Megascopic Quantum Phenomena
331
translons, follows from Eq. (11.27). However, the complete lack of interest from
the scientific community in this topic has indeed been astonishing, but the reason
may be quite banal. Whenever I speak with solid state physicists, they all admit that
they have always thought that the adiabatic approximation is synonymous with the
B-O approximation and that they see no difference between them, but it is in actual
fact the almost negligible difference between them that generates the 5 or 6 lost
Goldstone bosons! As a consequence the original concept of the electron-vibrational
field theory [103] must be abandoned and replaced by a new field theory with two
major upgrades:
The first upgrade: Eqs. (11.9)–(11.10) for the potential and the kinetic vibrational
energies, must be rewritten using a complete set of 3N Goldstone bosons—phonons,
rotons and translons—in agreement with the Born-Huang field transcription (11.28).
We shall call these bosons hyperphonons or hypervibrations.
E pot =
1
4
r ∈V
ω r B
+
r B r
(11.29)
E kin =
1
2
1
2
r ∈V
ω r +
r ∈R
ρ r +
r ∈T
τ r
B
+
r
B r
(11.30)
Defining the hyper-vibrational double-vector:
ω =
ω r
˜
ω r
=
ω r 0 0
ω r
2
ρ r
2
τ r
(11.31)
we get covariant expressions for the boson part of the hyper-vibrational Hamiltonian
with respect to all 3N Goldstone modes:
H B =
1
4
r
ω r B
+
r B r + ˜
ω r B
+
r
B r
(11.32)
Instead of the original total Hamiltonian (11.12) we obtain
H = E N N (B) − E
(2)
N N (B) − V
(2)
N (B) − W
(2)
N ( B) +
P Q
h P Q (B) a
+
P a Q
+
1
2
P Q RS
v
0
P Q RS a
+
P a
+
Q a S a R +
1
4
r
ω r B
+
r B r + ˜
ω r B
+
r
B r
(11.33)
The second upgrade: We will apply the same quasiparticle transformations
(11.13)–(11.16) as in the original electron-vibrational concept, but with entirely
different interpretation. In full analogy with Lorentz covariance, binding together
space and time coordinates, we introduce a new covariance, binding together the
internal and external degrees of freedom, i.e. Eqs. (11.13)–(11.16) will comprise all
331
translons, follows from Eq. (11.27). However, the complete lack of interest from
the scientific community in this topic has indeed been astonishing, but the reason
may be quite banal. Whenever I speak with solid state physicists, they all admit that
they have always thought that the adiabatic approximation is synonymous with the
B-O approximation and that they see no difference between them, but it is in actual
fact the almost negligible difference between them that generates the 5 or 6 lost
Goldstone bosons! As a consequence the original concept of the electron-vibrational
field theory [103] must be abandoned and replaced by a new field theory with two
major upgrades:
The first upgrade: Eqs. (11.9)–(11.10) for the potential and the kinetic vibrational
energies, must be rewritten using a complete set of 3N Goldstone bosons—phonons,
rotons and translons—in agreement with the Born-Huang field transcription (11.28).
We shall call these bosons hyperphonons or hypervibrations.
E pot =
1
4
r ∈V
ω r B
+
r B r
(11.29)
E kin =
1
2
1
2
r ∈V
ω r +
r ∈R
ρ r +
r ∈T
τ r
B
+
r
B r
(11.30)
Defining the hyper-vibrational double-vector:
ω =
ω r
˜
ω r
=
ω r 0 0
ω r
2
ρ r
2
τ r
(11.31)
we get covariant expressions for the boson part of the hyper-vibrational Hamiltonian
with respect to all 3N Goldstone modes:
H B =
1
4
r
ω r B
+
r B r + ˜
ω r B
+
r
B r
(11.32)
Instead of the original total Hamiltonian (11.12) we obtain
H = E N N (B) − E
(2)
N N (B) − V
(2)
N (B) − W
(2)
N ( B) +
P Q
h P Q (B) a
+
P a Q
+
1
2
P Q RS
v
0
P Q RS a
+
P a
+
Q a S a R +
1
4
r
ω r B
+
r B r + ˜
ω r B
+
r
B r
(11.33)
The second upgrade: We will apply the same quasiparticle transformations
(11.13)–(11.16) as in the original electron-vibrational concept, but with entirely
different interpretation. In full analogy with Lorentz covariance, binding together
space and time coordinates, we introduce a new covariance, binding together the
internal and external degrees of freedom, i.e. Eqs. (11.13)–(11.16) will comprise all