2 A Comment on the Question of Degeneracies in Quantum Mechanics
47
The following application, in accord with (2.15), of the two quasiparticle transformations, the first Q-dependent
a P =
Q
c P Q (B)a Q ,
b r = b r +
P Q
d rP Q (B)a
+
P a Q
(2.16)
with the unitary conditions
R
c P R (B)c
+
QR (B) = δ P Q ,
d rP Q =
R
c
+
RP (B)
b r , c RQ (B)
(2.17)
and the second P -dependent
a P =
Q
˜
c P Q ( ˜
B)a Q ,
b r = b r +
P Q
˜
d rP Q ( ˜
B)a
+
P a Q
(2.18)
with the unitary conditions
R
˜
c P R ( ˜
B) ˜
c
+
QR ( ˜
B) = δ P Q ,
˜
d rP Q =
R
˜
c
+
RP ( ˜
B)
b r , ˜
c RQ ( ˜
B)
(2.19)
will lead to new systems of fermions and bosons. The diagonalization procedures
permit choosing an optimal system, where we achieve a realistic separation into
individual (quasi) fermions and bosons with minimal interaction between them.
Looking at this problem from the standpoint of group theory, we realize that
we must adhere to the Poincaré group, as one of the most general group reflecting
the full symmetry of special relativity, a problem seldom treated in full generality.
Asking the question what would the most general group be that reflects the full
symmetry of the Fröhlich transformation, or in other words, what would be the
analogy of the Poincaré group for transformations carried out in the field theoretic
methods of quantum mechanics. It can be shown that the Fröhlich transformation
in Eq. (2.15) is decomposable into a product of two quasiparticle transformations:
the coordinate (adiabatic) and the momentum (non-adiabatic) ones. We can perform
the generalization to the case without the implied translational symmetry in a very
simple way by replacing the quasimomentum/spin notation, which Fröhlich used
in his original work, by the spinorbital notation. A further simple generalization
can also be attempted, given that the quasiparticle transformations remain valid,
by replacing the vibrational modes r, r ∈ V by the hypervibrational modes r, r ∈
{V , R, T }.
We can now show that (2.16), (2.18) form a group. First we write their inverse
transformations:
a P =
Q
c P Q (B)a Q ,
b r = b r +
P Q
d rP Q (B)a
+
P a Q
(2.20)
a P =
Q
˜
c P Q ( ˜
B)a Q ,
b r = b r +
P Q
˜
d rP Q ( ˜
B)a
+
P a Q .
(2.21)
47
The following application, in accord with (2.15), of the two quasiparticle transformations, the first Q-dependent
a P =
Q
c P Q (B)a Q ,
b r = b r +
P Q
d rP Q (B)a
+
P a Q
(2.16)
with the unitary conditions
R
c P R (B)c
+
QR (B) = δ P Q ,
d rP Q =
R
c
+
RP (B)
b r , c RQ (B)
(2.17)
and the second P -dependent
a P =
Q
˜
c P Q ( ˜
B)a Q ,
b r = b r +
P Q
˜
d rP Q ( ˜
B)a
+
P a Q
(2.18)
with the unitary conditions
R
˜
c P R ( ˜
B) ˜
c
+
QR ( ˜
B) = δ P Q ,
˜
d rP Q =
R
˜
c
+
RP ( ˜
B)
b r , ˜
c RQ ( ˜
B)
(2.19)
will lead to new systems of fermions and bosons. The diagonalization procedures
permit choosing an optimal system, where we achieve a realistic separation into
individual (quasi) fermions and bosons with minimal interaction between them.
Looking at this problem from the standpoint of group theory, we realize that
we must adhere to the Poincaré group, as one of the most general group reflecting
the full symmetry of special relativity, a problem seldom treated in full generality.
Asking the question what would the most general group be that reflects the full
symmetry of the Fröhlich transformation, or in other words, what would be the
analogy of the Poincaré group for transformations carried out in the field theoretic
methods of quantum mechanics. It can be shown that the Fröhlich transformation
in Eq. (2.15) is decomposable into a product of two quasiparticle transformations:
the coordinate (adiabatic) and the momentum (non-adiabatic) ones. We can perform
the generalization to the case without the implied translational symmetry in a very
simple way by replacing the quasimomentum/spin notation, which Fröhlich used
in his original work, by the spinorbital notation. A further simple generalization
can also be attempted, given that the quasiparticle transformations remain valid,
by replacing the vibrational modes r, r ∈ V by the hypervibrational modes r, r ∈
{V , R, T }.
We can now show that (2.16), (2.18) form a group. First we write their inverse
transformations:
a P =
Q
c P Q (B)a Q ,
b r = b r +
P Q
d rP Q (B)a
+
P a Q
(2.20)
a P =
Q
˜
c P Q ( ˜
B)a Q ,
b r = b r +
P Q
˜
d rP Q ( ˜
B)a
+
P a Q .
(2.21)