30 On the Causality Problem: Particle–Tachyon Collisions
327
tachyon that was emitted could also be absorbed. So, the deadly shot using a tachyon
and the end of causality a reality?
Well, in fact, the emission of a particle by a tachyon can be compatible to the
conservation laws. If we just remove the particle before the collision in Eq. (460),
we receive
μ sinh β
= cosh α − cosh α
,
μ cosh β
= sinh α − sinh α
.
No particle
before the collision
(470)
From 0 ≤ μ cosh β
= sinh α − sinh α
and the monotony of the hyperbolic sine
follows α
≤ α. Using cosh
2
β
− sinh
2
β
= 1 we eliminate β
by squaring and
subtraction of both Eq. (470) with the result
α
= α − arcosh
1 +
μ
2
2
.
(471)
According to the above, an arbitrary arriving tachyon, in other words α and m
T
∗
are arbitrarily given, can emit a particle with an arbitrary restmass m o = μ m
T
∗ . The
tachyon velocity u
after emission can be calculated from (471) keeping c o /u
=
v
/c o = tanh α
in mind, and the velocity w
of the emitted particle can be calculated
from (470) and (471) keeping w
/c o = tanh β
in mind.
All of these processes are formally possible according to the conservation laws
of energy and momentum. Do they however actually exist? Physically speaking, do
these processes exist as solutions of the fundamental equations of physics known
to us? If so, then we could only save our causal view of the world by referring
back to Treder’s [94] proposition of banning these processes into sub-microscopic
regions, or by simply formulating an additional condition stating that these solutions
are excluded—which would of course only be a stopgap used to bridge the holes in
the theory. This is the general situation in which we cannot discuss these problems
without actually coming to a consensus about which elementary particle theory is
the best to use in modern physics. Let us remain with that what we can elementarily
comprehend, the sine-Gordon equation with its particle and tachyon solutions inside
of a crystal.
The preceding considerations do not lead us to the conclusion that the particle
q
I
(x, t), the solution (111) of the sine-Gordon equation can in fact emit a tachyon
q
T
(x, t), the solution (261) of the sine-Gordon equation. Absolutely not! Due to the
non-linearity of the sine-Gordon equation, the sum of both solutions is not another
solution of the sine-Gordon equation. We cannot even state that the sum of the states
defined by the tachyon and the particle for t −→ ∞ results in a solution of the
sine-Gordon equation. The sine-Gordon equation therefore does not allow for such
an emission of tachyons.
Basically, the question with regards to causality in a non-linear field theory cannot
be formulated in its original form. In the original form, the action of one particle on
another particle was asked for. Such a situation however no longer exists, at least
not as strictly as before. We now have a composed state consisting of particle and
327
tachyon that was emitted could also be absorbed. So, the deadly shot using a tachyon
and the end of causality a reality?
Well, in fact, the emission of a particle by a tachyon can be compatible to the
conservation laws. If we just remove the particle before the collision in Eq. (460),
we receive
μ sinh β
= cosh α − cosh α
,
μ cosh β
= sinh α − sinh α
.
No particle
before the collision
(470)
From 0 ≤ μ cosh β
= sinh α − sinh α
and the monotony of the hyperbolic sine
follows α
≤ α. Using cosh
2
β
− sinh
2
β
= 1 we eliminate β
by squaring and
subtraction of both Eq. (470) with the result
α
= α − arcosh
1 +
μ
2
2
.
(471)
According to the above, an arbitrary arriving tachyon, in other words α and m
T
∗
are arbitrarily given, can emit a particle with an arbitrary restmass m o = μ m
T
∗ . The
tachyon velocity u
after emission can be calculated from (471) keeping c o /u
=
v
/c o = tanh α
in mind, and the velocity w
of the emitted particle can be calculated
from (470) and (471) keeping w
/c o = tanh β
in mind.
All of these processes are formally possible according to the conservation laws
of energy and momentum. Do they however actually exist? Physically speaking, do
these processes exist as solutions of the fundamental equations of physics known
to us? If so, then we could only save our causal view of the world by referring
back to Treder’s [94] proposition of banning these processes into sub-microscopic
regions, or by simply formulating an additional condition stating that these solutions
are excluded—which would of course only be a stopgap used to bridge the holes in
the theory. This is the general situation in which we cannot discuss these problems
without actually coming to a consensus about which elementary particle theory is
the best to use in modern physics. Let us remain with that what we can elementarily
comprehend, the sine-Gordon equation with its particle and tachyon solutions inside
of a crystal.
The preceding considerations do not lead us to the conclusion that the particle
q
I
(x, t), the solution (111) of the sine-Gordon equation can in fact emit a tachyon
q
T
(x, t), the solution (261) of the sine-Gordon equation. Absolutely not! Due to the
non-linearity of the sine-Gordon equation, the sum of both solutions is not another
solution of the sine-Gordon equation. We cannot even state that the sum of the states
defined by the tachyon and the particle for t −→ ∞ results in a solution of the
sine-Gordon equation. The sine-Gordon equation therefore does not allow for such
an emission of tachyons.
Basically, the question with regards to causality in a non-linear field theory cannot
be formulated in its original form. In the original form, the action of one particle on
another particle was asked for. Such a situation however no longer exists, at least
not as strictly as before. We now have a composed state consisting of particle and