324
M. Svrˇ cek
no counterpart in the form of the existence of the corresponding Goldstone bosons.
One may ask whether this is really true. Does the quantum field share the centre of
mass separation with quantum mechanics considering the same system? The answer
is no. We will provide a proof in the next section and show: first, qualitatively the
appearance of the 6 lost Goldstone bosons, and second, quantitatively their contributions to the quantum field equations together with the experimental evidence. It
means that every solid composed of N nuclei has precisely 3N spontaneously broken
symmetries with the exact correspondence to the emerging 3N Goldstone bosons.
This rule holds universally, regardless of the size of N, i.e. if it is small or great, if it
represents finite molecule or infinite crystal, and irrespective of the solid character,
i.e. if it is insulator, conductor, semiconductor, or superconductor.
Returning to Eq. (10.10) where the field ξ (x) relates to the degrees of freedom
connected with the angular displacement in the Mexican hat. The field corresponding
to the Lagrangian density in Eq. (10.11) results in a massless scalar particle that was
identified with the Goldstone boson, or is it really so? As just said above, every solid
produces exactly 3N Goldstone bosons. Thus the massless scalar particle descending
from the field ξ (x) cannot really be a Goldstone boson! It rather looks like the
following. Instead of loosing six authentic Goldstone bosons in the solid suddenly
one imposter Goldstone boson pops up in the superconductor. Thus we end up with
the illusive Eq. (10.11) without the chance to discuss the rest of the Higgs equations
which became misleading as well. This should be the end of Anderson’s textual
description about the gauge bosons eating the Goldstone bosons, on the basis of
which the anticipated Higgs mechanism was created.
We will close this section with a philosophical consideration of the Higgs mechanism. One of the best reflections I have found was written by Earman [96]: “Higgs
further showed that the gauge could be chosen so that the Goldstone bosons are
suppressed and that in this “unitary gauge” the new field had acquired a mass. As
the semi-popular presentations put it, “Particles get their masses by eating the Higgs
field.” Readers of Scientific American can be satisfied with these just-so stories.
But philosophers of science should not be. For a genuine property like mass cannot be gained by eating descriptive fluff, which is just what gauge is. Philosophers
of science should be asking the Nozick question: What is the objective (i.e. gauge
invariant) structure of the world corresponding to the gauge theory presented in the
Higgs mechanism? …consider the following three-tiered dilemma. First tier: Either
the gauge invariant content of the Higgs mechanism is described by local quantum
fields satisfying the standard assumptions of Poincaré invariance, local commutativity, spectrum condition, etc. or not. If not, then the implementation of the Higgs
mechanism requires a major overhaul of conventional QFT. If so, go to the second
tier. Second tier: Either the gauge invariant system admits a finite dimensional Lie
group as an internal symmetry group or not. If so, Noether’s first theorem applies
again. But (since the other standard assumptions are in place) Goldstone’s theorem
also applies and, hence, Goldstone bosons have not been suppressed after all. If not,
go to the third tier. Third tier: Either the gauge invariant system admits no non-trivial
symmetries at all or else it admits only discrete symmetries. In either case Goldstone
bosons are quashed. In the former case spontaneous symmetry breaking is not an
M. Svrˇ cek
no counterpart in the form of the existence of the corresponding Goldstone bosons.
One may ask whether this is really true. Does the quantum field share the centre of
mass separation with quantum mechanics considering the same system? The answer
is no. We will provide a proof in the next section and show: first, qualitatively the
appearance of the 6 lost Goldstone bosons, and second, quantitatively their contributions to the quantum field equations together with the experimental evidence. It
means that every solid composed of N nuclei has precisely 3N spontaneously broken
symmetries with the exact correspondence to the emerging 3N Goldstone bosons.
This rule holds universally, regardless of the size of N, i.e. if it is small or great, if it
represents finite molecule or infinite crystal, and irrespective of the solid character,
i.e. if it is insulator, conductor, semiconductor, or superconductor.
Returning to Eq. (10.10) where the field ξ (x) relates to the degrees of freedom
connected with the angular displacement in the Mexican hat. The field corresponding
to the Lagrangian density in Eq. (10.11) results in a massless scalar particle that was
identified with the Goldstone boson, or is it really so? As just said above, every solid
produces exactly 3N Goldstone bosons. Thus the massless scalar particle descending
from the field ξ (x) cannot really be a Goldstone boson! It rather looks like the
following. Instead of loosing six authentic Goldstone bosons in the solid suddenly
one imposter Goldstone boson pops up in the superconductor. Thus we end up with
the illusive Eq. (10.11) without the chance to discuss the rest of the Higgs equations
which became misleading as well. This should be the end of Anderson’s textual
description about the gauge bosons eating the Goldstone bosons, on the basis of
which the anticipated Higgs mechanism was created.
We will close this section with a philosophical consideration of the Higgs mechanism. One of the best reflections I have found was written by Earman [96]: “Higgs
further showed that the gauge could be chosen so that the Goldstone bosons are
suppressed and that in this “unitary gauge” the new field had acquired a mass. As
the semi-popular presentations put it, “Particles get their masses by eating the Higgs
field.” Readers of Scientific American can be satisfied with these just-so stories.
But philosophers of science should not be. For a genuine property like mass cannot be gained by eating descriptive fluff, which is just what gauge is. Philosophers
of science should be asking the Nozick question: What is the objective (i.e. gauge
invariant) structure of the world corresponding to the gauge theory presented in the
Higgs mechanism? …consider the following three-tiered dilemma. First tier: Either
the gauge invariant content of the Higgs mechanism is described by local quantum
fields satisfying the standard assumptions of Poincaré invariance, local commutativity, spectrum condition, etc. or not. If not, then the implementation of the Higgs
mechanism requires a major overhaul of conventional QFT. If so, go to the second
tier. Second tier: Either the gauge invariant system admits a finite dimensional Lie
group as an internal symmetry group or not. If so, Noether’s first theorem applies
again. But (since the other standard assumptions are in place) Goldstone’s theorem
also applies and, hence, Goldstone bosons have not been suppressed after all. If not,
go to the third tier. Third tier: Either the gauge invariant system admits no non-trivial
symmetries at all or else it admits only discrete symmetries. In either case Goldstone
bosons are quashed. In the former case spontaneous symmetry breaking is not an