Chapter 28
Particles and Tachyons
In Chap. 20, we examined solution (261), q
T
(x, t), of the sine-Gordon equation,
q
T
(x, t) =
2a
π
arctan exp
−π(x − u t)
L o κ
+
a
2
(261)
with
κ =
u
c o
1 −
v 2
c 2
o
= sign u
u 2
c 2
o
− 1 , u =
c
2
o
v
, |u| > c o .
(258)
Here, with this solution, an energy density also moves through the crystal; however,
here it moves with a velocity faster than the speed of sound, to be precise |u| > c o .
Solutions of this type are therefore defined by Eilenberger [11], as tachyons, and
we accepted in Chap. 20 that tachyons are indeed particles or quasi-particles in the
crystal, respectively. Firstly, q
T
(x, t) is a field, just as much so as the kink solution
(111), q
I
(x, t). Such a field creates an energy–momentum tensor (279). Only in
certain cases is it possible to attribute this tensor the definite physical parameters
of a particle, here a total energy E and a total momentum P. In Chap. 23, we
saw how we could understand the kink q
I
(x, t), using this method as a relativistic
particle, a quasi-particle in a crystal. But what of the tachyon? Is the tachyon’s
field (261) mechanically a particle, so that arbitrary collisions between particles and
tachyons must underlie the mechanical laws of energy and momentum conservation?
Mathematically, this leads to the following question: Can we attribute the tachyon’s
field (261) in any arbitrary reference system an energy E
T and a momentum P
T
in such a way that these quantities are linked together in different reference systems
by the Eq. (298).
How does the mass m
T of a tachyon and its velocity change when the tachyon
changes reference systems? There is no restmass here. The mass of a tachyon can
only be determined using the equation E
T
= m
T c
2
o . And even the term velocity
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_28
299
Particles and Tachyons
In Chap. 20, we examined solution (261), q
T
(x, t), of the sine-Gordon equation,
q
T
(x, t) =
2a
π
arctan exp
−π(x − u t)
L o κ
+
a
2
(261)
with
κ =
u
c o
1 −
v 2
c 2
o
= sign u
u 2
c 2
o
− 1 , u =
c
2
o
v
, |u| > c o .
(258)
Here, with this solution, an energy density also moves through the crystal; however,
here it moves with a velocity faster than the speed of sound, to be precise |u| > c o .
Solutions of this type are therefore defined by Eilenberger [11], as tachyons, and
we accepted in Chap. 20 that tachyons are indeed particles or quasi-particles in the
crystal, respectively. Firstly, q
T
(x, t) is a field, just as much so as the kink solution
(111), q
I
(x, t). Such a field creates an energy–momentum tensor (279). Only in
certain cases is it possible to attribute this tensor the definite physical parameters
of a particle, here a total energy E and a total momentum P. In Chap. 23, we
saw how we could understand the kink q
I
(x, t), using this method as a relativistic
particle, a quasi-particle in a crystal. But what of the tachyon? Is the tachyon’s
field (261) mechanically a particle, so that arbitrary collisions between particles and
tachyons must underlie the mechanical laws of energy and momentum conservation?
Mathematically, this leads to the following question: Can we attribute the tachyon’s
field (261) in any arbitrary reference system an energy E
T and a momentum P
T
in such a way that these quantities are linked together in different reference systems
by the Eq. (298).
How does the mass m
T of a tachyon and its velocity change when the tachyon
changes reference systems? There is no restmass here. The mass of a tachyon can
only be determined using the equation E
T
= m
T c
2
o . And even the term velocity
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_28
299