Megascopic Quantum Phenomena
323
Starting the examination of the Higgs mechanism with Anderson’s original paper
[95], we will pay special attention to the last sentence in the following quote: “I
should like to close with one final remark on the Goldstone theorem. This theorem was
initially conjectured, one presumes, because of the solid-state analogues, via the work
of Nambu and of Anderson. The theorem states, essentially, that if the Lagrangian
possesses a continuous symmetry group under which the ground or vacuum state
is not invariant, that state is, therefore, degenerate with other ground states. This
implies a zero-mass boson. Thus, the solid crystal violates translational and rotational
invariance, and possesses phonons; …”
One may ask whether Anderson, in view of what has been said here, really appreciated the content of the Goldstone theorem? Quasiparticles, representing collective
oscillations in condensed matter were known long before the Goldstone theorem
was formulated. When it was proved, it was sometimes easy to assign them to
Goldstone bosons and corresponding broken symmetries, and sometimes not. For
instance, magnons in ferromagnets are Goldstone bosons descending from broken
rotational symmetry, or longitudinal phonons in liquids are Goldstone bosons of
the broken Galilean symmetry. But what about solids? Anderson claims, that “the
solid crystal violates translational and rotational invariance, and possesses phonons”.
However, not so, phonons are the consequence of a broken Galilean symmetry, and
further, which Goldstone bosons correspond to the broken translational and rotational symmetry in crystals? Unfortunately one can find no answers to this question
in Anderson’s papers.
One might try to find an answer to the question: what are the Goldstone bosons
in solids in Wikipedia. A recent reply reads: “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 simple one-to-one correspondence between the
Goldstone modes and the broken symmetries.”
Although Wikipedia occasionally must be taken with a grain of salt, the sentence
“No simple one-to-one correspondence between the Goldstone modes and broken
symmetries” is surprising! Since the Goldstone theorem is a complement of Noether’s
first theorem, the claim is equivalent to saying that the conservation laws for linear
and angular momentum may not be in a simple one-to-one correspondence with
translational and rotational symmetry. Consequently a unique and unambiguous oneto-one correspondence between the Goldstone modes and spontaneously broken
symmetries in solids must necessarily exist! The conclusion must be that phonons
in solids do not represent the full set of Goldstone bosons, and some bosons are still
missing. It is better if we in the future would add this issue to the list of unsolved
problems of quantum physics rather than sweeping it under the carpet.
Principally we should concentrate our energies on hunting for the lost Goldstone
bosons in solids. Supposing that a solid is composed of N nuclei, the whole system has 3N degrees of freedom. The quantum mechanical treatment eliminates 6
degrees associated with the centre of mass, and the remaining 3N − 6 degrees after
the Galilean symmetry breaking represent the vibrational modes, the phonons. At
the first sight it seems that the violation of translational and rotational invariance has
323
Starting the examination of the Higgs mechanism with Anderson’s original paper
[95], we will pay special attention to the last sentence in the following quote: “I
should like to close with one final remark on the Goldstone theorem. This theorem was
initially conjectured, one presumes, because of the solid-state analogues, via the work
of Nambu and of Anderson. The theorem states, essentially, that if the Lagrangian
possesses a continuous symmetry group under which the ground or vacuum state
is not invariant, that state is, therefore, degenerate with other ground states. This
implies a zero-mass boson. Thus, the solid crystal violates translational and rotational
invariance, and possesses phonons; …”
One may ask whether Anderson, in view of what has been said here, really appreciated the content of the Goldstone theorem? Quasiparticles, representing collective
oscillations in condensed matter were known long before the Goldstone theorem
was formulated. When it was proved, it was sometimes easy to assign them to
Goldstone bosons and corresponding broken symmetries, and sometimes not. For
instance, magnons in ferromagnets are Goldstone bosons descending from broken
rotational symmetry, or longitudinal phonons in liquids are Goldstone bosons of
the broken Galilean symmetry. But what about solids? Anderson claims, that “the
solid crystal violates translational and rotational invariance, and possesses phonons”.
However, not so, phonons are the consequence of a broken Galilean symmetry, and
further, which Goldstone bosons correspond to the broken translational and rotational symmetry in crystals? Unfortunately one can find no answers to this question
in Anderson’s papers.
One might try to find an answer to the question: what are the Goldstone bosons
in solids in Wikipedia. A recent reply reads: “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 simple one-to-one correspondence between the
Goldstone modes and the broken symmetries.”
Although Wikipedia occasionally must be taken with a grain of salt, the sentence
“No simple one-to-one correspondence between the Goldstone modes and broken
symmetries” is surprising! Since the Goldstone theorem is a complement of Noether’s
first theorem, the claim is equivalent to saying that the conservation laws for linear
and angular momentum may not be in a simple one-to-one correspondence with
translational and rotational symmetry. Consequently a unique and unambiguous oneto-one correspondence between the Goldstone modes and spontaneously broken
symmetries in solids must necessarily exist! The conclusion must be that phonons
in solids do not represent the full set of Goldstone bosons, and some bosons are still
missing. It is better if we in the future would add this issue to the list of unsolved
problems of quantum physics rather than sweeping it under the carpet.
Principally we should concentrate our energies on hunting for the lost Goldstone
bosons in solids. Supposing that a solid is composed of N nuclei, the whole system has 3N degrees of freedom. The quantum mechanical treatment eliminates 6
degrees associated with the centre of mass, and the remaining 3N − 6 degrees after
the Galilean symmetry breaking represent the vibrational modes, the phonons. At
the first sight it seems that the violation of translational and rotational invariance has