Chapter 22
Particles and Fields
The typical characteristic for a system of particles is the number of particles contained
in the system. Every single particle has a velocity v and an inertial mass m attributed
to it. The typical physical parameters of a single particle that play a crucial role
in interaction between the particles are momentum p = m v and energy E. The
latter is determined by the mass in a relativistic theory contained in the Einsteinian
formula E = m c
2 , and it is expressed using the momentum and the mass in the nonrelativistic approximation according to E kin =
p
2
2 m
. Characteristic for an interaction
between two particles is a collision. During this process, all of the particle’s total
energy and momentum are available for exchange—upholding the laws of energy
and momentum conservation for these quantities; see the next chapter.
What do the solutions of the sine-Gordon equation q = q(x, t) have in common
with particles? The most important statement in Chap. 5 was that Newton’s mechanics does not make any statement concerning the number of masses involved in a
motion. On the basis of Newton’s axiomatics, we were able to, during our process
of deriving the sine-Gordon equation, immediately move on to the limit of infinitely
many particles. We thus got the field q = q(x, t) enabling us to describe motion of
the positions of the dislocations line in the proximity of a straight line. If we revert,
in respect of the above, back to talking about particles, of single masses and velocities, then these particles can only correspond to those masses that formed the starting
point of our considerations. These however would be the masses m α of the Newtonian
equations (79). These masses do not refer to our (outside) physical space, they are
inertias with respect to the crystal lattice, as we discussed in detail in Chap. 8. If we
therefore, with respect to the solutions of the sine-Gordon equation, talk of particles,
then these are the particles of the internal observers of the crystal, which has our full
attention.
1 The question that we have to answer here is: Does the field q = q(x, t)
that is composed of many, or even infinitely many particles behave as if it were one
1 Such particles with respect to the lattice in physics are called quasi-particles. They are called so
in order to differentiate them from the particles of our outside physical space.
© 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_22
237
Particles and Fields
The typical characteristic for a system of particles is the number of particles contained
in the system. Every single particle has a velocity v and an inertial mass m attributed
to it. The typical physical parameters of a single particle that play a crucial role
in interaction between the particles are momentum p = m v and energy E. The
latter is determined by the mass in a relativistic theory contained in the Einsteinian
formula E = m c
2 , and it is expressed using the momentum and the mass in the nonrelativistic approximation according to E kin =
p
2
2 m
. Characteristic for an interaction
between two particles is a collision. During this process, all of the particle’s total
energy and momentum are available for exchange—upholding the laws of energy
and momentum conservation for these quantities; see the next chapter.
What do the solutions of the sine-Gordon equation q = q(x, t) have in common
with particles? The most important statement in Chap. 5 was that Newton’s mechanics does not make any statement concerning the number of masses involved in a
motion. On the basis of Newton’s axiomatics, we were able to, during our process
of deriving the sine-Gordon equation, immediately move on to the limit of infinitely
many particles. We thus got the field q = q(x, t) enabling us to describe motion of
the positions of the dislocations line in the proximity of a straight line. If we revert,
in respect of the above, back to talking about particles, of single masses and velocities, then these particles can only correspond to those masses that formed the starting
point of our considerations. These however would be the masses m α of the Newtonian
equations (79). These masses do not refer to our (outside) physical space, they are
inertias with respect to the crystal lattice, as we discussed in detail in Chap. 8. If we
therefore, with respect to the solutions of the sine-Gordon equation, talk of particles,
then these are the particles of the internal observers of the crystal, which has our full
attention.
1 The question that we have to answer here is: Does the field q = q(x, t)
that is composed of many, or even infinitely many particles behave as if it were one
1 Such particles with respect to the lattice in physics are called quasi-particles. They are called so
in order to differentiate them from the particles of our outside physical space.
© 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_22
237