346
M. Svrˇ cek
the infinite volume limit and therefore separated by a “superselection rule” (see for
example Weinberg [111], pp. 164–165).”
Though this citation reflects Weinberg’s opinion, one can also find other interpretations, emphasizing the difference between the mechanical and the field descriptions,
rather than the condition of the volume infinity of fields, as e.g. in the paper of van
Dam [128]: “For the SSB in the rod and in the ferromagnet two comments were made
that apply equally to this case: each of the ground states has an equal chance to be the
ground state of the physical system, and the ground states are related to each other by
the U(1) symmetry of the Lagrangian. Relating this situation to quantum mechanics
might provoke suspicion: through tunnelling the real ground state could surely be a
superposition of the individual ground states, which would not be degenerate. But
although such a situation could occur in ordinary quantum mechanics, it does not
apply to quantum field theory. The fields live in an infinite volume and thus have
infinitely many degrees of freedom. Tunnelling cannot happen in this case. All vacuum states are orthogonal. This is extensively discussed by, for example, Weinberg
[111, pp. 163–167].”
The results presented in the previous section, testify in favour of van Dam’s
interpretation: there is no SSB in quantum mechanics, it can solely be described in
quantum field theory. Moreover, one might not need to rely on Weinberg’s proof,
which is valid only for infinite-volume fields, since the Goldstone bosons, associated with broken translational and rotational symmetries, earlier in this work called
translons and rotons [107], are responsible for singularities at symmetric points.
Hence the manifestation of SSB, in order to avoid these singularities, does appear
for any system, regardless of the system being finite or infinite.
As a matter of fact, if we have two legitimate physical descriptions, the quantum mechanical and the quantum field one, these two formulations must necessarily
yield the same energies for the system under examination. Therefore the classical
SSB archetype, here denoted as “falling down from the apex”, is not workable for
quantum systems. Quantum physics can only calculate the symmetry broken states,
while the SSB transition must somehow follow from the transitions between the
mechanical and the field states of the system proceeding without any energy gain or
loss, since symmetrical mechanical points and symmetry broken field points represent two descriptions with the same energy. A new law of nature is indispensable
for this purpose, since this type of transitions has no support within the Copenhagen
interpretation. As a consequence, from this new law, the origin of asymmetry must
emerge, and evidently, according to the second form of the van Fraassen’s argument,
this asymmetry cannot arise ex nihilo.
So where should one start searching for this new law of nature? One way would
be to focus on the simplest cases of SSB. We will, however, not start with complex
nonadiabatic phenomena such as superconductivity or the J-T effect, nor with the
adiabatic, but infinite systems like ferromagnets. The simplest example of SSB for
our purpose would be small adiabatic systems such as the formation of isomers. The
exact quantum mechanical solution, according to the Monkhorst-Cafiero-Adamowitz
approach [6, 9], cannot lead to isomerism. The latter appears only after the introduction of the B-O approximation, or the clamped-nuclei concept. It seems like this
M. Svrˇ cek
the infinite volume limit and therefore separated by a “superselection rule” (see for
example Weinberg [111], pp. 164–165).”
Though this citation reflects Weinberg’s opinion, one can also find other interpretations, emphasizing the difference between the mechanical and the field descriptions,
rather than the condition of the volume infinity of fields, as e.g. in the paper of van
Dam [128]: “For the SSB in the rod and in the ferromagnet two comments were made
that apply equally to this case: each of the ground states has an equal chance to be the
ground state of the physical system, and the ground states are related to each other by
the U(1) symmetry of the Lagrangian. Relating this situation to quantum mechanics
might provoke suspicion: through tunnelling the real ground state could surely be a
superposition of the individual ground states, which would not be degenerate. But
although such a situation could occur in ordinary quantum mechanics, it does not
apply to quantum field theory. The fields live in an infinite volume and thus have
infinitely many degrees of freedom. Tunnelling cannot happen in this case. All vacuum states are orthogonal. This is extensively discussed by, for example, Weinberg
[111, pp. 163–167].”
The results presented in the previous section, testify in favour of van Dam’s
interpretation: there is no SSB in quantum mechanics, it can solely be described in
quantum field theory. Moreover, one might not need to rely on Weinberg’s proof,
which is valid only for infinite-volume fields, since the Goldstone bosons, associated with broken translational and rotational symmetries, earlier in this work called
translons and rotons [107], are responsible for singularities at symmetric points.
Hence the manifestation of SSB, in order to avoid these singularities, does appear
for any system, regardless of the system being finite or infinite.
As a matter of fact, if we have two legitimate physical descriptions, the quantum mechanical and the quantum field one, these two formulations must necessarily
yield the same energies for the system under examination. Therefore the classical
SSB archetype, here denoted as “falling down from the apex”, is not workable for
quantum systems. Quantum physics can only calculate the symmetry broken states,
while the SSB transition must somehow follow from the transitions between the
mechanical and the field states of the system proceeding without any energy gain or
loss, since symmetrical mechanical points and symmetry broken field points represent two descriptions with the same energy. A new law of nature is indispensable
for this purpose, since this type of transitions has no support within the Copenhagen
interpretation. As a consequence, from this new law, the origin of asymmetry must
emerge, and evidently, according to the second form of the van Fraassen’s argument,
this asymmetry cannot arise ex nihilo.
So where should one start searching for this new law of nature? One way would
be to focus on the simplest cases of SSB. We will, however, not start with complex
nonadiabatic phenomena such as superconductivity or the J-T effect, nor with the
adiabatic, but infinite systems like ferromagnets. The simplest example of SSB for
our purpose would be small adiabatic systems such as the formation of isomers. The
exact quantum mechanical solution, according to the Monkhorst-Cafiero-Adamowitz
approach [6, 9], cannot lead to isomerism. The latter appears only after the introduction of the B-O approximation, or the clamped-nuclei concept. It seems like this