,

# Ultracold Neutron Storage Simulation Using the Kassiopeia Software Package

Z Bogorad1, P M Mohanmurthy1, and J A Formaggio1 1Laboratory for Nuclear
Science, Massachusetts Institute of Technology, Cambridge, MA 02139, United
States of America<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Kassiopeia software package was originally developed to simulate
electromagnetic fields and charged particle trajectories for neutrino mass
measurement experiments. Recent additions to Kassiopeia also allow it to
simulate neutral particle trajectories in magnetic fields based on their
magnetic moments. Two different methods were implemented: an exact method that
can work for arbitrary fields and an adiabatic method that is limited to
slowly-varying fields but is much faster for large precession frequencies.
Additional interactions to simulate reflection of ultracold neutrons from
material walls and to allow spin-flip pulses were also added. These tools were
used to simulate neutron precession in the Paul Scherrer Institute’s neutron
electric dipole moment experiment and predict the values of the longitudinal
and transverse relaxation times as well as the trapping lifetime. All three
parameters are found to closely match the experimentally determined values
when simulated with both the exact and adiabatic methods, confirming that
Kassiopeia is able to accurately simulate neutral particles. This opens the
door for future uses of Kassiopeia to prototype the next generation of atomic
traps and ultracold neutron experiments.

Keywords: simulation, software, particle tracking, ultracold neutrons

##  1 Introduction

At sufficiently low energies, neutrons can be reflected from material walls by
the coherent strong interaction [1, 2]. Such neutrons can be stored in
material traps, and are termed “ultracold neutrons” (UCNs).

The ability of UCNs to be stored for prolonged periods makes them of interest
in several experimental areas. They can be used to measure the neutron
lifetime [3, 4], which continues to be of interest due to the remaining
discrepancy between different experimental measurements [5]. UCNs can also be
used to constrain the neutron electric dipole moment, which is related to the
strong CP problem and can constrain beyond the Standard Model physics [6, 7,
8]. Currently, the best upper bound on the neutron electric dipole moment
comes from measurements on ultracold neutrons, described in [9, 10].

Several software packages have previously been used to simulate ultracold
neutrons, including Geant4 [11], PENTrack [12], STARucn [13] and MCUCN [14].
Some comparisons between these are made in [12, 15]. In this paper, we
demonstrate the ability of Kassiopeia [16], a software package originally
developed for neutrino experiments, to accurately simulate ultracold neutron
storage. In Section 2, we describe some general features of Kassiopeia, not
specific to UCNs. We then describe neutral particle tracking features that
have been added to Kassiopeia in order to simulate UCNs in Section 3. Next, we
describe a specific experiment that we simulated and the parameters of our
simulation in Section 4, and describe our results in Section 5. Finally, in
Section 6 we consider possible future applications of Kassiopeia’s neutral
particle tracking capabilities.

##  2 Simulation in Kassiopeia

Kassiopeia was created as a combined field solver and particle tracker for the
KATRIN collaboration [16]. It is developed in C++ using a modular design that
allows it to be adapted to various applications using different field solvers,
equations of motion and interactions. The standard distribution, available at
https://github.com/KATRIN-Experiment/Kassiopeia, implements a variety of these
features already, and Kassiopeia is designed for easy development of
additional modules for future applications. Individual simulations can then be
configured using an Extensible Markup Language (XML) file.

Kassiopeia has several implemented methods to calculate the electromagnetic
fields associated with a particular current geometry. Magnetic fields can be
calculated using zonal harmonic expansions [17] or the fast multipole method
[18], while electric fields can be computed using the boundary element method
[17]. Both types of fields can also be pre-calculated and imported, as is done
in this work.

Kassiopeia also has several modules, called “trajectories”, for solving
particles’ equations of motion. Different trajectories include different terms
in the equations of motion and use different representations of particle
states.

Early versions of Kassiopeia included two main trajectory modules: an exact
trajectory module, which describes particles via their position and momentum
in laboratory coordinates and is limited only by the integration step size,
and an adiabatic trajectory module, which calculates and records particles’
position and motion around the magnetic field lines that they adiabatically
follow at sufficiently low energies.

Both of these modules implemented equations of motion that assumed a non-zero
particle charge, which prevented Kassiopeia from simulating other particles of
interest, such as neutrons or neutral atoms. To remedy this, two new
trajectory modules were added to Kassiopeia: an exact spin trajectory and an
adiabatic spin trajectory, described below. Both modules describe particles’
position and momentum in laboratory coordinates, as in the original exact
trajectory, but then add additional terms based on particles’ magnetic
moments, determined by their spins. These additional terms are potentially the
only non-zero terms in the equations of motion for neutral particles.

##  3 Spin Features in Kassiopeia

At each integration step of a trajectory calculation, two parameters of the
particle’s evolution must be calculated: the time derivative of the spin, and
the force on the particle as a function of the spin and surrounding fields. We
therefore next describe how these are calculated for the exact spin
trajectory.

The relativistic generalization of a classical spin is the four-vector
$S=(S_{0},\bi{S})$ given by, for a particle with classical spin $\bi{s}$ and
velocity $\bi{v}=\boldsymbol{\beta}c$ and Lorentz factor $\Gamma$ (to avoid
confusion with the gyromagnetic ratio $\gamma$) [19],

$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\Gamma\boldsymbol{\beta}\cdot\bi{s}$ (1a) $\displaystyle\bi{S}$
$\displaystyle=$
$\displaystyle\bi{s}+\frac{\Gamma^{2}}{\Gamma+1}(\boldsymbol{\beta}\cdot\bi{s})\boldsymbol{\beta}.$
(1b)

The equations of motion for a relativistic spin with 4-velocity $U^{\alpha}$
in fields $F^{\mu\nu}$, using the particle’s proper time $\tau$, are given by
the BMT equation:

$\frac{\rmd
S^{\alpha}}{\rmd\tau}=\frac{\gamma}{c}\left[F^{\alpha\beta}S_{\beta}+\frac{1}{c^{2}}U^{\alpha}(S_{\lambda}F^{\lambda\mu}U_{\mu})\right]-\frac{1}{c^{2}}U^{\alpha}\left(S_{\lambda}\frac{\rmd
U^{\lambda}}{\rmd\tau}\right).$ (1b)

The spin-dependent contribution to the particle’s motion is given by the
magnetic dipole force term, which is implemented nonrelativistically:

$\mathbf{F}=\nabla(\gamma\mathbf{s}\cdot\mathbf{B})$ (1c)

This exact spin tracking method is completely general, except for the
assumption that relativistic corrections to the magnetic dipole force term are
irrelevant. However, the step size for numerically integrating these equations
of motion is limited by the precession rate of the spins,
$\gamma|\mathbf{B}|$. This is typically fast relative to other timescales of a
particle’s motion, which can make the exact spin tracking method impractically
slow for experiments with long timescales.

Given the high computational cost of simulating tracks with the exact spin
trajectory, an adiabatic spin trajectory was also implemented. The adiabatic
spin trajectory records particle spins using two variables instead of four: an
aligned spin $m$ that gives the component of the spin along the magnetic field
at the particle’s position, and a spin angle $\phi$ that gives the orientation
of the spin around that field. This spin angle is defined with respect to two
unit vectors orthogonal to the local magnetic field, $\bi{e_{1}}$ and
$\bi{e_{2}}$, defined, for a magnetic field $\bi{B}=(B_{x},B_{y},B_{z})$ in
laboratory coordinates, as:

$\displaystyle\bi{e_{1}}$ $\displaystyle:=$
$\displaystyle(B_{z},0,-B_{x})/\sqrt{B_{x}^{2}+B_{z}^{2}}$ (1da)
$\displaystyle\bi{e_{2}}$ $\displaystyle:=$
$\displaystyle\bi{B}\times\bi{e_{1}}/|\bi{B}|$ (1db)

Given these axes, $\phi$ gives the angle of the projection of the spin into
the plane perpendicular to $\bi{B}$ away from $\bi{e_{1}}$, in the direction
of $\bi{e_{2}}$:

$\bi{s}=(s^{2}-m^{2})^{1/2}(\bi{e_{1}}\cos\phi+\bi{e_{2}}\sin\phi)+m\bi{B}/|\bi{B}|$
(1de)

The relationship between these vectors is illustrated in Figure 1.

Figure 1: The vectors used to define the adiabatic coordinates $m$ and $\phi$
from the spin vector $\bi{s}$ in (1de).

These two parameters, $m$ and $\phi$, along with the position $x$, then give
the adiabatic spin equations of motion (excluding contributions independent of
spin) [20]:

$\displaystyle\frac{M}{\hbar}\ddot{\bi{x}}$ $\displaystyle=$
$\displaystyle\gamma
m\nabla|\bi{B}|+\gamma|\bi{B}|(s^{2}-m^{2})^{1/2}\nabla\bi{b}\cdot\bi{c}$
(1dfa) $\displaystyle\frac{\dot{m}}{s}$ $\displaystyle=$
$\displaystyle(s^{2}-m^{2})^{1/2}(\dot{\bi{x}}\cdot\nabla\bi{b}\cdot\bi{c})$
(1dfb) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-\gamma|\bi{B}|-\dot{\bi{x}}\cdot\bi{A}-\frac{sm}{(s^{2}-m^{2})^{1/2}}(\dot{\bi{x}}\cdot\nabla\bi{b}\cdot\bi{a})$
(1dfc)

where $x$ is position, $\gamma$ is the particle’s gyromagnetic ratio, $M$ is
its mass, $s$ is the magnitude of its total spin, and we have defined

$\displaystyle\bi{a}$ $\displaystyle=$
$\displaystyle-\bi{e_{1}}\cos\phi+\bi{e_{2}}\sin\phi$ (1dfga)
$\displaystyle\bi{b}$ $\displaystyle=$ $\displaystyle\bi{B}/|\bi{B}|$ (1dfgb)
$\displaystyle\bi{c}$ $\displaystyle=$
$\displaystyle\bi{e_{1}}\cos\phi+\bi{e_{2}}\sin\phi$ (1dfgc)
$\displaystyle\bi{A}$ $\displaystyle=$
$\displaystyle\nabla\bi{e_{1}}\cdot\bi{e_{2}}$ (1dfgd)

for convenience.

This adiabatic spin trajectory is accurate when particles are non-relativistic
and spin precession is fast compared to the rate at which the field at the
particle’s location changes. Assuming the spin precession rate is dominated by
the first term $\gamma|\bi{B}|$, this requires
$\gamma|\bi{B}|\gg|\dot{\bi{x}}\cdot\nabla\bi{B}|/|\bi{B}|$, which should hold
for all applications discussed in this work. During a single integration time
step, the exact trajectory requires the spin to rotate by much less than a
radian, so the left-hand side of that inequality approximately sets the
maximum step size for exact spin integration. The adiabatic trajectory can
handle large rotations per time step but assumes a constant field during the
step, so the right-hand side limits the step size for adiabatic spin
integration. In this limit, therefore, the adiabatic spin trajectory can be
much faster.

A comparison of the outputs from the adiabatic and exact trajectories is shown
in Figure 2. Note that the exact trajectory becomes increasingly inaccurate as
the integration time approaches the precession period $33\ \mu$s, even for
$10$ ms tracks.

Figure 2: A comparison of neutron spin angles around the local magnetic field
(in the coordinate system defined by (1de)), including an adiabatic trajectory
with an integration time step of $100\ \mu$s and exact trajectories with time
steps of $5$, $10$, and $20\ \mu$s. These were run in a version of the neutron
precession chamber described later in this work, but with the magnetic field
scaled up by a factor of $1000$, resulting in a precession frequency of
approximately $30$ MHz. Note that the plot necessarily suffers from aliasing,
but sampling was done simultaneously so that points at the same time can be
compared.

There may, however, be other limitations on the step size that prevent this
from being the case. In particular, for the ultracold neutron tracking
described in this work, weak magnetic fields mean that the limiting factor on
the time step is error in position, not spin, so adiabatic and exact spin
tracking require similar computation time.

##  4 Ultracold Neutron Storage

Having established that the exact and adiabatic spin tracking methods agree
under appropriate conditions, we next consider how well Kassiopeia can
reproduce experimental data from the Paul Scherrer Institute’s ultracold
neutron precession chamber [21].

The PSI neutron precession chamber is a cylinder of radius 23.5 cm and height
12 cm. The interior rounded surface is primarily deuterated polystyrene (dPS),
with small windows of deuterated polyethylene (dPE), while the two flat ends
of the cylinder (the electrodes) are covered in diamond-like carbon (DLC). For
purposes of our simulation, the entire surface was assumed to be dPS, since it
is the primary surface cover and any changes associated with including dPE and
DLC are likely to be smaller than the uncertainties in the values described
below.

UCN reflections from surfaces are described by three features: the reflection
probability, the depolarization probability, and the distribution of outgoing
angles for reflected neutrons. The first two are described in [22], while the
third is described in [23]. We will summarize these and discuss our estimates
of the associated parameters.

The UCN reflection probability at an energy $E$ and angle to the surface
normal $\theta$ is given by

$P(R)=1-\eta\left(\frac{E_{\perp}}{V_{f}-E_{\perp}}\right)^{1/2}$ (1dfgh)

where $E_{\perp}=E\cos^{2}\theta$ is the component of energy from normal
momentum, $V_{f}$ is the real part of the surface’s optical potential, and
$\eta$ is another constant of the surface. For dPS, these values are known
only approximately, with $\eta\sim(1-3)\times 10^{-4}$ and $V_{f}\sim
1.6\times 10^{-7}$ eV [22], while a range of values for DLC were found in
[24].

The UCN depolarization probability is the simplest to describe, and is given
by a constant $\beta$. For dPS, this is known very roughly as $\beta\sim
10^{-6}-10^{-5}$ [22], and again a range of values for DLC are given in [24].

Detailed theoretical treatments of the distribution of outgoing angles for
reflected UCNs can be found in [23, 25]. Since the behavior of UCNs in the
precession chamber should not depend significantly on the exact angular
distribution, we derive an approximate form. We start from the microroughness
model result which, up to a normalization constant, can be modeled as:

$p(\theta_{f},\Phi_{f})\propto|S(\theta_{f})|^{2}\exp\left[-\frac{(wk)^{2}}{2}(\sin^{2}\theta_{i}+\sin^{2}\theta_{f}-2\sin\theta_{i}\theta_{f}\cos\Phi_{f})\right]$
(1dfgi)

Here, $\theta_{i}$ and $\theta_{f}$ are the incident and outgoing angles to
the normal, $\phi_{f}$ is the change in direction around the normal, $w$ is
the surface roughness, $k$ is the neutron wave vector, and

$\displaystyle|S(\theta_{f})|$ $\displaystyle=$
$\displaystyle\left|\frac{2\cos\theta_{f}}{\cos\theta_{f}+\sqrt{\cos^{2}\theta_{f}-k_{c}^{2}/k^{2}}}\right|$
(1dfgja) $\displaystyle=$ $\displaystyle 2k\cos\theta_{f}/k_{c}$ (1dfgjb)

where $k_{c}=\sqrt{2mV_{f}/\hbar}$ is the wave number corresponding to the
optical potential and we consider only neutrons with $k<k_{c}$, since higher-
energy neutrons quickly escape the precession chamber. Then

$\displaystyle p(\theta_{f},\Phi_{f})$ $\displaystyle\propto$
$\displaystyle\cos^{2}\theta_{f}\exp\left[-\frac{1}{2}(wk)^{2}(\sin\theta_{i}-\sin\theta_{f})^{2}+(wk)^{2}\sin^{2}\theta_{i}(\cos\Phi_{f}-1)\right]$
(1dfgjka) $\displaystyle\approx$ $\displaystyle(1-\tan\theta_{i}\
\Delta\theta)\exp\left[-\frac{(wk)^{2}}{2}\cos^{2}\theta_{i}\Delta\theta^{2}-\frac{(wk)^{2}}{2}\sin^{2}\theta_{i}\Delta\theta^{2}\right]$
(1dfgjkb)

where we defined $\Delta\theta=\theta_{f}-\theta_{i}$. This gives a simple
approximate distribution of outgoing angles that can be efficiently sampled.

Note that this derivation assumes that the change in incident and outgoing
angles is small. This is a rough approximation in our case, but we found that
our simulations were insensitive to the exact parameters of the reflected
distribution, so this should not significantly affect our results.

For our simulations, we used $\beta=6\times 10^{-6}$, $\eta=1\times 10^{-4}$,
$V_{f}=150$ neV, and $w=30$ nm, which were consistent with the available
estimated ranges of each value (see [26, 27] for measurements of $w$). The
first two parameters were tuned so as to be most consistent with PSI’s data.
In the future, it might be possible to use our simulations as a means of
actually determining these values, but this was beyond the scope of this work.

For each test simulation, the precession chamber’s interior had an essentially
constant magnetic field of approximately $1\ \mu$T along the cylinder radius,
with $\mathcal{O}$(nT) deviations along each coordinate axis. In PSI’s
experiment, the apparatus would also contain an electric field in either axial
direction, but this was not included in our simulation since electric effects
are below the sensitivity of the experiment [9]. The axial magnetic fields
were different (and in particular were in opposite directions) for the two
electric field directions, and we received a set of magnetic field
measurements of each field configuration [28] and then used a combination of
natural neighbors interpolation (to reach a rectilinear lattice) and cubic
interpolation (within lattice cells) to estimate the magnetic field
everywhere.

During a particular run of our simulation, a number of neutrons would be added
to just above the bottom of the precession chamber. While neutrons may not
actually all start near the bottom surface, their motion around the trap is
sufficiently fast relative to the simulation times that this should not affect
the results significantly. Neutron kinetic energies were generated from a
Gaussian distribution of mean $150$ neV and standard deviation $50$ neV. This
is a higher energy distribution than the energy distribution estimated by PSI
[29]. It may be possible to match the measured energy distribution more
closely by varying the parameters described above, but this was beyond the
scope of this work. During a given run, all neutrons would also begin with the
same spin, which varied by experiment as described below. Three example tracks
inside of the simulated PSI precession chamber are shown in Figure 3.

Figure 3: A VTK image of three neutrons tracked for 1 second inside of the PSI
precession chamber. Colors correspond to kinetic energies, with higher
energies at the blue end of the spectrum. Note the approximately specular
reflections and the slight curvature in the tracks induced by gravity.

##  5 Results and Discussion

We compared our simulation’s results with three previously measured parameters
of PSI’s precession chamber: the longitudinal relaxation time T1 [30], the
transverse relaxation time T2 [31], and the effective trap lifetime $\tau$
including neutron decay and wall losses [30].

A total of 17280 neutrons were simulated up to maximum times of 500 seconds.
This was done using 4 parallel `c4.8xlarge` instances of Amazon Elastic
Compute Cloud (EC2), or 144 total cores. Kassiopeia does not directly support
parallelization of particle tracking, so we instead ran 144 separate
simulations of 120 neutrons each and then combined the outputs. This took
approximately 2500 CPU-hours, though further optimization is likely possible.

###  5.1 The Longitudinal Relaxation Time T1

The longitudinal relaxation time, T1, is the time constant of the decay of
average polarization for neutrons with initial spins that are either aligned
or anti-aligned relative to the local magnetic field. In particular, it is the
time constant for decay towards the thermal distribution of aligned spin. It
should be noted that, even for ultracold neutrons, the $\mu$T-order magnetic
fields in the precession chamber do not significantly affect neutron energies
since $\mu_{N}|\bi{B}|$ is far less than the kinetic energy, so the ultimate
thermal evolution of neutrons in the chamber is towards an essentially uniform
distribution of spins. On a microscopic level, longitudinal relaxation is
associated with inhomogeneities of the magnetic field, and with depolarization
due to reflections.

Since the adiabatic equation of motion (1dfb) for fully aligned or anti-
aligned spins always gives a zero derivative of the aligned spin, our
simulations instead started spins at $1^{\circ}$ or $179^{\circ}$ to the
magnetic field, though we found that any angle up to a few degrees led to
indistinguishable results, as the dominant contribution to T1 came from
reflection depolarization.

We simulated 4320 neutrons for 500 seconds using both the adiabatic and exact
trajectories with half initially aligned and half initially anti-aligned, and
compared our average polarization as a function of time to that obtained by
PSI. Our results can be seen in Figure 4, where a relative polarization of 1
indicates an average polarization equal to the initial value ($\pm$1 in
simulations, but smaller magnitudes in PSI’s experiments).

Figure 4: A comparison of the fractional remaining longitudinal polarization
measured at PSI with the results of our adiabatic and exact simulations,
including 1$\sigma$ confidence intervals for the simulations and 1$\sigma$
error bars for the experimental data. Here, a $0.95$ relative polarization
corresponds to an average spin along the z-axis equal to $0.95$ times the
initial average spin along the z-axis. Each data set averages over the two
field configurations and over initially aligned and anti-aligned neutrons. We
see good agreement over the entire period; the deviation of the exact data at
long times is not statistically significant.

As Figure 4 shows, both the adiabatic and exact simulations were able to
accurately reproduce the measured longitudinal depolarization rate. Both
simulations are within 1$\sigma$ of the experimental data (using the combined
error) until around 350 seconds, and the adiabatic simulations continue to
closely match the experiment for the entire run. The apparent discrepancy
between the experimental data and the exact trajectory at high times is not
statistically significant (see Table 1).

###  5.2 The Transverse Relaxation Time T2

The transverse relaxation time, T2, is the time constant of the decay of
average polarization for neutrons with initial spins that are aligned along an
axis perpendicular to the local magnetic field. The dominant contribution to
transverse relaxation of neutrons in the precession chamber is the variation
in the magnetic field over the trap volume. Note that spin-spin interactions,
which may be the dominant contribution to T2 in other contexts, should not be
significant for neutrons and cannot be implemented in Kassiopeia, which tracks
particles one at a time.

Figure 5: A comparison of the fractional remaining transverse polarization
measured at PSI with the results of our adiabatic and exact simulations,
including 1$\sigma$ confidence intervals for the simulations and 1$\sigma$
error bars for the experimental data. Here, a $0.95$ relative polarization
corresponds to an average spin along the axis of average precession equal to
$0.95$ times the initial average spin along the initial polarization axis.
Each data set averages over the two field configurations and over two initial
spin directions. We see good agreement over the entire period; the deviation
of the exact data at long times is not statistically significant.

In PSI’s T2 measurements, neutrons began with polarizations aligned or anti-
aligned with the magnetic field and then T2 was measured between two $\pi/2$
pulses that rotated them into and out of the perpendicular plane. While ideal
spin rotating pulses are implemented in Kassiopeia, they were not used for
this simulation since we were interested only in the behavior occurring within
the field-perpendicular plane.

We simulated 4320 neutrons for 500 seconds using both trajectory types and two
different initial spin directions perpendicular to the magnetic field. We then
compared our average polarization as a function of time to that obtained by
PSI. Our results can be seen in Figure 5, with relative polarization defined
as for T1, but along the axis corresponding to average precession over that
time instead of along the z-axis.

As Figure 5 shows, both simulations were also able to accurately reproduce the
measured transverse depolarization rate. The adiabatic simulation matches all
of the experimental values, however the exact simulation results appear to
deviate from the data at long times. However, this discrepancy is again not
statistically significant (see Table 1), as our simulation errors are highly
correlated: neutrons that deviate significantly from the average polarization
(due to depolarization on reflection, for example) are likely to remain in the
trap for some time, lowering the average polarization at those later times as
well.

###  5.3 The Effective Trap Lifetime $\tau$

Since neutron reflection from material surfaces is not guaranteed, especially
at higher energies (see (1dfgh)), the effective lifetime of neutrons in the
precession chamber is significantly lower than their decay lifetime. Neutrons
may also be lost due to holes or other imperfections in a physical trap, which
should be included in the estimated value of $\eta$.

Figure 6: A comparison of the fraction of neutrons remaining in the trap
measured at PSI with the results of our adiabatic and exact simulations,
including 1$\sigma$ confidence intervals for the simulations and 1$\sigma$
error bars for the experimental data. Averages for each data set are taken as
in Figure 4 and Figure 5. Good agreement is seen along the entire PSI data
set. Note that the exact data uses a shifted y-axis for visual clarity. The
experimental values are arbitrarily normalized as the initial neutron count is
not known.

Lifetimes may be extracted from both the T1 and T2 simulations, since the
lifetime should be nearly independent of the neutrons’ spins in a $\mu$T
field. The results from both sets of simulations, as well as from PSI’s
experiments, are shown in Figure 6

Figure 6 once again shows good agreement between each of the four simulations
and the experimental data. The adiabatic T1 simulation deviates significantly
from the experimental data (see Table 1), but the lifetime is strongly
dependent on the chosen value of $\eta$ and the initial energy
distribution–neither of which is precisely known for the PSI trap–so this
discrepancy is not practically significant.

###  5.4 Summary of Results

A summary of our simulated data’s agreement to the PSI experimental data is
presented in Table 1, showing generally good agreement. The apparent
overestimation of errors is expected to be the result of correlations between
our errors at different times.

| T1 or T2 | Lifetime
---|---|---
Simulation |  $\chi^{2}$/ndf | $p$ |  $\chi^{2}$/ndf | $p$
T1 Ad | 0.22 | 1.00 | 2.43 | 0.02
T1 Ex | 0.66 | 0.84 | 0.53 | 0.90
T2 Ad | 0.13 | 1.00 | 0.56 | 0.88
T2 Ex | 1.29 | 0.29 | 0.85 | 0.66
Table 1: The results of $\chi^{2}$ testing of our results against the PSI
experimental data, including left-sided p-values. Note that ndf=11 for the T1
and T2 measurements and ndf=9 for the lifetime measurements. These values
suggest that we are over-estimating our errors, likely due to our results
being strongly correlated. The T1 adiabatic lifetime data’s disagreement with
the experimental data may indicate that our assumed value of $\eta$ was not
correct.

This demonstrates that Kassiopeia can reproduce the effective lifetime as well
as both decoherence times for neutrons within the PSI precession chamber,
confirming that it can accurately simulate neutral particles.

##  6 Other Applications

Having established that Kassiopeia can accurately simulate neutral particles’
motion in at least one context, we next consider a few potential future
applications of these features.

The application most directly stemming from the simulations in this paper is
simulated prototyping of future ultracold neutron experiments. In this work,
we simulated neutron precession using a previously measured magnetic field,
but Kassiopeia also includes extensive field calculation methods, which can be
used to calculate the magnetic fields inside of a trap based on its design
[17, 18]. Kassiopeia could therefore be used to estimate the effective
lifetime and relaxation times of future ultracold neutron experiments.

Another potential application is simulating tritium storage methods in the
Project 8 neutrino mass experiment [32]. Currently, atomic tritium is stored
using a purely magnetic trap. We have previously simulated versions of this
trap in order to obtain estimates of necessary currents and effective
lifetimes, but development of the final trap configuration is ongoing and
further simulation work could be done in this area. Furthermore, one current
proposal is to use a combined magneto-gravitational trap instead, which has
not been simulated as of writing.

Neutral particle simulation can also be applied to atomic experiments.
Experiments such as ACME [33] and others such as [34] use molecular beams to
measure the electric dipole moment. Molecules are prepared in particular spin
states and then precess through constant electric and magnetic fields; a
dependence of precession on the electric field can indicate an electric dipole
moment. Kassiopeia may be able to simulate molecular evolution within these
fields to aid with estimating systematic errors.

The authors would like to thank Mathieu Guigue for some helpful discussion. We
would also like to acknowledge the KATRIN and Project 8 collaborations as well
as the Paul Scherrer Institute’s nEDM collaboration for their support of this
project. This work was supported by the MIT Undergraduate Research
Opportunities Program, by the Department of Energy Office of Nuclear Physics
under award number DE-SC0011091, and by SERI-FCS under award number 2015.0594.
Other allied works were supported by Sigma Xi under grant number
G2017100190747806.

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# The spin group of fermions

K. S. Karplyuk<EMAIL_ADDRESS>Department of Radiophysics, Taras
Shevchenko University, Academic Glushkov prospect 2, building 5, Kyiv 03122,
Ukraine<EMAIL_ADDRESS>O. O. Zhmudskyy<EMAIL_ADDRESS>Department of Physics, University of Central Florida, 4000 Central Florida
Blvd. Orlando, FL, 32816<EMAIL_ADDRESS>
###### Abstract

It was shown that in the small Wigner group there is a one-parameter subgroup
of the Lorentz transformations, which leave unchanged not only the momentum of
the fermion with spin $\hbar/2$, but also its spin characteristics. This is
the group of symmetry transformations of the moving fermion. These
transformations consist of rotations around the projection axis of the spin
and boosts coordinated with them. In other words, there are infinitely many
frames of reference in which the fermion has the same momentum and the same
spin projection onto the selected axis, although these frames move relative to
each other and have been rotated relative to each other.

###### pacs:

12., 12.20.-m, 12.15.-y, 13.66.-a

##  I Introduction

It is known that a fixed fermion with spin projection $\frac{\hbar}{2}$ onto
the direction of the three-dimensional unit pseudovector $\bm{\mathfrak{s}}$
($\mathfrak{s}^{2}_{x}+\mathfrak{s}^{2}_{y}+\mathfrak{s}^{2}_{z}=1$) is
described by the bispinor $\psi^{(0)}$, which is the solution to the Dirac
equation for the fermion at rest

$\displaystyle p_{0}\gamma^{0}\psi^{(0)}=mc\psi^{(0)}$ (1)

and is a eigen bispinor of the operator
$\frac{\hbar}{2}\frac{i\mathfrak{s}_{kl}\sigma^{kl}}{2}$:

$\displaystyle\frac{\hbar}{2}\frac{i\mathfrak{s}_{kl}\sigma^{kl}}{2}\psi^{(0)}=\frac{\hbar}{2}\psi^{(0)}.$
(2)

Here $\sigma^{kl}=\frac{1}{2}(\gamma^{k}\gamma^{l}-\gamma^{l}\gamma^{k})$,
$\mathfrak{s}_{kl}$ are coefficients antisymmetric in $k$ and $l$,
$\mathfrak{s}_{23}=\mathfrak{s}_{x},\mathfrak{s}_{31}=\mathfrak{s}_{y},\mathfrak{s}_{12}=\mathfrak{s}_{z}$.
As usual, the summation is performed on the indices, the Latin indices take
the values 1,2,3, the Greek ones - 0,1,2,3. The operator
$\frac{\hbar}{2}\frac{i}{2}\mathfrak{s}_{kl}\sigma^{kl}$ is the operator of
the projection of the spin of a fermion at rest (or moving slowly) onto the
direction $\bm{\mathfrak{s}}$.

For an observer moving with velocity $-\bm{v}$ relative to the fermion, the
fermion has velocity $\bm{v}$ and is described by the equations that can be
obtained from bispinor equations (1) - (2) by boost transformations:

$\displaystyle
e^{b_{0k}\sigma^{0k}}p_{0}\gamma^{0}e^{-b_{0k}\sigma^{0k}}e^{b_{0k}\sigma^{0k}}\psi^{(0)}=mce^{b_{0k}\sigma^{0k}}\psi^{(0)},$
(3) $\displaystyle
e^{b_{0k}\sigma^{0k}}\frac{\hbar}{2}\frac{i\mathfrak{s}_{kl}{2}}{\sigma}^{kl}e^{-b_{0k}\sigma^{0k}}e^{b_{0k}\sigma^{0k}}\psi^{(0)}=\frac{\hbar}{2}e^{b_{0k}\sigma^{0k}}\psi^{(0)}.$
(4)

That is, by the equations

$\displaystyle p_{\alpha}\gamma^{\alpha}\psi=mc\psi,$ (5)
$\displaystyle\frac{\hbar}{2}\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}\psi=\frac{\hbar}{2}\psi.$
(6)

We have

$\displaystyle e^{b_{0k}\sigma^{0k}}=\cosh b+\frac{b_{0k}\sigma^{0k}}{b}\sinh
b,$ (7) $\displaystyle\frac{b_{01}}{b}=\frac{v_{x}}{v},\hskip
8.53581pt\frac{b_{02}}{b}=\frac{v_{y}}{v},\hskip
8.53581pt\frac{b_{03}}{b}=\frac{v_{z}}{v},\hskip 8.53581pt\tanh
2b=\frac{v}{c},$ (8) $\displaystyle\psi=e^{b_{0k}\sigma^{0k}}\psi^{(0)},$ (9)
$\displaystyle\frac{\hbar}{2}\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}=\frac{\hbar}{2}e^{b_{0k}\sigma^{0k}}\frac{i\mathfrak{s}_{kl}\sigma^{kl}}{2}e^{-b_{0k}\sigma^{0k}}.$
(10)

The hypercomplex number $\mathfrak{s}_{\alpha\beta}\sigma^{\alpha\beta}$ when
using the Lorentz transformations in the form (10) represents an antisymmetric
tensor of the second order, similar to the electromagnetic field tensor
$F_{\alpha\beta}$. More precisely, the tensor of the electromagnetic field,
which in the fermion’s own system is only a magnetic field, since
$\mathfrak{s}_{0i}=0,\mathfrak{s}_{kl}\neq 0$. Let’s call
$\mathfrak{s}_{\alpha\beta}$ the spin tensor. The representation of the tensor
$\mathfrak{s}_{\alpha\beta}$ in the form
$\mathfrak{s}_{\alpha\beta}\sigma^{\alpha\beta}$ is similar to the
representation of the 3-vector $(a_{x},a_{y},a_{z})$ by quaternion
$a_{x}\bm{i}+a_{y}\bm{j}+a_{z}\bm{k}$, or bivector $(a_{x},a_{y})$ with a
complex number $a_{x}+a_{y}i$. Note that, in addition to the spin tensor
$\mathfrak{s}_{\alpha\beta}$, exists the spin pseudovector or the Pauli-
Lubanski pseudovector Lubanski (1942)

$\displaystyle\lambda^{\alpha}=\varepsilon^{\alpha\beta\mu\nu}p_{\beta}s_{\mu\nu}.$
(11)

However, in this article we don’t need it.

The constituents of the tensor $s_{\alpha\beta}$, defined by relation (10),
can be conveniently written in 3-dimensional form as a 3-vector $\bm{s}_{1}$
and a 3-pseudovector $\bm{s}_{2}$, similarly to writing the tensor
$F_{\alpha\beta}$ in the form of an electric field vector $\bm{E}$ and a
pseudovector of a magnetic field $\bm{B}$:

$\displaystyle\frac{\hbar}{2}i\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}=\frac{\hbar}{2}i(\bm{s}_{1}\bm{\varsigma}_{1}+\bm{s}_{2}\bm{\varsigma}_{2}),$
(12)
$\displaystyle\bm{s}_{1}\bm{\varsigma}_{1}=s_{01}\sigma^{01}+s_{02}\sigma^{02}+s_{03}\sigma^{03},$
(13)
$\displaystyle\bm{s}_{2}\bm{\varsigma}_{2}=s_{23}\sigma^{23}+s_{31}\sigma^{31}+s_{12}\sigma^{12},$
(14) $\displaystyle\bm{s}_{1}=\frac{\bm{p}\times\bm{\mathfrak{s}}}{mc},$ (15)
$\displaystyle\bm{s}_{2}=\frac{p_{0}}{mc}\bm{\mathfrak{s}}-\frac{\bm{p(\bm{p\cdot\bm{\mathfrak{s}}})}}{mc(mc+p_{0})}.$
(16)

Here we use 3-dimensional notation

$\displaystyle\bm{\mathfrak{s}}=(\mathfrak{s}_{x},\mathfrak{s}_{y},\mathfrak{s}_{z})=(\mathfrak{s}_{23},\mathfrak{s}_{31},\mathfrak{s}_{12}),$
(17) $\displaystyle\bm{s}_{1}=(s_{1x},s_{1y},s_{1z})=(s_{01},s_{01},s_{03}),$
(18) $\displaystyle\bm{s}_{2}=(s_{2x},s_{2y},s_{2z})=(s_{23},s_{31},s_{12}).$
(19)

Using (12) - (16) it is easy to be convinced that $p_{\alpha}\gamma^{\alpha}$
and $\frac{1}{2}s_{\mu\nu}\sigma^{\mu\nu}$ commute:

$\displaystyle
p_{\alpha}\gamma^{\alpha}\frac{1}{2}s_{\mu\nu}\sigma^{\mu\nu}-\frac{1}{2}s_{\mu\nu}\sigma^{\mu\nu}p_{\alpha}\gamma^{\alpha}=p^{\alpha}s_{\alpha\beta}\gamma^{\beta}=0.$
(20)

According to equations (5) - (6), bispinor (9) describes a fermion with
velocity $\bm{v}$, and operator (12) is the operator of the projection of the
spin of a moving fermion. But in what direction is the spin projected in the
system moving relative to the fermion? In this system, with the tensor
$s_{\alpha\beta}$, two 3-directions are connected: the direction of the vector
$\bm{s}_{1}$ and the direction of the pseudovector $\bm{s}_{2}$. It is natural
to assume that the spin is projected onto $\bm{s}_{2}$, since $\bm{s}_{2}$
tends to $\bm{\mathfrak{s}}_{2}$ as $\bm{p}\to 0$, while $\bm{s}_{1}$ tends to
zero. The question is: what role does $\bm{s}_{1}$ play? Considering the spin
group will help answer this question.

##  II The spin group

Consider the transformation
$e^{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}$, whose
infinitesimal operator is $\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}$. It
is clear that this transformation does not change the spin projection operator
(12):

$\displaystyle
e^{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}\,\frac{\hbar}{2}\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}\,e^{-\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}=$
$\displaystyle\Bigl{(}\cosh
s\frac{\varphi}{2}+\frac{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}{s\frac{\varphi}{2}}\sinh
s\frac{\varphi}{2}\Bigr{)}\frac{\hbar}{2}\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}\times$
$\displaystyle\Bigl{(}\cosh
s\frac{\varphi}{2}-\frac{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}{s\frac{\varphi}{2}}\sinh
s\frac{\varphi}{2}\Bigr{)}=\frac{\hbar}{2}\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}.$
(21)

Here
$s=\sqrt{s_{\alpha\beta}\sigma^{\alpha\beta}s_{\alpha\beta}\sigma^{\alpha\beta}}=\sqrt{-\bm{\mathfrak{s}}^{2}}=i$.
It is less clear that this transformation leaves unchanged not only operator
(12), but also the 4-momentum of the fermion. It is not difficult to see that

$\displaystyle
e^{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}\,p_{\mu}\gamma^{\mu}\,e^{-\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}=(\cos\frac{\varphi}{2}+\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\sin\frac{\varphi}{2})\times$
$\displaystyle
p_{\mu}\gamma^{\mu}(\cos\frac{\varphi}{2}-\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\sin\frac{\varphi}{2})=p_{\mu}\gamma^{\mu}.$
(22)

Thus, this is a transformation belonging to the small Wigner group Wigner
(1939) for the momentum $p_{\mu}$.

We apply the transformation
$e^{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}$ to the
eigen bispinor $\psi$ of the operator
$\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}$, i.e. to the solution $\psi$
of equation (6):

$\displaystyle
e^{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}\psi=\Bigl{(}\cosh
s\frac{\varphi}{2}+\frac{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}{s\frac{\varphi}{2}}\sinh
s\frac{\varphi}{2}\Bigr{)}\psi=$
$\displaystyle(\cos\frac{\varphi}{2}+\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\sin\frac{\varphi}{2})\psi=(\cos\frac{\varphi}{2}-i\sin\frac{\varphi}{2})\psi=$
$\displaystyle e^{-i\frac{\varphi}{2}}\psi.$ (23)

As we see, after such a transformation, the bispinor $\psi$ acquires the
factor $e^{-i\frac{\varphi}{2}}$. This is due to the double-valuedness of the
bispinor during rotations. After the rotation by $\varphi=2\pi$, the bispinor
changes its sign, and transformation (23), as we will see below, contains a
rotation. However, the bispinor $e^{-i\frac{\varphi}{2}}\psi$ describes a
fermion with the same spin characteristics as the bispinor $\psi$. I.e., the
transformation
$e^{\frac{s_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}\psi$
leaves unchanged both the momentum of the fermion $\psi$ and its spin
characteristics. In this sense, it is the symmetry transformation of the
fermion.

Note that when performing transformation (22) (keeping the momentum unchanged)
only relations (12) - (16) are used, connecting $s_{\alpha\beta}$ with
$\bm{\mathfrak{s}}$, and the condition $\bm{\mathfrak{s}}^{2}=1$. Therefore,
all transformations
$e^{\frac{w_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}$ belong to
the small Wigner group, in which the parameters $s_{\alpha\beta}$ are replaced
by the parameters $w_{\alpha\beta}$ connected with the arbitrary unit spatial
3-pseudovector
$\bm{\mathfrak{w}}=(\mathfrak{w}_{x},\mathfrak{w}_{y},\mathfrak{w}_{z})$ with
the same relations as (15)-(16)

$\displaystyle\bm{w}_{1}=\frac{\bm{p}\times\bm{\mathfrak{w}}}{mc},$ (24)
$\displaystyle\bm{w}_{2}=\frac{p_{0}}{mc}\bm{\mathfrak{w}}-\frac{\bm{p(\bm{p\cdot\bm{\mathfrak{w}}})}}{mc(mc+p_{0})}.$
(25)

Here $\bm{w}_{1}=(w_{1x},w_{1y},w_{1z})=(w_{01},w_{02},w_{03})$,
$\bm{w}_{2}=(w_{2x},w_{2y},w_{2z})=(w_{23},w_{31},w_{12})$,
$\bm{\mathfrak{w}}$ \- is an arbitrary unit 3-pseudovector not related to spin
in any way. However, these transformations, while conserving the momentum, do
not preserve the spin properties of the fermion. Transformations in which
$w_{\alpha\beta}=s_{\alpha\beta}$ constitute a subgroup of the small Wigner
group. We will call this subgroup the spin group. The transformations of the
spin group, in contrast to other transformations of the small group, conserve
not only the momentum, but also the spin characteristics of the fermion.

Transformations of the small group
$e^{\frac{w_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}$ are
neither spatial rotations nor boosts. However, they, like all the Lorentz
transformations, can be represented as a sequence of two transformations - a
spatial rotation and a boost

$\displaystyle
e^{\frac{w_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}=e^{b_{0k}\sigma^{0k}}e^{\frac{r_{kl}\sigma^{kl}}{2}}$
(26)

or boost and spatial rotation

$\displaystyle
e^{\frac{w_{\alpha\beta}\sigma^{\alpha\beta}}{2}\frac{\varphi}{2}}=e^{\frac{r^{\prime}_{kl}\sigma^{kl}}{2}}e^{b^{\prime}_{0k}\sigma^{0k}}.$
(27)

Performing this factorization, we obtain

$\displaystyle e^{b_{ok}o^{ok}}=$
$\displaystyle\sqrt{\frac{1+\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\tan^{2}\frac{\varphi}{2}}}\Bigl{(}1+\frac{\bm{w}_{1}\tan\frac{\varphi}{2}-(\bm{w}_{1}\times\bm{w}_{2})\tan^{2}\frac{\varphi}{2}}{1+\bm{w}_{2}^{2}\tan^{2}\frac{\varphi}{2}}\bm{\varsigma}_{1}\Bigr{)}=$
$\displaystyle\sqrt{\frac{1+\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\tan^{2}\frac{\varphi}{2}}}\Bigl{(}1+\frac{\bm{w}_{1}\tan\frac{\varphi}{2}+\frac{\bm{p}_{\perp}}{p_{0}}\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\bm{w}_{2}^{2}\tan^{2}\frac{\varphi}{2}}\bm{\varsigma}_{1}\Bigr{)},$
(28) $\displaystyle e^{b^{\prime}_{ok}o^{ok}}=$
$\displaystyle\sqrt{\frac{1+\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\tan^{2}\frac{\varphi}{2}}}\Bigl{(}1+\frac{\bm{w}_{1}\tan\frac{\varphi}{2}+(\bm{w}_{1}\times\bm{w}_{2})\tan^{2}\frac{\varphi}{2}}{1+\bm{w}_{2}^{2}\tan^{2}\frac{\varphi}{2}}\bm{\varsigma}_{1}\Bigr{)}=$
$\displaystyle\sqrt{\frac{1+\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\tan^{2}\frac{\varphi}{2}}}\Bigl{(}1+\frac{\bm{w}_{1}\tan\frac{\varphi}{2}-\frac{\bm{p}_{\perp}}{p_{0}}\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\bm{w}_{2}^{2}\tan^{2}\frac{\varphi}{2}}\bm{\varsigma}_{1}\Bigr{)},$
(29) $\displaystyle
e^{\frac{1}{2}r_{kl}o^{kl}}=e^{\frac{1}{2}r^{\prime}_{kl}o^{kl}}=\frac{1+\bm{w}_{2}\bm{\varsigma}_{2}\tan\frac{\varphi}{2}}{\sqrt{1+\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}}.$
(30)

Here, to abbreviate the notation, we used the notation

$\displaystyle\bm{a}\bm{\varsigma}_{1}=a_{x}\sigma^{01}+a_{y}\sigma^{02}+a_{z}\sigma^{03},$
(31)
$\displaystyle\bm{a}\bm{\varsigma}_{2}=a_{x}\sigma^{23}+a_{y}\sigma^{31}+a_{z}\sigma^{12},$
(32)

and $\bm{p}_{\perp}$ denotes the component of the vector $\bm{p}$
perpendicular to $\bm{w}_{2}$ and lying in the plane $(\bm{p},\bm{w}_{2})$:

$\displaystyle\bm{p}_{\perp}=\bm{p}-\frac{\bm{w}_{2}(\bm{w_{2}\cdot\bm{p}})}{\bm{w}^{2}_{2}}.$
(33)

It is easy to check that equalities (26) - (27) hold. Thus, the
transformations of the small Wigner group are represented as a sequence of two
transformations one with a unitary operator, the other with a Hermitian
operator.

Unitary operators
$e^{\frac{1}{2}r_{kl}o^{kl}}=e^{\frac{1}{2}r^{\prime}_{kl}o^{kl}}$ carry out a
spatial rotation around an arbitrary axis $\bm{w}_{2}$ by an angle

$\displaystyle 2r=2\arctan\Bigl{(}|\bm{w}_{2}|\tan\frac{\varphi}{2}\Bigr{)}.$
(34)

In the partickle’s own system, this is a rotation around an arbitrary axis
$\bm{\mathfrak{w}}$ on an angle $\varphi$. This rotation changes the spin
characteristics of the resting fermion, but does not change its 4-momentum
$(mc,0,0,0)$. In the case of a moving particle, rotation around the
$\bm{w}_{2}$ axis changes (rotates) the momentum component perpendicular to
$\bm{w}_{2}$.

Hermitian operators $e^{b_{ok}o^{ok}}$ or $e^{b^{\prime}_{ok}o^{ok}}$ perform
boosts at the same speed $v=v^{\prime}$

$\displaystyle\tanh 2\frac{v}{c}=\tanh
2\frac{v^{\prime}}{c}=\Bigl{|}\frac{\bm{w}_{1}\tan\frac{\varphi}{2}+\frac{\bm{p}_{\perp}}{p_{0}}\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\bm{w}_{2}^{2}\tan^{2}\frac{\varphi}{2}}\Bigr{|}=$
$\displaystyle=\Bigl{|}\frac{\bm{w}_{1}\tan\frac{\varphi}{2}-\frac{\bm{p}_{\perp}}{p_{0}}\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}}{1+\bm{w}_{2}^{2}\tan^{2}\frac{\varphi}{2}}\Bigr{|},$
(35)

but in different directions
$\bm{w}_{1}\tan\frac{\varphi}{2}+\frac{\bm{p}_{\perp}}{p_{0}}\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}$
or
$\bm{w}_{1}\tan\frac{\varphi}{2}-\frac{\bm{p}_{\perp}}{p_{0}}\bm{w}^{2}_{2}\tan^{2}\frac{\varphi}{2}$
perpendicular to $\bm{w}_{2}$. In its own reference frame, $v=v^{\prime}=0$
even these transformations are absent,
$e^{b_{ok}o^{ok}}=e^{b^{\prime}_{ok}o^{ok}}=1$. In the case of a moving
particle, these boosts change the component of the momentum perpendicular to
$\bm{w}_{2}$ in such a way as to compensate for the change caused by the
rotation. As a result of two transformations - rotation and boost (or vice
versa) - the momentum remains unchanged, but the spin characteristics change.

In order to return from the small Wigner group to the spin group, it is
necessary in relations (26) - (36) to replace an arbitrary axis
$\bm{\mathfrak{w}}_{\alpha\beta}$ by the axis
$\bm{\mathfrak{s}}_{\alpha\beta}$ onto which the spin of the fixed fermion is
projected.

In the case of a fermion at rest, we see that the transformation of the spin
group is a spatial rotation of the system through an arbitrary angle $\varphi$
around the $\bm{\mathfrak{s}}$ axis, onto which the spin of the resting
fermion is projected. This transformation does not change the spin
characteristics and does not change the 4-momentum of the resting fermion
$(mc,0,0,0)$. When the parameter $\varphi$ changes continuously, the turn
becomes rotation. The representation of a fermion as a rotating particle
reflects this possibility of rotation of the fermion by an arbitrary angle
around the $\bm{\mathfrak{s}}$ axis without changing the properties of the
fermion.

In the case of a moving fermion, the transformation of the spin group, which
does not change the properties of the fermion, is a sequence of spatial
rotation and a boost coordinated with this rotation (or vice versa).

Spatial rotation is performed at an angle

$\displaystyle 2r=2\arctan\Bigl{(}|\bm{s}_{2}|\tan\frac{\varphi}{2}\Bigr{)}$
(36)

around the $\bm{s}_{2}$ axis, which is the $\bm{\mathfrak{s}}$ axis
transformed into a moving system. If the parameter $\varphi$ changes
continuously, instead of a turn, we obtain rotation around the axis onto which
the spin of the moving fermion is projected.

This turn, in the case of a moving fermion, is accompanied by boosts in the
direction
$\bm{s}_{1}\tan\frac{\varphi}{2}+\frac{\bm{p}_{\perp}}{p_{0}}(1+\bm{s}^{2}_{1})\tan^{2}\frac{\varphi}{2}$
or
$\bm{s}_{1}\tan\frac{\varphi}{2}-\frac{\bm{p}_{\perp}}{p_{0}}(1+\bm{s}^{2}_{1})\tan^{2}\frac{\varphi}{2}$,
perpendicular to $\bm{s}_{2}$. The components of these directions are
determined by the vector $\bm{s}_{1}$ perpendicular to $\bm{s}_{2}$ and the
component of the momentum perpendicular to $\bm{p}_{\perp}$. The speed of the
boosts is the same, $v=v^{\prime}$. It is determined from the equation

$\displaystyle\tanh 2\frac{v}{c}=\tanh
2\frac{v^{\prime}}{c}=\Bigl{|}\frac{\bm{s}_{1}\tan\frac{\varphi}{2}+\frac{\bm{p}_{\perp}}{p_{0}}(1+\bm{s}^{2}_{1})\tan^{2}\frac{\varphi}{2}}{1+(1+\bm{s}_{1}^{2})\tan^{2}\frac{\varphi}{2}}\Bigr{|}=$
$\displaystyle\Bigl{|}\frac{\bm{s}_{1}\tan\frac{\varphi}{2}-\frac{\bm{p}_{\perp}}{p_{0}}(1+\bm{s}^{2}_{1})\tan^{2}\frac{\varphi}{2}}{1+(1+\bm{s}_{1}^{2})\tan^{2}\frac{\varphi}{2}}\Bigr{|}.$
(37)

Since the boosts are changes in velocity, for a moving fermion the rotations
around the $\bm{s}_{2}$ axis are complemented by changes in velocity
perpendicular to this axis. The velocities $\bm{v}$ or $\bm{v}^{\prime}$ are
periodic functions of the parameter $\varphi$. Therefore, continuous changes
in the parameter $\varphi$, which do not change the properties of the fermion,
correspond to rotation around the $\bm{s}_{2}$ axis and the oscillating
motions associated with them, perpendicular to this axis $\bm{s}_{2}$.

As we see, in the depiction reflecting the symmetry of a moving fermion, it is
not only a rotating fermion, but also an oscillating one perpendicular to the
rotation axis. Of course, the depiction of a rotating particle is only an
image that helps to relate the existence of spin with the classical ideas of
angular momentum. The concept of an oscillating motion perpendicular to the
axis of rotation is a similar image. The question arises: the existence of
which physical characteristic reflects the image of the oscillating motion?

Let us turn to experiments that directly convince us of the existence of spin
in a charged particle. These are experiments similar to that of Stern-Gerlach
experiment Gerlach,Stern (1921), Gerlach,Stern (1922). That is, observing the
interaction of the spin-related intrinsic magnetic moment of a particle with
an external magnetic field. Since the directions of the spin and magnetic
moment are related to the direction of the component $\bm{s}_{2}$ of the spin
tensor $s_{\alpha\beta}$, it is this component that is observed in such
experiments.

In a similar way, the observation of the polarization of a moving charged
particle with spin in an electric field convinces us of the existence of an
electric moment of such a particle. This kind of polarization accompanies
spin-orbit interaction or electron scattering at the Coulomb center
Bjorken,Drell (1964). Usually, to explain it, the process is considered in the
particle’s own system. In this system, the particle is stationary, and the
external electric field generates an induction magnetic field, which interacts
with the magnetic moment of the particle at rest and polarizes it. If we
remain in the nuclear reference frame or Coulomb center system, in which there
is only an electric field, then the polarization can be explained only by the
appearance of an electric dipole moment in the moving particle. Normally, it
is generated by the magnetic moment of a stationary particle in the transition
from its own system to the system in which it moves. It is perpendicular to
the magnetic moment of the particle. It is similar to the generation of an
induction electric field in a system moving relative to the magnetic field.
Since the spin tensor $s_{\alpha\beta}$ and the electromagnetic field tensor
$F_{\alpha\beta}$ are transformed in the same way, these two ways of
explaining the polarization of a moving fermion in an electric field are
consistent and equivalent.

Returning to the oscillating motion of a moving charged particle with spin, we
can consider it as an image that helps us understand the appearance of an
electric moment in such a particle, just as the image of a rotating charged
particle does so with respect to the magnetic moment.

##  III Acknowledgements

The authors would like to thank Prof. Denis Kovalenko for stimulating
discussions.

##  IV Summary

In this way we have shown that there is a symmetry group of moving fermions
with spin $\frac{\hbar}{2}$. This is a spin group. The transformations of this
group do not change the fermion momentum and its spin characteristics. These
transformations are sequences of a rotation and a boost coordinated with it
(or vice versa). This spin group is a subgroup of the small Wigner group. The
spin projection operator
$\frac{\hbar}{2}\frac{is_{\alpha\beta}\sigma^{\alpha\beta}}{2}$ is the
infinitesimal operator of this spin group, that is, the symmetry group of the
moving fermion.

## References

  * Lubanski (1942) J. K. Lubanski, Phisica (in French) 9, 310 (1942).
  * Wigner (1939) E. P. Wigner, Ann. Math. 40, 149 (1939).
  * Gerlach,Stern (1921) W. Gerlach, O. Stern, Zeitschrift fur Physik 8, 110 (1921).
  * Gerlach,Stern (1922) W. Gerlach, O. Stern, Zeitschrift fur Physik 9, 349 (1922).
  * Bjorken,Drell (1964) J. D. Bjorken S. D. Drell, _Relativistic quantum mechanics_ (Mc-Graw Hill Book Company, 1964).