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Therefore, within the range of small wave vectors (continuum approximation)
dispersion law (7) fulfils the relationship
ω
2
l
=
ω
2
0
−
c
2
k
2
.
(6)
Group velocity of the waves corresponding to the dispersion law (8) can
be written as follows:
dω
dk
=
−
c
2
k
ω
=
−
c
2
k
p
ω
2
0
−
c
2
k
2
.
(7)
According to the relationship (9), in this case phase and group velocities
have mutually opposite directions. Effective mass
m
and rest mass
m
0
of
the longitudinal photons according to expressions (8) and (9) are negative:
m
=
~
d
2
ω
dk
2
=
−
~
(
ω
2
0
−
c
2
k
2
)
3
2
c
2
ω
2
;
m
0
=
m
(0) =
−
~
ω
0
c
2
.
Along with translational degrees of freedom, clusters of the low-temperature
phase of vacuum have respiratory and axion degrees of freedom. Therefore,
two additional dispersion branches corresponding to paraphotons and
axions exist in vacuum besides the branches of the vector photons (see
Fig. 6). The dispersion laws for paraphotons and axions have relativistic
form:
ω
2
=
ω
2
par
+
c
2
0
k
2
;
ω
2
=
ω
2
ax
+
c
2
0
k
2
.
Photon-boson conversion. Generation of paraphotons, axions,
and longitudinal photons in the laboratory.
Quantum description of
the photon-boson conversion processes in the material media and in
physical vacuum is performed on the basis of introduction of Hamiltonian
anaharmonical terms
H
=
H
01
+
H
02
+
V
;
V
=
f
1
[
a
(
a
0
)
+
b
+
+
a b
(
a
0
)
+
] +
f
2
[
a
(
a
0
)
b
+
+
b a
+
(
a
0
)
+
]
.
Here
Н
01
and
Н
02
are Hamiltonian functions of photon and boson fields in
harmonic approximation;
V
is an excitation operator;
a
,
a
0
,
b
,
b
,
a
+
,
(
a
0
)
+
,
b
,
b
+
are operators of annihilation and genesis of photon and boson fields.
Probability of the respective processes of photon-boson conversion is
given by the expressions
W
fi
=
2
π
~
f
2
1
[
n
0
(
n
0
+ 1)(
m
b
+ 1) +
n
0
m
b
(
n
0
+ 1)] ;
W
fi
=
2
π
~
f
2
2
[
n
2
0
(
m
b
+ 1) +
m
b
(
n
0
+ 1)(
n
0
+ 1)]
.
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2015. No. 1
43