions, either as a mechanical system
or as a field system, and that only the second one recognizes SSB, one may ask if
these two descriptions are still equivalently valid or whether only one of them is
fundamentally correct. There is also the temptation to believe that the field approach
supersedes the mechanical one, i.e. that the quantum field description should be
seen as a generalization of quantum mechanics in the same way as general relativity becomes a generalization of special relativity. But listen to what Einstein said
about this topic [19]: “Newton’s theory deserves the name of a classical theory. It
has nevertheless been abandoned since Maxwell and Hertz have shown that the idea
of forces at a distance has to be relinquished and that one cannot manage without
the idea of continuous “fields.” The opinion that continuous fields are to be viewed
as the only acceptable basic concepts, which must also be assumed to underlie the
theory of the material particles, soon won out. Now this conception became, so to
speak, “classical”; but a proper, and in principle complete, theory has not grown out
of it. Maxwell’s theory of the electric field remained a torso, because it was unable to
set up laws for the behaviour of electric density, without which there can, of course,
be no such thing as an electro-magnetic field. Analogously the general theory of relativity furnished then a field theory of gravitation, but no theory of the field-creating
masses.”
Einstein’s quote above indicates that we must accept both the mechanical- and
the field description as equally valid. However, the mechanical description pictures
every system as isolated with no SSB, while the field formulation allows SSB at the
same time being responsible for the individual character of the system. The question becomes whether there is any mathematical transformation between them, as
inquired by Sutcliffe and Woolley. This reminds on the first stage of the development
of quantum physics, when it turned out that elementary entities had both mechanical
properties (particles) and field properties (waves). Even if Bohr had introduced the
concept of complementarity into physics, some physicists developed a negative attitude towards it and attempted to find direct transformations between particles and
waves without the need for any complementarity, such as Einstein or de Broglie. In
analogy with just how a century ago the different aspects of the mechanical and the
field descriptions of elementary entities led to Bohr’s concept of complementarity,
we are now in a similar situation with respect to the descriptions of the many-body