Abstract
Controlling motion of a charged particle is to control its phase space, in particular the momentum part. Particle with relativistic velocity is one model, when owing to relativistic effects the phase space is almost stationary, except for translational degrees of freedom. Relativistic motion is an example of the universal recipe to control motion of, charged, particles; they should be confined to a small volume of space and then manipulated by electromagnetic field. Confinement is achieved by static and time dependent magnetic field and electromagnetic waves of various properties. Extreme confinement of charges when their motion is in the relativistic regime, have specific features that are manifested in distribution of their momenta and energies when the system decays, and also in distribution of charges densities within the bound system, for which two body Dirac equation is used.
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Notes
- 1.
\(\Delta u\) is standard deviation, which is defined as \(\left( \Delta u\right) ^{2}=\left( u^{2}\right) _{aver}-\left( u_{aver}\right) ^{2}\) and it is used in calculation of the uncertainty relationship.
- 2.
Using bubble chamber is only one way of determining trajectories of charged elementary particles, however, the same argument applies to the other methods. The essence is to trace trajectories by measuring with high accuracy the sequence of their positions.
- 3.
Classical results here are almost identical (negative energy trajectories not included for simplicity) with quantum for free particles. This treatment appears to be more in line with Klein-Gordon solution rather than Dirac one, because in the latter the spin degree of freedom is taken into account. However, for a free particle spin is not significant and the results are identical in both cases. More detailed relativistic quantum analysis is given in a separate chapter.
- 4.
For a general \(\rho _{0}\left( \overrightarrow{r}_{1},\overrightarrow{r}_{2}, \overrightarrow{p}_{1},\overrightarrow{p}_{2}\right) \) one takes N random choices of the four coordinates, with the weight of the absolute value for initial phase space density. Sampling is made by adding or subtracting the outcome depending on the sign of \(\rho _{0}\), which allows for the possibility that it could have negative values. For more details see [9].
- 5.
Being “bound” is here used for the modelling purpose. The pair is created in collision and its precise history prior to that is essentially not known. In this context the term “bound” means that between separating and creation the two particles were confined in a certain space.
- 6.
If the mass of antineutrino is assumed to be not zero, but small, the final results do not change in essential way.
- 7.
For potentials, scalar or vector, that are linear in spatial coordinates, regardless of time dependence, phase space and quantum dynamics give identical results. In this example phase space analysis is simpler.
- 8.
The phase of a plane standing wave in time and coordinate dependence is removed by a suitable choice of the initial instant in time and initial position of a charge.
- 9.
Function a(z) may have several maxima but here the assumption is that there is only one.
- 10.
For simplicity it is assumed that the effects due to the motion in the x direction are not essential for investigating confinement in the z direction, and so the choice \(x=0\) is made.
- 11.
This conclusion does not take into account other effects in detection of elementary particles, which are not reviewed here.
- 12.
It should be noted here that the tracks which the charged elementary particles produce in detectors are analyzed precisely in this way.
- 13.
Again an approximation because this is not relativistically invariant interaction. Justification for it is that one of the partner particles has large mass and therefore moves non relativistically.
- 14.
It should be noted that the total energy, both in nonrelativistic and relativistic dynamics, has term with the total momentum of the particles. In the former this term could be omitted, being replaced by explicit reference in the phase space density by the total momentum. In the latter this is not possible and therefore the total momentum term appears in two places.
- 15.
One should be careful when using the term “physical significance”, in this context it is meant that velocity does not have arbitrary signature whilst momentum does.
- 16.
Chapter 2 is devoted to relativistic dynamics, and here the emphasis is on bound states.
- 17.
When the probability amplitude is strictly zero outside certain finite nterval then it should also be zero at the end points of the interval, otherwise uncertainty principle is violated. For example, constant probability amplitude within the interval is not a physically acceptable choice.
- 18.
This choice is inspired by measurement of the charge density in neutron [20], and as the model it is assumed that it is result of bound state of the electron and proton.
- 19.
Scalar interaction could be Lorentz invariant if it is function of the Lorentz scalar, which in this case this is not the case. The interaction is function of the relative distance of two particles.
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Bosanac, S.D. (2016). Confinement of Charge. In: Electromagnetic Interactions. Springer Series on Atomic, Optical, and Plasma Physics, vol 94. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52878-5_5
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DOI: https://doi.org/10.1007/978-3-662-52878-5_5
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