text:
\section*{Nomenclature}

\section{Introduction}

We present here a neo-classical relativistic mechanics model for an electron that extends Einstein's special relativity to include its spin as a natural part of its space-time path as a \emph{point} particle.
The theory for this model is built primarily on earlier work of {\color{Green} Barut and Zanghi (1984), Hestenes (1985,1990, 2010), Rivas (1989, 1994, 2001, 2003, 2008)} and
{\color{Green} Salesi (2002, 2005)}, with new connections between their work and new results, including a classical derivation of Dirac's wave equation for a free electron for the first time.

In the model, the space-time path of an electron of mass $m$ is the sum of the motion of an auxiliary point, the \emph{spin center} that describes the \emph{global} motion
corresponding to the bodily transport of the electron, plus an inherent \emph{local} circular motion with respect to a reference frame at the spin center.
This perpetual circular motion around the spin center is the source of the phenomenon described by {\color{Green} Schr\"{o}dinger (1930)} as \emph{zitterbewegung} (trembling motion) that he derived
from the free electron solution of Dirac's wave equation. It occurs at the speed of light $c$ with angular frequency $\omega_0 = 2mc^2 / \hbar$ and so has radius $r_0 = c/\omega_0 = \hbar/(2mc)$,
half of the reduced Compton wavelength, and an angular momentum about the spin center of $mcr_0 = \hbar / 2$.
It is also the source of the \emph{rest energy} $mc^2 =  \hbar \omega_0 /2$ of the electron, which is the kinetic energy of this local spin motion in its \emph{rest frame} fixed at the spin center so
that the global velocity of the electron (not its total velocity) is zero. Proper time $\tau$ is the time measured by a clock fixed in this rest frame.
Note that a reference frame attached to the electron itself could never be an inertial frame because the electron is always accelerating relative to any observer inertial frame.

In Section 2, we introduce the fundamental equations of motion for the electron for the neo-classical theory that are essentially those of {\color{Green} Rivas (2003)} but expressed
using proper time $\tau$ rather than observer time $t$.  All physical quantities are completely specified in a similar way to classical mechanics (hence the terminology ``neo-classical").
We show that the 4-vector equation for the \emph{total} (local plus global) motion of the electron is a fourth-order differential equation in proper time $\tau$
that is obtained by simply adding a spin energy term to the Lagrangian function from special relativity.
This equation of motion can be expressed as two coupled second-order differential equations, one for the position of the point electron and the other for its spin center.
The latter equation describing the sub-luminal motion of the spin center is just Newton's Second Law using the Lorentz force on the point charge.
The inertial resistance to acceleration of the electron is therefore associated only with its \emph{global} motion and there is no requirement for a centrifugal force to produce
the local inherent circular motion creating the spin. We show that a coupling between spin and the electromagnetic field arises naturally from the Lorentz force acting on the spin motion,
leading to a neo-classical energy equation with a spin-field interaction term that is absent in Einstein's special relativity.
The spin motion also creates an apparent wave action when viewed through a Lorentz transformation from an observer inertial reference frame.

In Section 3, we introduce a spin tensor representing the angular momentum from the total velocity about the spin center.
Using this spin tensor, the equations of motion can be re-written in a form that mimics the operator equations derived from the Heisenberg picture of the proper-time Dirac wave equation
given in {\color{Green} Barut (1987)}. The same equations have been derived by others using quite different approaches than starting with the fundamental neo-classical equations of motion
presented in  Section 2, as done here (e.g. {\color{Green} Weyssenhoff and Raabe (1947), Grossmann and Peres (1963), Barut and Zanghi (1984)}, and {\color{Green} Salesi (2002)}).
Using the spin tensor, the spin-field interaction term in the neo-classical energy equation can be expressed as the sum of two energy terms due to magnetic and electric dipole moments,
although the implied magnetic moment is one half of that in Dirac's theory.

In Section 4, a neo-classical four-dimensional complex \emph{state function} is introduced that provides through the usual Hermitian inner products
with self-adjoint linear operators the \emph{actual} dynamic quantities (not their expected values) as a function of proper time.
The evolution of this state function is described by a first-order differential equation with respect to proper time given by the spinor equation in the theory of
{\color{Green} Barut and Zhangi (1984)}. This equation is similar to the Feynman proper-time parameterization of the Dirac-Schr\"{o}dinger equation in QM (quantum mechanics).
We show that the spin tensor form of the equations of motion can be expressed in operator form by using the Dirac velocity and spin tensor operators that act on the spinor state function.

Also in Section 4, we show that in the rest frame for a free electron, a superposition of the spin-up and spin-down neo-classical state functions produces the state function for spin about an axis whose
direction is specified by the superposition coefficients. The implied velocities from the Dirac velocity operators (the Dirac matrices) are shown to lie in a plane perpendicular to the spin axis and agree with the basic hypothesis
that the electron's spin motion is circular at the speed of light $c$ with  angular frequency $\omega_0 = 2mc^2 /\hbar$.
These spin state functions are in turn superpositions of the so-called ``positive energy'' and ``negative energy'' solutions, which actually both have positive energy; the latter are just the conjugate solutions needed
to properly produce the spin motion under the action of the velocity operators, commonly described in QM as due to ``interference'' with the ``positive energy'' solutions.

Finally in Section 4, we show by using a Lorentz transformation for proper time that the state function solutions of the neo-classical Dirac- \linebreak Schr\"{o}dinger equation for a free electron
become neo-classical \emph{wave functions} that satisfy Dirac's wave equation where the independent variables are the 4-vector position coordinates with respect to an observer inertial reference frame.
Thus, for the first time, Dirac's wave equation is derived directly from a modified relativistic mechanics theory that incorporates spin.
These wave functions exhibit wave-like behavior with a frequency and wavenumber proportional to the electron's energy and global kinetic momentum as in De Broglie's theory.
However, minimal coupling in the neo-classical Lagrangian function does not produce the same field coupling implied by the usual QM minimal coupling prescription in Dirac's equation
since it produces one half of the magnetic moment for the electron compared with Dirac's derivation.
Although the present neo-classical theory provides a consistent hidden-variable model for Dirac's relativistic wave equation for the free electron, it therefore needs further development
for the coupling between an electron and an electro-magnetic field.

 In Section 5, some remarks are made about introducing probability into the neo-classical theory since its wave functions currently have no probability implications, unlike in QM theory.
Following concluding remarks in Section 6, three appendices give some details of a few results stated without proof in the main sections.

\section{Neo-Classical Spin Model and its Fundamental Equations}

In this section, the equations of motion for an electron in an em-field that describe the neo-classical model for spin are presented, along with an explicit solution of
these equations for the free electron showing that its dynamics naturally exhibits spin through its space-time motion that explains Schr\"{o}dinger's zitterbewegung.
We derive two important energy equations and then show that the equations of motion can also be derived by adding a term for the spin kinetic energy to the usual classical relativistic Lagrangian function for the electron.

\subsection{Equations of Motion}

Using 4-vectors relative to an observer inertial reference frame $X_o$ with its origin at $O$, the \emph{total} motion $x(\tau)$ of the electron of mass $m$ and charge $q = -e$ is modeled as the sum of
(i) a \emph{local} spin motion $z(\tau)$ where the electron moves in a closed path relative to its spin center $C$, and
(ii) a \emph{global} motion $y(\tau)$ of the electron corresponding to the motion of this spin center.
In the \emph{rest frame} for the global motion, which corresponds to a reference frame $X_c$ fixed at $C$ (not inertial if there is an electromagnetic field present),  the total motion is just the spin motion where a free electron moves
at the speed of light $c$ in a circle of radius $r_0 = \hbar/(2mc) \sim 1.93 \times 10^{-13}$m so that the spin angular momentum of the electron is $r_0mc = \hbar/2 = h/(4\pi)$ where $h$ denotes Planck's constant.
The angular frequency of the circular spin motion is $\omega_0 = c/r_0 = 2mc^2 / \hbar \sim 1.55 \times 10^{21}$s$^{-1}$ and its kinetic energy is $mc^2 = \frac{1}{2} \hbar \omega_0$, the rest energy of Einstein's special relativity.
No centrifugal force is required for the circular motion, which is an inherent perpetual motion, implying that the electron's mass is not directly an inertial mass.
Instead, the inertia to moving the electron comes from accelerating the  \emph{global} motion, $y(\tau)$, of the spin center $C$ (also called the \emph{center of mass} for this reason),
which is described by Newton's Second Law using \emph{proper time} $\tau$, the time for a clock fixed at $C$.
Clearly, an electron must only emit and absorb electromagnetic energy when the global motion is accelerating.
Furthermore, the spin motion, $z(\tau)$, cannot be observed directly because it is much too fast and so the location of the electron cannot be known more accurately than a position vector error
of magnitude $r_0 = \hbar/(2mc)$. The motion $z(\tau)$ is essentially a hidden variable but it has consequences that are observable, such as the electron's magnetic moment.

The equations of motion for the four components of the total and global motions, $x(\tau)$ and $y(\tau)$, are therefore:
\begin{align}\nonumber
	\ddot{x}^{\mu} & = - \omega_0^2 \left( x^{\mu} - y^{\mu} \right) \  \\ \label{Eq_1_1}
	\ddot{y}^{\mu} & = \frac{1}{m} f^{\mu} = \frac{q}{m} F^{\mu \nu} (x) \dot{x}_{\nu}
\end{align}
\noindent 	Here $F(x)$ is the em-field (electromagnetic field) tensor at the electron's space-time coordinates $x$, an overhead dot denotes a derivative with respect to proper time $\tau$, and
$x^{\mu} (\tau)$ and $y^{\mu} (\tau)$ ($\mu = 0,1,2,3$) are the contravariant components of the 4-vector position coordinates $x = (ct,\mathbf{x})$ for the total motion
and $y = (ct_y,\mathbf{y})$ for the spin center motion relative to the inertial reference frame $X_o$ at $O$.
The local spin motion $z = (ct_z,\mathbf{z})$ about the spin center $C$ relative to $X_o$ then has the contravariant components $z^{\mu} = x^{\mu} - y^{\mu}$.
Throughout the paper, bold letters are 3-vectors with ${\bf a} \cdot {\bf b}$ and ${\bf a} \times {\bf b}$ denoting the scalar and vector products (${\bf a}^2 = {\bf a} \cdot {\bf a}$ ) and
Einstein's summation convention for repeated indices is used. The Minkowski metric tensor is $G = \diag(1,-1,-1,-1)$ with components $g_{\mu\nu} = g^{\mu\nu}$, and
the space-time inner product of $y$ and $z$ is denoted $y*z = y_{\mu} g^{\mu\nu} z_{\nu} = y_{\mu} z^{\mu} = y^0z^0 - y^1z^1 - y^2z^2 - y^3z^3$.

Using the usual expression for the em-field tensor $F$, the Lorentz force $f=(f^0,{\bf f})$ in Eqs.~\eqref{Eq_1_1} can be expressed as:
\begin{equation} \label{Eq_1_1b}
	f^0 = \frac{q}{c} \mathbf{E} \cdot {\bf \dot{x}} \text{  and  } {\bf f} = q \left( \dot{t} \mathbf{E} + {\bf \dot{x}} \times  \mathbf{B} \right)
\end{equation}
\noindent where $ \mathbf{E} (x)$ and $ \mathbf{B} (x)$ are the electric and magnetic fields and SI units are used.
Introducing the 4-vector \emph{global} momentum $\pi = m \dot{y} = \left( \frac{E}{c}, \mathbf{P} \right)$, the last equation in Eqs.~\eqref{Eq_1_1} can be written as $\dot{\pi} = f$ and so:
\begin{equation} \label{Eq_1_1c}
	\dot{E} = c f^0 = q \mathbf{E} \cdot {\bf \dot{x}} \text{  and  } \dot{\mathbf{P}} = {\bf f} = q \left( \dot{t} \mathbf{E} + \dot{\mathbf{x}} \times  \mathbf{B} \right)
\end{equation}
\noindent implying that $\dot{E} = \dot{\mathbf{P}} \cdot \dot{\mathbf{x}}/ \dot{t} = {\bf f}  \cdot \frac{d \mathbf{x}}{dt}$, so the proper time rate of energy change is equal to the observed rate of work done by the em-field on the electron, as in special relativity.

The equations of motion in Eqs.~\eqref{Eq_1_1}, which are remarkably simple when expressed using proper time $\tau$, require specification of 16 initial conditions: $x(0)$, $\dot{x}(0)$, $y(0)$, $\dot{y}(0)$, for a unique solution.
In addition, there are two constraints:
\begin{align}\nonumber
	C1: \dot{x}*\dot{x} & = \dot{x}_{\mu} \dot{x}^{\mu} = 0 \  \\ \label{C1_C2}
	C2: \ddot{x}*\ddot{x} & = \ddot{x}_{\mu} \ddot{x}^{\mu} = - c^2 \omega_0^2
\end{align}
\noindent 	Constraint $C1$ specifies that the electron always moves at the speed of light $c$ while constraint $C2$ reflects that it travels in a circle of radius $r_0 = c/ \omega_0$ relative to the rest frame at $C$ with
angular frequency $\omega_0$. Under Eqs.~\eqref{Eq_1_1}, $C2$ can be expressed as $z*z = - r_0^2$.
It is hypothesized that the motion of all fundamental particles with spin satisfy these two constraints with an appropriate choice of $\omega_0$.

The equations of motion have several interesting features.
The first equation expresses the kinematics for the electron's acceleration in terms of what the inertial observer fixed at O sees of the circular motion in the rest-frame fixed at the spin center $C$
where it is simply $\ddot{\mathbf{r}} = - \omega_0^2 \mathbf{r}$ where $\mathbf{r}$ is the position vector of the electron. This observer sees a distortion of the circular spin motion due to the motion $\mathbf{y}$ of the spin center.
The equation of motion denotes an oscillatory motion with an ultra-high frequency $\omega_0 \sim 10^{21}$s$^{-1}$ representing the phenomenon of zitterbewegung revealed by
{\color{Green}Schr\"{o}dinger (1930)} in his analysis of Dirac's wave equation. He showed that for a free electron, the ``positive'' and ``negative'' energy solutions of the wave function ``interfere''
to produce a motion with the high frequency $\omega_0 = 2 mc^2 / \hbar$ (twice De Broglie's frequency) and a spatial extent of $r_0 = c/\omega_0 = \hbar/ (2mc)$  related to Compton's wavelength.
We will show that the equations in Eqs.~\eqref{Eq_1_1} ultimately imply Dirac's wave equation, giving it a derivation from a classical relativistic mechanics theory for the first time.

In the second equation of Eq.~\eqref{Eq_1_1}, Newton's Second Law describes the sub-luminal motion of the spin center that gives the \emph{global} motion of the electron
where the Lorentz force $f$ due to the em-field is evaluated at the position $x$ of the charge, as might be expected.
The \emph{total} (local plus global) motion $x(\tau)$ of the electron is described by an equivalent single fourth-order differential equation given later,
which implies that the local circular motion is inherent and so there is no requirement for a centrifugal force to produce it.
The equations reveal that the inertia is not a property of the mass per se but instead it is associated with resistance to accelerating the ``ring'' of the electron's spin motion.
The spin center $C$ therefore acts like a center of mass, the terminology used by others (e.g. {\color{Green}Rivas (2001)} and {\color{Green}Salesi (2002)}).

In his book, {\color{Green}Rivas (2001)} derives the equations of motion in Eqs.~\eqref{Eq_1_1} from a Lagrangian function and extensively analyzes them for the non-relativistic case where the overhead dot
corresponds to a derivative with respect to observer time $t$, not our proper time $\tau$.
Actually, in this non-relativistic case, the Galileo transformation replaces the Lorentz one so the time parts ($\mu = 0$) of Eqs.~\eqref{Eq_1_1} are not needed because then $t = t_y = \tau$ in the theory presented in this paper.
Rivas shows that the equations of motion exhibit the electron's spin, zitterbewegung and magnetic moment (but with a gyroscopic factor $g=1$), as well as the ability to jump a potential barrier that a classical particle without spin could not do.
He also studies the case of an electron in a uniform magnetic field $\mathbf{B}$ and finds that the orbital motion exhibits a circular frequency that is essentially the expected cyclotron frequency $\omega_c = eB/m$ but that
the precession of the spin has the Larmor frequency $\frac{1}{2} \omega_c$. His equations for the motion are the same as the spatial part of Eqs.~\eqref{Eq_1_1} except that $t$ replaces $\tau$, so his results are applicable to the
relativistic spin model presented here. In QM theory, the spin precession frequency is $ \omega_c$ (see, for example, {\color{Green}Barut and Bracken (1982)} who solve the Heisenberg equations corresponding to Dirac's equation).
In the non-relativistic case of {\color{Green}Rivas (2001)}, as in the relativistic spin model here, the spin-field coupling comes from the Lorentz force alone and the magnetic dipole moment is produced by the path of the electron's spin motion.
We will see later that a rigorous derivation of the magnetic moment from a new energy equation does indeed produce a gyroscopic factor $g=1$ rather than the value $g=2$ of Dirac's theory.

For the relativistic case, {\color{Green}Rivas (2001)} investigates various Lagrangian functions, all having a similar structure but he again uses observer time $t$ rather than proper time $\tau$ used in this paper.
One special case that is similar to the relativistic spin equations presented in this paper corresponds to the electron moving at the speed of light (called the lumen case) where the equations of motion derived from the
Lagrangian function exhibit similar behavior to his non-relativistic case.

\subsection{Free Electron Motion}

In the case of a free electron, the equations in Eqs.~\eqref{Eq_1_1} become $\ddot{z} = - \omega_0^2 z$ and $\ddot{y} = 0$. Therefore, $\dot{y}$ = constant and the 4-vector spin motion $z$ satisfies:
\begin{align} \nonumber
	z(\tau)  & = z(0) \cos (\omega_0 \tau) + \left( \dot{z}(0) / \omega_0 \right) \sin (\omega_0 \tau) \\ \label{Eq_1_3}
	\dot{z}(\tau)  & = - \left( z(0) \omega_0 \right) \sin (\omega_0 \tau) + \dot{z}(0) \cos (\omega_0 \tau)
\end{align}

The 4-vector global momentum $\pi = m \dot{y} = \left( \frac{E}{c}, \mathbf{P} \right)$ is constant in this free electron case and so $y(\tau) = (\pi/m) \tau + y(0)$.
Using Eqs.~\eqref{Eq_1_1} and~\eqref{Eq_1_3}, the total 4-vector velocity and 4-vector position can be written as:
\begin{align} \nonumber
	\dot{x}(\tau) & = \dot{y}(\tau) + \dot{z}(\tau)  = \frac{1}{m} \pi + \left[ \dot{x}(0) - \frac{1}{m} \pi \right] \cos (\omega_0 \tau) + \frac{1}{\omega_0} \ddot{x}(0)  \sin (\omega_0 \tau)  \\ \label{Eq_1_4}
	x(\tau) & = y(0) + \frac{1}{m} \pi \tau + \frac{1}{\omega_0} \left[ \dot{x}(0) - \frac{1}{m} \pi \right] \sin (\omega_0 \tau) + \left[ x(0) - y(0) \right] \cos (\omega_0 \tau)
\end{align}
\noindent These solutions show that the electron's free motion in space is a helix where the circling at the speed of light and angular frequency $\omega_0$ corresponds to Schr\"{o}dinger's zitterbewegung
and represents the local spin contribution $z(\tau)$ while the global velocity $\dot{y} = \frac{1}{m}\pi$ is given in terms of the electron's global momentum.
Since $x^0(\tau) = ct$, the last equation in Eq.~\eqref{Eq_1_4} shows that $t$ is a non-linear function of proper time $\tau$ along the electron's space-time path.

The electron's kinetic energy $E = c \pi_0 = m c \dot{y}_0 = mc^2  \dot{t}_y = mc^2 \gamma$, defining $\gamma = \dot{t}_y$.
The global momentum components for $\mu \neq 0$ are $P_{\mu} =  \pi_{\mu} = m \dot{y}_{\mu} = m \frac{dy_{\mu}}{dt_y} \frac{dt_y}{d\tau} = \gamma m V_{\mu}$ where $V_{\mu} = \frac{dy_{\mu}}{dt_y}$,
which becomes in 3-vector notation, $\mathbf{P} =  \gamma m \mathbf{V}$. These results also show that $\dot{y} = \left(\gamma c, \gamma \mathbf{V} \right)$.
Note that $\mathbf{V}$ is the velocity of the electron's global motion (the motion of its spin center) relative to an observer fixed with respect to the inertial reference frame $X_o$.
For a free electron where $\pi$ is constant, $\gamma$ and $\mathbf{V}$ are constant.
The factor $\gamma m$ is sometimes viewed as an effective relativistic mass, although $\gamma$ is due to the different time scales, $t_y$ and $\tau$, at the center of spin $C$ when relative to
observers fixed in the inertial reference frames $X_o$ and $X_c$, respectively.

The inner product $y*z$ is invariant under a Lorentz transformation between inertial reference frames and so it is independent of the choice of inertial reference frame used to express the 4-vectors.
Consider the inertial reference frame $X_c$ with its origin at the spin center $C$, then $x'=(c\tau,\mathbf{r})$ with $y'=(c\tau,\mathbf{0})$ and $z'=(0,\mathbf{r})$ the time-like and space-like components of $x'$, respectively.
Thus, $\dot{y}*\dot{y} = \dot{y'}*\dot{y'} = (c,\mathbf{0})*(c,\mathbf{0}) = c^2$. Substituting for $\dot{y}$ gives $\gamma = \left[ 1 - \mathbf{V}^2/c^2 \right]^{-1/2}$,
which corresponds to the well-known Lorentz transformation factor between the inertial reference frames $X_o$ and $X_c$.
This is not always the case, however, because $\gamma$, which denotes $\dot{t}_y$ in this paper, gets modified by a factor when an em-field is present, as shown in the next subsection.

These results also give the usual relativistic energy equation $m^2c^2 = m\dot{y}*m\dot{y} = \pi*\pi = E^2/c^2 - \mathbf{P}^2$ and an additional one that is a feature of the spin model,
$\pi*\dot{x} = m \left( \dot{y}*\dot{y} + \dot{y}*\dot{z} \right) = m \dot{y}*\dot{y} = mc^2$ where we have used the invariance of the inner product $\dot{y}*\dot{z} = \dot{y'}*\dot{z'} = (c,\mathbf{0})*(0,\dot{\mathbf{r}}) = 0$.
As shown in the next subsection, the second energy equation also holds in an em-field but the first energy equation requires another term due to the coupling between the spin and the em-field.

Using the inner product invariance between reference frames $X_o$ and $X_c$ again to evaluate the motion constraints in Eq.~\eqref{C1_C2}, we get:
\begin{align}\nonumber
	C1: & ~0 = \dot{x}*\dot{x} = \dot{x}' *\dot{x}' = (c,\dot{\mathbf{r}})*(c,\dot{\mathbf{r}}) = c^2 - \dot{\mathbf{r}}^2 \  \\ \label{C1_C2b}
	C2: & - c^2 \omega_0^2 = \ddot{x}*\ddot{x} = \omega_0^4 z*z =  \omega_0^4 z'*z' = \omega_0^4 (0,\mathbf{r})*(0,\mathbf{r}) = - \omega_0^4 \mathbf{r}^2
\end{align}
\noindent  We see that the two constraints therefore imply $\dot{\mathbf{r}}^2 = c^2$ and $\mathbf{r}^2 =  r_0^2$,
giving the consistent result that the circular motion of the free electron about the spin center $C$, relative to the rest frame $X_c$, has speed $c$ and radius $r_0 = c/\omega_0$, respectively.
If this circular motion lies in the $r^1-r^2$ plane, then the spatial part of one solution in Eqs.~\eqref{Eq_1_3} is $\mathbf{r}(\tau) = \left( r_0 \cos (\omega_0 \tau + \varphi_0), r_0 \sin (\omega_0 \tau + \varphi_0), 0 \right)$ and
$\dot{\mathbf{r}}(\tau) = \left( -c \sin (\omega_0 \tau + \varphi_0), c \cos (\omega_0 \tau + \varphi_0), 0 \right)$ where $\varphi_0$ is the initial phase angle specifying the electron's position on the circle $\mathbf{r}^2 = r_0^2$ at $\tau = 0$.
The spin vector due to this circular motion is $\mathbf{s} = \mathbf{r} \times m \dot{\mathbf{r}} = \left( 0, 0, r_0 mc \right) = \left( 0, 0, \hbar/2 \right)$, which has magnitude $\hbar/2$ and its direction is orthogonal to the plane of circular motion.
Therefore, this corresponds to the electron being in a spin-up state in the $x^3$ direction.
Similarly, $\mathbf{r}(\tau) = \left( r_0 \cos (\omega_0 \tau - \varphi_0), -r_0 \sin (\omega_0 \tau - \varphi_0), 0 \right)$ and $\dot{\mathbf{r}}(\tau) = \left( -c \sin (\omega_0 \tau - \varphi_0), -c \cos (\omega_0 \tau - \varphi_0), 0 \right)$
is another solution with spin vector $\mathbf{s} = \mathbf{r} \times m \dot{\mathbf{r}} = \left( 0, 0, - r_0 mc \right) = \left( 0, 0, - \hbar/2 \right)$, which corresponds to the electron being in a spin-down state.

The Lorentz transformation between the reference frames $X_o$ and $X_c$ gives alternative but equivalent expressions to those in Eqs.~\eqref{Eq_1_3} and~\eqref{Eq_1_4} for
the time and space components (for convenience, assume that the direction of the spatial axes of $X_o$ and $X_c$ are aligned, so no rotation matrix is needed):
\begin{align} \nonumber
	 t(\tau) & =  \left[ \frac{\gamma}{c^2} \left(\mathbf{V} \cdot \mathbf{r}(\tau) \right) \right] + \left[ \gamma \tau + a \right] = t_z(\tau) + t_y(\tau) =  \frac{1}{c} z^0(\tau) + \frac{1}{c} y^0(\tau)
	 \\ \label{Eq_1_4b}
	\mathbf{x}(\tau) & = \left[ \mathbf{r}(\tau) + \frac{\gamma^2}{(1+\gamma)c^2} \left(\mathbf{V} \cdot \mathbf{r}(\tau) \right) \mathbf{V} \right] + \left[ \gamma \tau \mathbf{V} + \mathbf{b} \right] = \mathbf{z}(\tau) + \mathbf{y}(\tau)
\end{align}
\noindent where $\ddot{\mathbf{r}} = - \omega_0^2 \mathbf{r}$ so that $\mathbf{r}(\tau) = \mathbf{r}(0) \cos (\omega_0 \tau) + \left( \dot{\mathbf{r}}(0) / \omega_0 \right) \sin (\omega_0 \tau)$,
while $a = t_y(0)$ and $\mathbf{b} = \mathbf{y}(0)$ are constants.
This implies that $\dot{t}(\tau) = \gamma \left( 1 + \frac{1}{c^2} \mathbf{V} \cdot \dot{\mathbf{r}}(\tau) \right) > 0$ since $ | \mathbf{V} \cdot \dot{\mathbf{r}}(\tau) | <c^2$,
so $t$ always increases monotonically with $\tau$, despite its harmonic variation $t_z(\tau)$ with frequency $\omega_0$ about the linear trend $ t_y(\tau) = \gamma \tau + a $.

Consider the inverse Lorentz transformation relating the electron's coordinates $(c \tau, \mathbf{r})$ using the reference frame $X_c$ to the coordinates $(ct, \mathbf{x})$ using the reference frame $X_o$:
\begin{equation}\label{Eq_1_5}
	d\tau = \gamma dt - \frac{\gamma}{c^2} \mathbf{V} \cdot d\mathbf{x}
\end{equation}
\begin{equation}
	d\mathbf{r} = d\mathbf{x} - \gamma \mathbf{V} dt + \frac{\gamma^2}{c^2(1+ \gamma)} (\mathbf{V} \cdot\ d\mathbf{x}) \mathbf{V}
\end{equation}
\noindent The first equation can be rewritten using the previous expressions derived for the energy and 3-vector momentum, $E = \gamma mc^2$ and $\mathbf{P} =  \gamma m \mathbf{V}$, as:
\begin{equation}\label{Eq_1_6}
	mc^2d\tau = Edt - \mathbf{P}\ \cdot d\mathbf{x} = \pi*dx
\end{equation}
\noindent This equation implies the previously derived energy equation:
\begin{equation}\label{Eq_1_7}
	C3: E\dot{t} - \mathbf{P} \cdot \dot{\mathbf{x}} = \pi*\dot{x} = mc^2
\end{equation}
\noindent We consider this equation to be a constraint; however, only two of the constraints $C1$-$C3$ are independent, as shown in the next subsection.

Since $\pi$ is constant for the free electron case, the Lorentz transformation in Eq.~\eqref{Eq_1_6} can be integrated to give
$mc^2 \tau = \pi*x = Et - \mathbf{P} \cdot \mathbf{x}$ (take $\tau = 0$ when $t = 0$ at $\mathbf{x}=\mathbf{0}$), or, equivalently,
\begin{equation}\label{Eq_1_10b}
	\omega_0 \tau = \left(\pi*x \right)/h^* = \left(Et - \mathbf{P} \cdot \mathbf{x} \right)/h^* = \omega t - \mathbf{k} \cdot \mathbf{x} = \gamma \omega_0 \left( t - \frac{1}{c^2} \mathbf{V} \cdot \mathbf{x} \right)
\end{equation}
\noindent where $h^* = \hbar/2$, $\omega = E/h^* = \gamma \omega_0$ and $\mathbf{k} = \mathbf{P}/h^* = \gamma \frac{\omega_0}{c^2} \mathbf{V}$,
giving the well-known De Broglie frequency-energy and wavenumber-momentum relations except for a factor of two.
Thus, the inherent circular motion with phase $\omega_0 \tau$ in Eqs.~\eqref{Eq_1_3} and~\eqref{Eq_1_4} gives an apparent plane-wave characteristic to the electron's motion
with De Broglie's apparent wave propagation speed of $\frac{c^2}{V} > c$ when viewed by an observer fixed with respect to the reference frame $X_o$.

\subsection{Energy Equations}

Consider now the general case of an electron in an em-field. Under $C1$ and Eqs.~\eqref{Eq_1_1}, the energy constraint $C3$ holds if and only if $C2$ holds, which can be shown as follows.
Let $u=\dot{x}$ and differentiate $C1$, then $0 = u*\dot{u} = - \omega_0^2 u*z$. Then using the derivative of $u*z = 0$:
\begin{equation}\label{Eq_1_7b}
	- \frac{1}{\omega_0^2} \dot{u}*\dot{u}  = \dot{u}*z = -u*\dot{z} = -u*(u - \dot{y}) = u*(\pi/m)
\end{equation}
\noindent by using $C1$ again. Therefore, $\pi*u = mc^2$ if and only if $\dot{u}*\dot{u} = - c^2 \omega_0^2$, proving that $C2$ and $C3$ are equivalent constraints.

From $C3$: $0 = \dot{\pi}*\dot{x} + \pi*\ddot{x} = -\omega_0^2\pi*z$ by using Eqs.~\eqref{Eq_1_1} and the anti-symmetry of the tensor
$F^{\mu\nu}$ so that $ \dot{\pi}*\dot{x} = m\ddot{y}*\dot{x}  = q F^{\mu\nu} \dot{x}_{\mu} \dot{x}_{\nu} = 0$. Thus,
\begin{equation}\label{Eq_1_8}
	\pi*z = 0
\end{equation}
\noindent Differentiating: $0 = f*z + \pi*\dot{z}$.
Using C3 again, $\pi*\dot{z} = \pi*\dot{x} - \pi*\dot{y} = mc^2 - \frac{1}{m} \pi*\pi$, giving the \emph{energy equation} that generalizes the previous free electron case:
\begin{equation}\label{Eq_1_9}
	\frac{1}{m} \left( E^2/c^2 - \mathbf{P}^2 \right) = \frac{1}{m} \pi*\pi = mc^2 - \Phi
\end{equation}
\noindent where $\Phi = \pi*\dot{z} = -f*z$ is a spin-field interaction energy. As shown later, this term comes from the electric and magnetic dipole moments set up by the spin motion of the electron.
Eq.~\eqref{Eq_1_9} shows that the kinetic energy $E = c \pi_0$ comes from the ``rest" kinetic energy $mc^2$ and the energy $\Phi$ from the Lorentz force, both energies coming from the spin motion,
plus the kinetic energy $\frac{1}{m} \mathbf{P} \cdot \mathbf{P}$ from the global momentum.

\subsubsection{Flywheel analogy for kinetic energy storage}

The kinetic energy of a free electron can be viewed as being stored in the spin kinetic energy as if it were a flywheel.
For a larger global velocity of the electron relative to one inertial observer compared with another, the spin of the electron
about its center C will appear to be faster.
The kinetic energy associated with this spin is equal to the sum of the rest energy $mc^2$ and the translational kinetic energy due to the global velocity of the electron.
Quantitatively, we have from the previous subsection that the apparent spin frequency of the electron's local motion for an observer fixed relative to the inertial reference frame $X_o$ is given by
$\omega = \gamma \omega_0$ where $\omega_0$ is the spin frequency for an observer fixed relative to the electron (i.e. relative to the inertial reference frame $X_c$).
The kinetic energy corresponding to the spin frequency $\omega$ is $E = h^* \omega = \gamma h^* \omega_0 = \gamma mc^2 \approx mc^2 + \frac{m}{2} \mathbf{V}^2$ to first order in the small quantity
$\frac{1}{c^2} \mathbf{V}^2$ for a non-relativistic approximation.

In the case of an electron in an external em-field, by substituting $E = \gamma mc^2$ and $\mathbf{P} =  \gamma m \mathbf{V}$ into the energy equation in Eq.~\eqref{Eq_1_9},
which includes the energy term $\Phi$ from the interaction between the spin and the external field, we get:
\begin{equation}\label{Eq_1_10c}
	\gamma^2 = \left( 1 - \frac{1}{mc^2} \Phi \right) / \left( 1 - \frac{1}{c^2} \mathbf{V}^2 \right)
\end{equation}
\noindent The spin frequency  $\omega = \gamma \omega_0$ is therefore affected
by the spin-field interaction in such a way that the total kinetic energy $E = \gamma mc^2 = h^* \omega$ also includes a contribution from the interaction energy $\Phi$.
The hypothesis that the spin-field interaction affects the time dilation $\gamma$ has been proposed by {\color{Green}van Holten (1992)}, although not explicitly through the flywheel analogy given here.

\subsection{Lagrangian Function for Spin Model}

Consider the Lagrangian function:
\begin{equation}\label{Eq_1_11}
	L = \frac{m} {2} u^{\mu} u_{\mu} + q A^{\mu}(x) u_{\mu} - \frac{m} {2 \omega_0^2} \dot{u}^{\mu} \dot{u}_{\mu}
\end{equation}
\noindent where $u = \dot{x}$ and $A = \left( \frac{V}{c}, \mathbf{A} \right)$ is the 4-vector potential for the em-field.
This Lagrangian function is just the classical relativistic one with minimal coupling to the em-field (e.g. {\color{Green}Goldstein (1959)}) except for the additional last term, which is the contribution from the spin kinetic energy.
For a free electron, $L$ is just the scaled sum of the two constraints $C1$ and $C2$ introduced in Eq.~\eqref{C1_C2}, where $C2$ implies that the last term in $L$ is just $\frac{m} {2} c^2$.
For the classical relativistic Lagrangian function, $u = \dot{y}$ and it is the first term that is $\frac{m} {2} c^2$ whereas in $L$, the first term is zero by $C1$.

The four Euler-Lagrange equations for stationarity of the action integral involving $L$ are:
\begin{equation}\label{Eq_1_12}
	\frac{d^2}{d \tau^2} \left(\frac{\partial L}{\partial \dot{u}_{\mu}} \right) - \frac{d}{d \tau} \left(\frac{\partial L}{\partial u_{\mu}} \right) + \frac{\partial L}{\partial x_{\mu}} = 0
\end{equation}
\noindent This produces a fourth-order equation for each of the four components of the total motion:
\begin{equation}\label{Eq_1_2}
	\ddddot{x\hspace{0pt}}^{\mu} + \omega_0^2 \ddot{x}^{\mu} - \frac{q \omega_0^2}{m} F^{\mu\nu} (x) \dot{x}_{\nu} = 0
\end{equation}
\noindent 	For complete specification of the solution, these equations of motion require the 16 initial conditions: $x(0)$, $\dot{x}(0)$, $\ddot{x}(0)$, $\dddot{x\hspace{0pt}}(0)$.

The pair of second-order equations of motion in Eq.~\eqref{Eq_1_1} clearly imply the fourth-order equation in Eq.~\eqref{Eq_1_2} (just differentiate the first equation twice with respect to proper time).
The converse is also seen to hold by setting $z = - \frac{1} {\omega_0^2}\ddot{x}$ and $y = x-z$, which gives immediately the first equation of Eq.~\eqref{Eq_1_1} and
then the second equation follows by using $\ddot{y} = \ddot{x}-\ddot{z}$ in Eq.~\eqref{Eq_1_2}.

The Lagrangian function $L$ in Eq.~\eqref{Eq_1_11} is considered by {\color{Green}Riewe (1972)} for the free electron to derive a Hamiltonian function with spin that he used to perform a canonical quantization
but the resulting equation was not directly related to Dirac's equation for the electron's wave function.
{\color{Green}Salesi (2002)} also presents $L$ as  a special case of a more general Lagrangian function for a charged particle in an em-field defined by a scalar potential.
He remarks that Eq.~\eqref{Eq_1_11} is of special interest for what he called a Classical Dirac particle. His Hamiltonian function for this case is equivalent to that of {\color{Green}Riewe (1972)}.

\section{Spin Tensor Form of Equations of Spin Model}

\subsection{Spin Tensor Definition and Associated Equations}

We introduce a spin 4-tensor $S$ to describe the angular momentum of the \emph{total} momentum $mu = m \dot{x}$ of the electron about the spin center $C$ by defining its 16 components for $\mu , \nu = 0,1,2,3$ as:
\begin{equation}\label{Eq_1_13}
	S^{\mu\nu} = - m \left( z^{\mu}u^{\nu} - z^{\nu}u^{\mu} \right)
\end{equation}
\noindent By using Eq.~\eqref{Eq_1_1}, this new expression is equivalent to the definition given by {\color{Green}Salesi (2002)} who used invariance of the Lagrangian function $L$
in Eq.~\eqref{Eq_1_11} with $A = 0$ under infinitesimal 4-rotations to derive the spin tensor as:
\begin{equation}\label{Eq_1_13b}
	S^{\mu\nu} = \frac{m} {\omega_0^2} \left( \dot{u}^{\mu}u^{\nu} - \dot{u}^{\nu}u^{\mu} \right)
\end{equation}
\noindent From  Eq.~\eqref{Eq_1_13}, the proper time derivative of the spin is $\dot{S}^{\mu \nu} = -m \left( \dot{z}^{\mu} u^{\nu} - u^{\mu} \dot{z}^{\nu} \right)$ because
$z^{\mu} \dot{u}^{\nu} = \dot{u}^{\mu} z^{\nu}$ from Eq.~\eqref{Eq_1_1}. Thus,
\begin{equation} \label{Eq_1_20}
	\dot{S}^{\mu \nu} = - \left[ \left( m u^{\mu} - \pi^{\mu} \right) u^{\nu} -  \left( m u^{\nu} - \pi^{\nu} \right) u^{\mu} \right] =  \pi^{\mu} u^{\nu} - \pi^{\nu} u^{\mu}
\end{equation}
\noindent If we take the tensor for the orbital angular momentum about origin O, $L^{\mu\nu} = x^{\mu}\pi^{\nu} - x^{\nu}\pi^{\mu}$, then:
\begin{equation}\label{Eq_1_20b}
	\dot{L}^{\mu\nu} + \dot{S}^{\mu\nu} = x^{\mu}f^{\nu} - x^{\nu}f^{\mu} = M^{\mu\nu}
\end{equation}
where $M^{\mu\nu}$ is the moment tensor about O of the em-force on the electron; in particular, the total angular momentum $J^{\mu\nu} = L^{\mu\nu} + S^{\mu\nu}$ is conserved for a free electron.

Since $S$ is anti-symmetric ($S^{\nu\mu} = - S^{\mu\nu}$), the four diagonal components are zero and of the remaining 12 components, only 6 are independent, giving it the contravariant form:
\begin{equation}\label{Eq_1_14}
S =
\left[  \begin{array}{cccc}
0 & d^1 & d^2 & d^3 \\
-d^1 & 0 & -s^3 & s^2 \\
-d^2 & s^3 & 0 & -s^1 \\
-d^3 & -s^2 & s^1 & 0
\end{array} \right]
\end{equation}
Here, $\mathbf{s}$ $= \left(s^1, s^2, s^3 \right)$ is the spin vector and Eqs.~\eqref{Eq_1_13} and~\eqref{Eq_1_14} imply:
\begin{equation}\label{Eq_1_15}
	\mathbf{s} = \mathbf{z} \times m \mathbf{u}
\end{equation}
\noindent where $\mathbf{u} = \dot{\mathbf{x}}$, showing that $\mathbf{s}$ is the 3-vector angular momentum for the total linear momentum $m\mathbf{u}$ about the spin center $C$. Differentiating and using Eqs.~\eqref{Eq_1_1}, we get:
\begin{equation}\label{Eq_1_15b}
	\dot{\mathbf{s}} = \mathbf{u} \times \mathbf{P}
\end{equation}
\noindent which is just the spatial part of Eq.~\eqref{Eq_1_20}. It implies that the spin axis will precess if the observer is not fixed relative to the rest frame so that $\mathbf{P} = 0$.
Similarly, vector $\mathbf{d} = \left(d^1, d^2, d^3 \right)$ is given by:
\begin{equation}\label{Eq_1_16}
	\mathbf{d} = mc ( \dot{t} \mathbf{z} - t_z \mathbf{u} )
\end{equation}
\noindent where $u*u = 0$ from $C1$ and $z*u = 0$, previously derived in Section 2.3, are used. Eq.~\eqref{Eq_1_20} implies that $\dot{\mathbf{d}} = mc \dot{t}_y (\mathbf{u} - \dot{t} \mathbf{V})$.

Notice that the negative sign is needed out front in the definition of spin in Eq.~\eqref{Eq_1_13} if the total angular momentum is to be conserved for a free electron.
The chosen representation of the anti-symmetric tensor $S$ in terms of 3-vectors in Eq.~\eqref{Eq_1_14} then leads to the spin 3-vector $\mathbf{s}$ as the angular momentum about the spin center $C$, as in Eq.~\eqref{Eq_1_15}.
Using 3-vectors, the spatial part of Eq.~\eqref{Eq_1_20b} can now be written as:
\begin{equation}\label{Eq_1_14a}
	 \dot{\mathbf{J}} = \frac{d}{d\tau} \left( \mathbf{x} \times \mathbf{P} - \mathbf{s} \right) =  \mathbf{x} \times {\bf f}
\end{equation}
where $\mathbf{J} = \mathbf{x} \times \mathbf{P} - \mathbf{s} = \mathbf{y} \times \mathbf{P} - \mathbf{z} \times m\dot{\mathbf{z}}$ may be viewed as the 3-vector total angular momentum about O, which is conserved for a free electron.
However, it is the difference between the 3-vector orbital angular momentum and the 3-vector spin, rather than the sum. This feature has been noted previously.
For example, rather than using  Eq.~\eqref{Eq_1_15} to define the spin vector $\mathbf{s}$, {\color{Green}Weyssenhoff (1947)} and {\color{Green}Rivas (2001)} define it to be the negative of the angular momentum of the total linear momentum.

It can be readily shown that the spin tensor $S$ satisfies two additional constraints:
$\mathbf{s} \cdot \mathbf{s} = \left(\frac{\hbar}{2}\right)^2 \dot{t}^2$ and ${\bf d} \cdot {\bf d} = \left(\frac{\hbar}{2}\right)^2 \dot{t}^2$, so that $S$ has effectively only 4 independent components.
These constraints imply that $S^{\mu \nu} S_{\mu \nu} = 2 \mathbf{s} \cdot \mathbf{s} - 2 {\bf d} \cdot {\bf d} = 0$.
Furthermore, $\{ \frac{2}{\hbar \dot{t}} \mathbf{d}, \frac{1}{c \dot{t}}\mathbf{u}, \frac{2}{\hbar \dot{t}} \mathbf{s} \}$ is a rotating right-handed orthonormal triad of 3-vectors because it is readily shown that:
\begin{equation}\label{Eq_1_16a}
	\text{(i) } \mathbf{d} = (\mathbf{u} \times \mathbf{s}) / (c \dot{t}) \hspace{0.2in} \text{(ii) } \mathbf{u} = \frac{4c^2}{\hbar^2} (\mathbf{s} \times \mathbf{d}) / (c \dot{t}) \hspace{0.2in} \text{(iii) } \mathbf{s} =(\mathbf{d} \times \mathbf{u} ) / (c \dot{t})
\end{equation}

Using the definition of $S$ in Eq.~\eqref{Eq_1_13} and the first of Eqs.~\eqref{Eq_1_1} along with the constraints $C1$, $C2$ and $C3$ one at a time,
the following identities for each $\mu = 0,1,2,3$ are readily proved using a line or two :
\begin{align} \nonumber
	\text{(i) } & S^{\mu\nu} u_{\nu} = 0 \\ \nonumber
	\text{(ii) } & S^{\mu\nu} \dot{u}_{\nu} = mc^2 u^{\mu} \\ \nonumber
	\text{(iii) } & S^{\mu\nu} \pi_{\nu} = -(mc)^2 z^{\mu} \\  \nonumber
	\text{(iv) } & S^{\mu\nu} z_{\nu} = - \frac{\hbar}{2 \omega_0} u^{\mu} \\
	\text{(v) } & S^{\mu\nu} \dot{z}_{\nu} = mc^2 z^{\mu} \label{Eq_1_16b}
\end{align}
\noindent For example, (iii) follows from $S^{\mu \nu} \pi_{\nu} = -m \left( z^{\mu} u^{\nu}  \pi_{\nu} - u^{\mu} z^{\nu}  \pi_{\nu} \right) = - (mc)^2 z^{\mu}$
using $C3$ given in Eq.~\eqref{Eq_1_7} and its consequence in Eq.~\eqref{Eq_1_8}.
The identity in (i) is often applied as a constraint when developing classical models of spin, although some authors impose $S^{\mu\nu} \pi_{\nu}  = 0$ instead,
which does not hold in the neo-classical spin model presented here, as (iii) shows.
It can be shown by using  Eqs.~\eqref{Eq_1_1} and $C3$ that (i) holds if and only if $C1$ holds, that is,
the electron moves at the speed of light. Eq.~\eqref{Eq_1_16a}(i) is just the spatial part of Eq.~\eqref{Eq_1_16b}(i).

For a \emph{free} electron and using the rest-frame fixed at the spin center $C$ where $\mathbf{P} = {\bf 0}$ and, as noted earlier, $\mathbf{x} = \mathbf{z} = \mathbf{r}, ~\mathbf{u} = \dot{\mathbf{r}}, ~t_z = 0$ and $t = \tau$,
then (i) $\mathbf{d} = mc \mathbf{r} = \frac{\hbar}{2} {\bf e}_1$ where ${\mathbf{r}} = r_0 {\bf e}_1$, (ii) $\mathbf{s} = \mathbf{r} \times m \dot{\mathbf{r}} = mcr_0 {\bf e}_1 \times {\bf e}_2 = \frac{\hbar}{2} {\bf e}_3$ where $\dot{\mathbf{r}} = c {\bf e}_2$,
and (iii) $\dot{{\bf s}} = {\bf 0}$ so ${\bf e}_3$ is constant. Here, we have introduced $\{ {\bf e}_1, {\bf e}_2, {\bf e}_3 \}$ as a rotating right-handed orthonormal triad of 3-vectors,
and used the consequences of the constraints $C1$ and $C2$. It is readily shown that $\dot{{\bf e}}_1 = \omega_0 {\bf e}_2 =$ \mbox{\boldmath $\Omega$} $\times {\bf e}_1$ where
\mbox{\boldmath $\Omega$} $ = \omega_0  {\bf e}_3$ and $\dot{{\bf e}}_2 = - \omega_0 {\bf e}_1 =$ \mbox{\boldmath $\Omega$} $ \times {\bf e}_2$.
Thus, the circular spin motion takes place in the plane containing the orthogonal unit vectors ${\bf e}_1$ and ${\bf e}_2$,
which is orthogonal to the constant spin direction ${\bf e}_3$ and it has a circular frequency of $\omega_0 = c/r_0$.

\subsection{Magnetic and Electric Dipole Moments}

The spin-field interaction term $\Phi$ in Eq.~\eqref{Eq_1_9} can be expressed in terms of the spin and em-field tensors by using Eq.~\eqref{Eq_1_13}:
\begin{equation} \label{Eq_1_21}
	\Phi = - z_{\mu} f^{\mu} = - \frac{1}{2} \left(z_{\mu} q F^{\mu \nu} u_{\nu} - z_{\nu} q F^{\mu \nu} u_{\mu} \right) = \frac{q}{2m} F^{\mu \nu}S_{\mu \nu}
\end{equation}
\noindent since $F^{\mu \nu} = - F^{\nu \mu}$. Substituting for the em-field and spin tensors:
\begin{equation} \label{Eq_1_22}
	\Phi = \frac{q}{m} \left( \mathbf{B} \cdot \mathbf{s} +\frac{1}{c} \mathbf{E} \cdot \mathbf{d} \right)
\end{equation}
\noindent showing that $\Phi$ is the sum of two energy terms due to electric and magnetic dipole moments. Thus, the energy equation in Eq.~\eqref{Eq_1_9} becomes:
\begin{equation}\label{Eq_1_9b}
	\frac{1}{mc^2} E^2 = mc^2 + \frac{1}{m} \mathbf{P} ^2 - \frac{q}{m} \left( \mathbf{B} \cdot \mathbf{s} +\frac{1}{c} \mathbf{E} \cdot \mathbf{d} \right)
\end{equation}
\noindent Since $E=\gamma mc^2$, this equation can be expressed as:
\begin{equation} \label{Eq_1_9c}
	E = \gamma^{-1} mc^2 + \mathbf{P} \cdot \mathbf{V} - \left( \frac{q \gamma^{-1}}{m} \mathbf{s} \right) \cdot \mathbf{B} - \left( \frac{q \gamma^{-1}}{mc} \mathbf{d} \right) \cdot \mathbf{E}
\end{equation}
\noindent and it looks as though the magnetic moment is \mbox{\boldmath $\mu$} $= \frac{q \gamma^{-1}}{m} \mathbf{s}$ and the electric moment is
\mbox{\boldmath $\epsilon$} $= \frac{q \gamma^{-1}}{mc} \mathbf{d}$ ($= \gamma^{-1} q \mathbf{r}$ in the rest frame), as expected.
However, as shown in Eq.~\eqref{Eq_1_10c}, $\gamma$ depends on $\Phi$ and a first-order expansion in small quantities gives $\gamma^{-1} mc^2 \approx mc^2 - \frac{m}{2} \mathbf{V}^2 + \frac{1}{2} \Phi$
and $\mathbf{P} \cdot \mathbf{V} \approx m \mathbf{V}^2$, so:
\begin{equation} \label{Eq_1_9d}
	E \approx mc^2 + \frac{1}{2} m \mathbf{V}^2 - \frac{1}{2} \Phi
\end{equation}
\noindent giving orthogonal magnetic and electric moment 3-vectors of approximately $\frac{1}{2}$ \mbox{\boldmath $\mu$} and $\frac{1}{2}$ \mbox{\boldmath $\epsilon$}, respectively.
This agrees with classical electromagnetic theory if one argues that there is an effective current loop created by the fast circular spin motion of the electron,
which gives it a magnetic moment of magnitude $ \frac{q}{2\pi r_0/c} (\pi r_0^2) = \frac{q}{2m}(mc r_0) = \frac{q \hbar}{4m} = \frac{1}{2} \mu$.

For a free electron, relative to the rest frame at C,  the spin vector $\mathbf{s} =  \frac{\hbar}{2} {\bf e}_3$ is constant from Eq.~\eqref{Eq_1_15b}
while the vector $\mathbf{d} = \frac{\hbar}{2} {\bf e}_1 = mc \mathbf{r}$ rotates at the ultra-high spin frequency $\omega_0$ due to the electron going around its circular path $\mathbf{r} = r_0 {\bf e}_1$ at speed $c$.
In an electro-magnetic field, an ultra-high spin frequency will still be present and so the second energy term in Eq.~\eqref{Eq_1_22} will have a temporal average of essentially zero for most electric fields.

{\color{Green} Dirac (1928, 1958)} in his analysis using the "square'' of his wave equation for the electron in an em-field, also gets Eqs.~\eqref{Eq_1_22} and~\eqref{Eq_1_9b} in an operator form
but his operator $\Phi$ corresponding to Eq.~\eqref{Eq_1_21} is a factor of two larger.
If a scaling factor $g$ is added to the definition of $\Phi$, the neo-classical theory here has $g=1$ but Dirac's theory for the electron gives $g=2$, implying a magnetic moment that is close to experimental results for the electron.
{\color{Green} Dirac (1928)} states: "The electric moment, being a pure imaginary, we should not expect to appear in the model. It is doubtful whether the electric moment has any physical meaning."
In his book, {\color{Green} Dirac (1958)} drops this term. The derivation of Eq.~\eqref{Eq_1_22} here gives a clear physical basis for the presence of both electric and magnetic dipoles for the electron.
This physical basis has also been noted by {\color{Green} Rivas (2001)} and {\color{Green} Hestenes (2010)}.

{\color{Green}Salesi and Recami (2000)} use the spin tensor form of the equations of motion from {\color{Green}Barut and Zanghi (1984)} to study the implied behavior of an electron in a constant uniform magnetic field.
These equations of motion are shown in the next subsection (along with Appendix A) to be equivalent to those in Eqs.~\eqref{Eq_1_1}.
However, the derivation of Salesi and Recami leads to an altered form of these equations because they use an analog of Dirac's operator form of the energy equation corresponding to Eq.~\eqref{Eq_1_9} with $\Phi$
having a factor $g=2$. The correct value is $g=1$ for the theory of Barut and Zanghi, as we derived in Eq.~\eqref{Eq_1_21}.
A consequence of forcing $g=2$ is that the quantity $G=z*\pi$ is nonzero but it is proved that $G=0$ in Appendix A. Because of this contradiction, the main conclusion of {\color{Green}Salesi and Recami (2000)} is questionable,
namely, that the cyclotron frequency differs slightly for the two cases of the electron's spin being in the same or opposite direction to the magnetic field vector.

{\color{Green}Hestenes (2010)} produces an energy equation like Eq.~\eqref{Eq_1_9b} for a classical model of the electron dynamics using Clifford space-time algebra by explicitly adding a dipole energy term
to his Lagrangian function that has a Pauli spin-field interaction form.
In contrast, in the neo-classical model presented here, the spin-field interaction terms exhibited in Eq.~\eqref{Eq_1_9b}
arise solely from the minimal field-coupling term in the Lagrangian function in Eq.~\eqref{Eq_1_11}, which leads to the Lorentz force acting on the electron's spin motion.
Although it is not widely accepted that the electron exhibits an electric dipole moment in addition to its magnetic dipole moment,
{\color{Green} Hestenes (2010)} argues that there is experimental evidence ({\color{Green} Gouan\`ere et al. (2005)}) suggesting that the electron does indeed exhibit an ultra-fast
rotating electric dipole moment due to its spin (''zitter") motion and that it can cause resonance when an electron is propelled through a crystal ion channel at an appropriate speed.

\subsection{Equations of Motion using Spin Tensor}

The eight second-order equations of motion in Eq.~\eqref{Eq_1_1} can be re-written as 16 equations in dynamic state-space form using the state $(x, u, y, \pi)$:
\begin{align} \nonumber
	\dot{x}^{\mu} & = u^{\mu} \\ \nonumber
	\dot{u}^{\mu} & = - \omega_0^2 \left( x^{\mu} - y^{\mu} \right)  \\ \nonumber
	\dot{y}^{\mu} & = \frac{1}{m} \pi^{\mu}  \\
	\dot{\pi}^{\mu} & = q F^{\mu \nu} (x) u_{\nu} \label{Eq_1_17}
\end{align}
\noindent Specification of the initial state at some time, make it $\tau = 0$, defines a unique solution to this set of equations for a specified field tensor $F$.
They are linear except for the last equation, which is nonlinear unless there is no em-field or the electric and magnetic fields ${\bf E}(x)$ and ${\bf B}(x)$ are constant with respect to $x$.

We now show that we can re-write these state-space equations in the following equivalent form using the state $(x, u, S, \pi)$:
\begin{align}  \nonumber
	\dot{x}^{\mu} & = u^{\mu}  \\ \nonumber
	\dot{u}^{\mu} & =  \frac{4c^2}{\hbar^2} S^{\mu \nu} \pi_{\nu}  \\ \nonumber
	\dot{S}^{\mu \nu} & = \pi^{\mu} u^{\nu} - \pi^{\nu} u^{\mu}  \\
	\dot{\pi}^{\mu} & = q F^{\mu \nu} (x) u_{\nu} \label{Eq_1_18}
\end{align}
\noindent The third equation is just Eq.~\eqref{Eq_1_20} that we proved using Eq.~\eqref{Eq_1_1}. The second equation comes from substituting (iii) of Eqs.~\eqref{Eq_1_16b} into Eq.~\eqref{Eq_1_1}:
\begin{equation} \label{Eq_1_19}
	\dot{u}^{\mu} =  - \omega_0^2 z^{\mu} =   \frac{4c^2}{\hbar^2} S^{\mu \nu} \pi_{\nu}
\end{equation}
\noindent This derivation shows that the equations of motion in Eqs.~\eqref{Eq_1_1}, or equivalently Eqs.~\eqref{Eq_1_17}, imply the equations of motion in Eqs.~\eqref{Eq_1_18}.
The converse is proved in Appendix A where the mapping between the initial conditions for~\eqref{Eq_1_17} and~\eqref{Eq_1_18} is also given.
Notice that the spin form of the set of equations of motion~\eqref{Eq_1_18} is nonlinear in the state except for a free electron where $\pi$ is a constant parameter.
Also, the only property of the electron appearing in these equations is its charge $q=-e$.

Eqs.~\eqref{Eq_1_18} reveal the first connection of the neo-classical spin theory to Dirac's equation. These four equations are identical to the equations in the Appendix of {\color{Green}Grossmann and Peres (1963)} who derive
them by replacing the commutator expression for the proper time derivatives of the operators representing dynamic quantities in the relativistic Heisenberg picture by an appropriate Poisson bracket for the dynamic quantities.
They relegate these equations to an appendix because of reservations about their appropriateness. The equations that they present in the main body of their paper use observer time $t$ in the derivatives, rather than
proper time $\tau$, and their sinusoidal dependence $\sin \omega t$ of the zitterbewegung part of the free electron motion is not consistent with the $\sin \omega_0 \tau$ terms in Eq.~\eqref{Eq_1_4} except in the rest frame at $C$
where $\omega = \omega_0$ and $t = \tau$.

{\color{Green}Krylovetskii (1978)} derives the Grossmann-Peres equations of motion using observer time $t$ by postulating a Lagrangian function involving a neo-classical spinor of the form investigated by {\color{Green}Proca (1954)}.
He then shows that these equations using observer time $t$ are indeed covariant when moving from one inertial reference frame to another.
{\color{Green}Barut and Zanghi (1984)} use a proper time version of Krylovetskii's spinor Lagrangian function, although apparently unaware of his work, and show that the resulting equations imply Eq.~\eqref{Eq_1_18}.
We return to their theory in the next section.

{\color{Green}Hestenes (2010)} presents a similar Lagrangian function involving spinors in proper time but formulated using Clifford space-time algebra.
His derived equations of motion are equivalent to those in Eqs.~\eqref{Eq_1_18} if the explicit dipole moment term that he adds to his Lagrangian function is dropped.
As we showed in the previous subsection, the Lagrangian function in Eq.~\eqref{Eq_1_11} has implicitly electric and magnetic dipole moments,
although half of the expected strength, because of the action of the Lorentz force on the spin motion arising from its spin energy term.

{\color{Green}Barut (1987)} shows that the relativistic Heisenberg operator equations for the proper-time Dirac equation have a correspondence with the spin tensor form in Eqs.~\eqref{Eq_1_18}
where the dynamic variables in the state correspond to their operator representations. A similar correspondence is exhibited later in the operator equations in Eqs.~\eqref{Eq_1_34}.

Finally, {\color{Green}Weyssenhoff and Raabe (1947)} develop equations of motion for an electron moving at the speed of light that are very similar to Eqs.~\eqref{Eq_1_18}
except that they have an implicit equation for $\dot{u}$ rather than the explicit one in Eqs.~\eqref{Eq_1_18}.
They derive their equations by applying linear and angular momentum conservation to a fluid that has volume elements that spin,
then they let the elements shrink to a point in the limit of an infinitely large mass density and angular momentum density.
Their paper, and the others just referenced, show that there is something special about Eqs.~\eqref{Eq_1_18}, which are apparently applicable to any fundamental charged particle with spin.

\section{Operator Form of Equations of Spin Model}

\subsection{Neo-classical Dirac-Schr\"{o}dinger Equation}

It is shown in this section that the spin tensor form of the equations of motion can be represented using operators on a four-dimensional complex state function that is a solution in proper time of a neo-classical Dirac-Schr\"{o}dinger equation.

Based on {\color{Green}Barut and Zanghi (1984)}, we introduce a \emph{state function} $\phi(\tau) \in {\cal C}^4$, a 4-dimensional complex column vector function, satisfying the neo-classical spinor Dirac-Schr\"{o}dinger equation:
\begin{equation} \label{Eq_1_23}
	 i \hbar \dot{\phi} = \hat{H} \phi
\end{equation}
\noindent where the $4 \times 4$ complex matrix $\hat{H} = c \pi_{\mu} \gamma^{\mu}$ is the Hamiltonian operator and $\gamma^{\mu}, \mu = 0,1,2,3,$ are the usual $4 \times 4$ Dirac matrices.
Notice that this spinor equation represents 8 real-valued equations involving the real and imaginary parts of the complex column vector $\phi$.
As shown later, the state function $\phi$ corresponds to Dirac's wave function after a Lorentz transformation.
Eq.~\eqref{Eq_1_23} shows that $\phi$ depends on the electron's motion only through its global momentum $\pi$.

We also introduce the two $4 \times 4$ constant matrix operators:
\begin{equation} \label{Eq_1_24}
\hat{u}^{\mu} = c \gamma^{\mu}, \hspace{0.2in} \hat{S}^{\mu\nu} = - \frac{i \hbar}{4} \left[ \gamma^{\mu}\gamma^{\nu} - \gamma^{\nu}\gamma^{\mu} \right]
\end{equation}
\noindent corresponding to the total velocity component $u^{\mu} = \bar{\phi} \hat{u}^{\mu} \phi$ and the spin tensor  component $S^{\mu\nu} = \bar{\phi} \hat{S}^{\mu\nu} \phi$, where, as in QM theory,
we define $\bar{\phi} = \phi^{*} \gamma^0$ where $\phi^{*}$ is the Hermitian (conjugate) transpose of $\phi$.
Notice the negative sign at the front of the expression for $\hat{S}$, which is needed for consistency with $S$ in Eq.~\eqref{Eq_1_13}.
Notice also that the matrix operators $\hat{u}^{\mu}$ and $\hat{S}^{\mu\nu}$ are not Hermitian but $\gamma_0 \hat{u}^{\mu}$ and $\gamma_0 \hat{S}^{\mu\nu}$ are, which are the actual operators in the inner product
$\phi^{*} (.) \phi$ that defines the usual Hilbert space on ${\cal C}^4$. The time operator $\hat{u}^0 = c \gamma^0$ gives $c \dot{t} = u^0 = \bar{\phi} \hat{u}^0 \phi = c \phi^* \phi$.
Thus, as noted after Eq.~\eqref{Eq_1_4b}, observer time $t$ increases monotonically with proper time $\tau$ because $\dot{t} = \phi^* \phi$ is always positive.

If $\hat{Q}(\tau)$ is an operator and $Q(\tau)$ is the corresponding dynamical variable defined by the inner product $Q = \phi^{*} (\gamma^0 \hat{Q}) \phi = \bar{\phi} \hat{Q} \phi$, then the derivative of $Q(\tau)$ is given by:
\begin{align}  \nonumber
	 \dot{Q} & = \dot{\bar{\phi}} \hat{Q} \phi + \bar{\phi} \hat{Q} \dot{\phi} + \bar{\phi} \dot{\hat{Q}} \phi \\
	             & = \frac{i}{\hbar}  \bar{\phi} \left[ \hat{H}\hat{Q}  - \hat{Q} \hat{H}  \right]  \phi + \bar{\phi} \dot{\hat{Q}} \phi \label{Eq_1_27}
\end{align}
\noindent where we use Eq.~\eqref{Eq_1_23} in the first two terms. This proves a result that is familiar from QM for the time derivative of an observable in terms of a commutator with the Hamiltonian operator:
\begin{equation} \label{Eq_1_28}
	\hat{\dot{Q}} = \frac{i}{\hbar} \left[ \hat{H}, \hat{Q} \right] + \dot{\hat{Q}}
\end{equation}
\noindent where from Eq.~\eqref{Eq_1_27}, $\dot{Q} = \bar{\phi} \hat{\dot{Q}} \phi$. Note that $\hat{H}$ could be time-varying because $\pi$ can be if the electron is in an em-field
but the Hamiltonian function $H$ corresponding to $\hat{H}$ is constant since:
\begin{equation} \label{Eq_1_29}
	\dot{H} = \bar{\phi} \hat{\dot{H}} \phi =  \bar{\phi} \frac{i}{\hbar} \left[ \hat{H}, \hat{H} \right] \phi +  \bar{\phi} \dot{\hat{H}} \phi = \dot{\pi}_{\mu}u^{\mu} = 0
\end{equation}
\noindent by using the definition of $\hat{H}$ and $\hat{u}^{\mu}$ above and then using Eq.~\eqref{Eq_1_17} for $\dot{\pi}$ in terms of the anti-symmetric tensor $F$.
The time invariance of $H$ can also be shown directly:
\begin{equation} \label{Eq_1_30}
	H = \bar{\phi} \hat{H} \phi = \pi_{\mu} \bar{\phi} \hat{u}^{\mu} \phi = \pi_{\mu} u^{\mu} = mc^2
\end{equation}
\noindent using constraint $C3$. We use this equation to provide a normalization of the solution $\phi$ of the homogeneous linear equation~\eqref{Eq_1_23}.

\subsection{Operator Form of Equations of Motion}

Consider the following state-space equations using the state $(x, \phi, \pi)$:
\begin{align}  \nonumber
	\dot{x}^{\mu} & = \bar{\phi} \hat{u}^{\mu} \phi  \\ \nonumber
	\dot{\phi} & = - \frac{i}{\hbar} \hat{H} \phi = - \frac{ic}{\hbar} \pi_{\nu}  \gamma^{\nu} \phi   \\
	\dot{\pi}^{\mu} & = q F^{\mu\nu}(x) \bar{\phi} \hat{u}_{\nu} \phi   \label{Eq_1_31}
\end{align}
\noindent We now show that a solution of these equations (equivalent to 16 real-valued equations) gives the solution of Eqs.~\eqref{Eq_1_18} with corresponding initial conditions:
$x(0)$, $\pi(0)$, $u(0) = \bar{\phi}(0) \hat{u} \phi(0)$ and $S(0) = \bar{\phi}(0) \hat{S} \phi(0)$.

First, consider $u^{\mu} = \dot{x}^{\mu} = \bar{\phi} \hat{u}^{\mu} \phi$, then $\dot{u}^{\mu} = \bar{\phi} \hat{\dot{u}}^{\mu} \phi$ where from Eq.~\eqref{Eq_1_28}:
\begin{align}  \nonumber
	\hat{\dot{u}}^{\mu} & = \frac{i}{\hbar} \left[ \hat{H}, \hat{u}^{\mu} \right]  \\ \nonumber
	                      & = \frac{ic}{\hbar} \left[ \hat{H}, \gamma^{\mu} \right]  \\ \nonumber
	                      & = \frac{ic^2}{\hbar} \left[\pi_{\nu}  \gamma^{\nu}  \gamma^{\mu} - \pi_{\nu}  \gamma^{\mu}  \gamma^{\nu} \right]  \\
	                      & = \frac{4c^2}{\hbar^2} \hat{S}^{\mu\nu} \pi_{\nu}  \label{Eq_1_32}
\end{align}
\noindent Thus, $\dot{u}^{\mu} = \bar{\phi} \hat{\dot{u}}^{\mu} \phi = \frac{4c^2}{\hbar^2} \bar{\phi} \hat{S}^{\mu\nu} \phi \pi_{\nu} = \frac{4c^2}{\hbar^2} S^{\mu\nu} \pi_{\nu}$, as in Eqs.~\eqref{Eq_1_18}.

Second, consider $\dot{S}^{\mu\nu} = \bar{\phi} \hat{\dot{S}}^{\mu\nu} \phi$. The Dirac matrices satisfy the identity $\gamma^{\mu}\gamma^{\nu} + \gamma^{\nu}\gamma^{\mu} = 2 g^{\mu\nu} I_4$
where $I_4$ is the $4 \times 4$ identity matrix and so $\hat{S}^{\mu\nu} = - \frac{i \hbar}{2} \left[ \gamma^{\mu}\gamma^{\nu} - g^{\mu\nu} I_4 \right]$.
If $\mu = \nu$, then $\gamma^\mu \gamma^\mu = g^{\mu\mu} I_4$ and $\hat{S}^{\mu\nu} = 0$.
If $\mu \neq \nu$, then $\gamma^\mu \gamma^\nu = - \gamma^\nu \gamma^\mu$ and $\hat{S}^{\mu\nu} = - \frac{i\hbar}{2} \gamma^\mu \gamma^\nu$, so from Eq.~\eqref{Eq_1_28}:
\begin{align}  \nonumber
	\hat{\dot{S}}^{\mu\nu} & = \frac{i}{\hbar} \left[ \hat{H}, \hat{S}^{\mu\nu} \right]  \\ \nonumber
	                           & = \frac{1}{2} \left[ \hat{H}, \gamma^{\mu} \gamma^{\nu} \right]  \\ \nonumber
	                           & = \frac{c}{2} \left[ \pi_{\sigma} \gamma^{\sigma}\gamma^{\mu}\gamma^{\nu} - \pi_{\sigma} \gamma^{\mu}\gamma^{\nu}\gamma^{\sigma} \right]  \nonumber
\end{align}
\noindent But if $\sigma \neq \mu, \nu$, then $\gamma^\sigma \gamma^\mu \gamma^\nu = - \gamma^\mu \gamma^\sigma \gamma^\nu = \gamma^\mu \gamma^\nu \gamma^\sigma$,
so only the terms with $\sigma = \mu,\nu$ appear in the sum over $\sigma$, implying that:
\begin{align}  \nonumber
	\hat{\dot{S}}^{\mu\nu} & = \frac{c}{2} \pi_\mu \left[ \gamma^\mu \gamma^\mu \gamma^\nu - \gamma^\mu \gamma^\nu \gamma^\mu \right]
						+ \frac{c}{2} \pi_\nu \left[ \gamma^\nu \gamma^\mu \gamma^\nu - \gamma^\mu \gamma^\nu \gamma^\nu \right]  \\ \nonumber
	                           & = c \pi_{\mu} g^{\mu\mu} \gamma^{\nu} - c \pi_{\nu} g^{\nu\nu} \gamma^{\mu}  \\
	                           & = \pi^{\mu}\hat{u}^{\nu} - \pi^{\nu}\hat{u}^{\mu}   \label{Eq_1_33}
\end{align}
\noindent Thus, $\dot{S}^{\mu\nu} = \bar{\phi} \hat{\dot{S}}^{\mu\nu} \phi = \pi^{\mu} u^{\nu} - \pi^{\nu} u^{\mu}$, as in Eqs.~\eqref{Eq_1_18}.
In effect, the neo-classical Dirac-Schr\"{o}dinger equation for the evolution of the spinor state function $\phi$ in Eqs.~\eqref{Eq_1_31} replaces the evolution equations for the total 4-velocity $u$
and the spin tensor $S$ in Eqs.~\eqref{Eq_1_18}, so that the two sets of state-space equations produce the same solution for all dynamic variables.

We note that Eqs.~\eqref{Eq_1_18} can now be equivalently expressed as operator equations:
\begin{align}  \nonumber
	\hat{\dot{x}}^{\mu} & = \hat{u}^{\mu}  \\ \nonumber
	\hat{\dot{u}}^{\mu} & =  \frac{4c^2}{\hbar^2} \hat{S}^{\mu \nu} \pi_{\nu}  \\ \nonumber
	\hat{\dot{S}}^{\mu \nu} & = \pi^{\mu} \hat{u}^{\nu} - \pi^{\nu} \hat{u}^{\mu}  \\
	\hat{\dot{\pi}}_{\mu} & = q F_{\mu \nu} (x) \hat{u}^{\nu} \label{Eq_1_34}
\end{align}
\noindent where a dynamic variable $Q(\tau) = \bar{\phi} \hat{Q} \phi$ and $\phi(\tau)$ is given by Eq.~\eqref{Eq_1_23} with $\phi(0)$ specified.
These equations are identical to the Heisenberg equations from Dirac's equation in proper time ({\color{Green}Barut (1987)}), except that the actual global momentum $\pi$ is involved,
not an operator for it. We introduce an operator for $\pi$ for the neo-classical theory in a later subsection where a wave function is defined in terms of the state function.

\subsection{Superposition of State Functions for a Free Electron}

For the free electron, $\hat{H}$ in Eq.~\eqref{Eq_1_23} is not a function of $\tau$, so:
\begin{equation} \label{Eq_1_41}
	 \ddot{\phi} = \frac{1}{i \hbar} \frac{d}{d\tau} \left( \hat{H} \phi \right) =   - \frac{1}{\hbar^2} \hat{H} \left(i \hbar \dot{\phi} \right) = - \frac{1}{\hbar^2} \hat{H}^2 \phi = - \omega_1 ^2 \phi
\end{equation}
\noindent setting $\omega_1 = \frac{mc^2}{\hbar} = \frac{1}{2} \omega_0$ after using the identity:
\begin{equation} \label{Eq_1_39}
	\hat{H}^2 = c^2 \pi_{\mu} \pi_{\nu} \gamma^{\mu} \gamma^{\nu} = \frac{1} {2} c^2 \pi_{\mu} \pi_{\nu} \left( \gamma^{\nu} \gamma^{\mu} + \gamma^{\mu} \gamma^{\nu} \right) = c^2 \pi_{\mu} \pi_{\nu} g^{\nu\mu} I_4 = (mc^2)^2  I_4
\end{equation}
\noindent where the last equality is from Eq.~\eqref{Eq_1_9} with $\Phi = 0$. This simple linear oscillator equation is a neo-classical spin theory version of the \emph{Klein-Gordon equation} in QM and
it has the solution:
\begin{align}  \nonumber
	\phi(\tau) & = \cos (\omega_1 \tau) \phi(0) + \frac{1}{\omega_1} \sin (\omega_1 \tau) \dot{\phi}(0) \\
	               & = \left[ \cos (\omega_1 \tau) I_4 - \frac{i}{mc^2} \sin (\omega_1 \tau) \hat{H} \right] \phi(0) \label{Eq_1_42}
\end{align}
\noindent using Eq.~\eqref{Eq_1_23} in the second equation.
This solution of Eq.~\eqref{Eq_1_41} is readily shown to also be the solution of Eq.~\eqref{Eq_1_23}.
It was first given by {\color{Green}Barut and Zanghi (1984)} in their Eq.\  (4) along with an expression for $u^{\mu} = \bar{\phi} \hat{u}^{\mu} \phi$ (see Appendix B for a detailed derivation),
which agrees with $\dot{x} (\tau)$ in Eq.~\eqref{Eq_1_4}, as expected. (Their Eq.\  (4) has three typographical errors).
It is readily shown that if $\phi(0)$ satisfies the energy normalization in Eq.~\eqref{Eq_1_30}, so does $\phi(\tau)$ in Eq.~\eqref{Eq_1_42} for all proper time $\tau$.

Expressing the sine and cosine terms in Eq.~\eqref{Eq_1_42} as complex exponentials:
\begin{equation} \label{Eq_1_45}
	\phi(\tau) = \left[ \frac{1}{2} \left(I_4 + \frac{1}{mc^2} \hat{H} \right) \exp (-i \omega_1 \tau) + \frac{1}{2} \left(I_4 - \frac{1}{mc^2} \hat{H} \right) \exp (i \omega_1 \tau) \right] \phi(0)
\end{equation}
\noindent For arbitrary $\phi(0) = A \in {\cal C}^4$ satisfying the energy normalization $\bar{A} \hat{H} A = mc^2$ from Eq.~\eqref{Eq_1_30}, define:
\begin{equation} \label{Eq_1_45a}
	A^+ = \frac{1}{2} \left(I_4 + \frac{1}{mc^2} \hat{H} \right) A, \hspace{0.2in} A^- = \frac{1}{2} \left(I_4 - \frac{1}{mc^2} \hat{H} \right) A
\end{equation}
\noindent then using the identity~\eqref{Eq_1_39}, $\hat{H} A^+ = mc^2 A^+$ and $\hat{H} A^- = - mc^2 A^-$. Thus, $A^+$ and $A^-$ are eigenvectors of $\hat{H}$ belonging
to the two-dimensional eigenspaces corresponding to eigenvalues $mc^2$ and $-mc^2$,
respectively, that depend on the global momentum $\pi$. From Eq.~\eqref{Eq_1_45}, any solution of Eq.~\eqref{Eq_1_23} for the free electron is a superposition:
\begin{equation} \label{Eq_1_46a}
	\phi (\tau) = \phi^+ (\tau) + \phi^- (\tau), \hspace{0.2in} \phi^+ (\tau) = \exp (- i\omega_1 \tau) A^+,  \hspace{0.2in} \phi^- (\tau) = \exp ( i\omega_1 \tau) A^-
\end{equation}
\noindent where these special state function solutions satisfy $\hat{H} \phi^+ = mc^2 \phi^+$ and $\hat{H} \phi^- = - mc^2 \phi^-$.
They correspond to the so-called \emph{positive energy} and \emph{negative energy} solutions, respectively, in relativistic QM theory but this is misleading terminology here.
As stand-alone solutions of Eq.~\eqref{Eq_1_23}, the normalization in~\eqref{Eq_1_30} implies that $\phi^+$ and $\phi^-$ both have positive energy, as expected
for a neo-classical relativistic mechanics theory, and $\bar{A}^+ A^+ = 1$ and $\bar{A}^- A^- = -1$. In general, Eqs.~\eqref{Eq_1_45a} imply that
$\bar{A}^+ A^+ = \frac{1}{2}(\bar{A} A + 1), \bar{A}^- A^- = \frac{1}{2}(\bar{A} A - 1)$ and $\bar{A}^+ A^-, \bar{A}^- A^+$ are both zero.

To investigate spin states, consider the special case where the observer's inertial reference frame coincides with the rest frame fixed at the spin center C, then $\hat{H} = mc^2 \gamma^0$ and
so the eigenvectors of $\hat{H}$ have the form:
\begin{equation} \label{Eq_1_47a}
	A^+ = \left[ A_1, A_2, 0,0 \right]^T, \hspace{0.2in} A^- = \left[ 0,0, A_3, A_4 \right]^T
\end{equation}
\noindent From Eq.~\eqref{Eq_1_46a}, we can express the general solution of Eq.~\eqref{Eq_1_23} for the free electron case as the superposition:
\begin{equation} \label{Eq_1_48a}
	\phi(\tau) = \phi^+ (\tau) + \phi^- (\tau) = \left[ A_1 e^{-i \omega_1 \tau}, A_2 e^{-i \omega_1 \tau}, A_3 e^{i \omega_1 \tau}, A_4 e^{i \omega_1 \tau} \right]^T
\end{equation}
\noindent The complex amplitudes $A_i = \phi_i(0)$ must satisfy the previous normalization condition that gives positive energy $mc^2$, so $mc^2 = \bar{\phi} \hat{H} \phi = mc^2 \phi^* \phi$, which implies:
\begin{equation} \label{Eq_1_49a}
	1 = \phi^* \phi = |A_1|^2 + |A_2|^2 + |A_3|^2 + |A_4|^2
\end{equation}
\noindent This is consistent with the requirement from Eq.~\eqref{Eq_1_24} that any state function must satisfy $\phi^*  \phi = \frac{1}{c} \bar{\phi} \hat{u}^0 \phi = \frac{1}{c} u^0 = \dot{t} =  1$ because $t=\tau$ in the rest frame at $C$.

In Appendix C, the total velocity components $u^j$ and spin components $s^j$ $(j = 1,2,3)$ are given for the state function $\phi(\tau)$ in Eq.~\eqref{Eq_1_48a}.
Consider the choice $A_1 = \frac{1}{\sqrt{2}} \exp (-i \varphi/2), A_2 = 0, A_3 = 0, A_4 = \frac{1}{\sqrt{2}} \exp (i \varphi/2)$,
then the normalization in Eq.~\eqref{Eq_1_49a} is satisfied and from Appendix C:
\begin{align}  \nonumber
	u^1 & = c \cos (\omega_0 \tau + \varphi), \hspace{0.1in} u^2 = c \sin (\omega_0 \tau + \varphi), \hspace{0.1in} u^3 = 0 \\
	s^1 & = s^2 = 0, \hspace{0.3in} s^3 = \hbar/2  \label{Eq_1_50a}
\end{align}
\noindent This choice for the state function $\phi$ corresponds to spin-up in the direction of the $x^3$ axis where the electron is moving in a circle in the $x^1-x^2$ plane at speed $c$
with angular momentum $\hbar/2$ and so radius $r_0 = \frac{\hbar}{2mc}$.
Now consider the choice $A_1 = 0, A_2 = = \frac{1}{\sqrt{2}} \exp (i \varphi/2), A_3 = - \frac{1}{\sqrt{2}} \exp (-i \varphi/2), A_4 = 0$ in  Eq.~\eqref{Eq_1_48a}, then:
\begin{align}  \nonumber
	u^1 & = - c \cos (\omega_0 \tau - \varphi), \hspace{0.1in} u^2 = c \sin (\omega_0 \tau - \varphi), \hspace{0.1in} u^3 = 0 \\
	s^1 & = s^2 = 0, \hspace{0.3in} s^3 = - \hbar/2  \label{Eq_1_51a}
\end{align}
\noindent This choice for $\phi$ corresponds to spin-down for the $x^3$ direction.
The solutions in Eqs.~\eqref{Eq_1_50a} and~\eqref{Eq_1_51a} agree with those given in Subsection 2.2 for the initial phase $\varphi_0 = \varphi - \pi/2$.
Notice also that the spin-up and spin-down state functions here are both linear superpositions of the ``positive energy" and ``negative energy" solutions in Eq.~\eqref{Eq_1_46a}.

Consider the electron spinning about an axis defined by a unit vector $\mathbf{n} = (n^1,n^2,n^3) = n^j \mathbf{e}_j$ (summation convention over $j=1,2,3$) where $\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \}$ is an orthonormal triad of 3-vectors.
The spin vector is then $\mathbf{s}_n = s_n \mathbf{n}$ where $s_n = \mathbf{s}_n \cdot \mathbf{n} = (s^j \mathbf{e}_j) \cdot \mathbf{n} = s^j n^j$.
Using the operator for each $s^j$ given in Appendix C, the operator for $s_n$ is therefore:
\begin{equation} \label{Eq_1_52a}
	\hat{s}_n = n^j \hat{s}^j = \frac{\hbar}{2} \left[  \begin{array}{cc}
		\sigma_n & 0 \\
		0 & \sigma_n
\end{array} \right]
\end{equation}
\noindent where $\sigma_n = n^j \sigma^j$ and the $\sigma^j$'s are the three Pauli spin matrices. Using spherical coordinates with polar angle $\theta$ and the $x^3$-axis as the polar axis:
\begin{equation} \label{Eq_1_53a}
	n^1 = \sin \theta \cos \varphi, \hspace{0.1in} n^2 = \sin \theta \sin \varphi, \hspace{0.1in} n^3 = \cos \theta
\end{equation}
\noindent and so
\begin{equation} \label{Eq_1_54a}
	\sigma_n = \left[  \begin{array}{cc}
		\cos \theta & \exp (-i \varphi) \sin \theta \\
		\exp (i \varphi) \sin \theta & - \cos \theta
\end{array} \right]
\end{equation}

For any state $\phi (\tau)$, $s_n = \bar{\phi} \hat{s}_n \phi$, so:
\begin{equation} \label{Eq_1_55a}
	s_n = \frac{\hbar}{2} \phi^* \Sigma \phi =
		\frac{\hbar}{2} \phi^*
		\left[  \begin{array}{cc}
		\sigma_n & 0 \\
		0 & - \sigma_n
	\end{array} \right] \phi
\end{equation}
\noindent where $\phi^* \phi = 1$ and $\Sigma$ is a Hermitian matrix.
By Rayleigh's Quotient Theorem, $s_n$ has its maximum value $\frac{\hbar}{2}$ at the maximum eigenvalue $\lambda = 1$ of $\Sigma$ and its minimum value $- \frac{\hbar}{2}$ at the minimum eigenvalue $\lambda = -1$.
These values occur at the corresponding normalized eigenvectors of $\Sigma$. This implies that the state function $\phi_n (\tau) \in {\cal C}^4$ that gives the spin vector $\mathbf{s}_n = s_n \mathbf{n} =  \frac{\hbar}{2} \mathbf{n}$
is the eigenvector corresponding to eigenvalue $\lambda = 1$, which is given by the special case of Eq.~\eqref{Eq_1_48a} with coefficients:
\begin{align}  \nonumber
	A_1 & = \frac{1}{\sqrt{2}} \exp (-i \varphi/2) \cos (\theta/2), \hspace{0.1in} & A_2 = \frac{1}{\sqrt{2}} \exp (i \varphi/2) \sin (\theta/2) \\
	A_3 & = - \frac{1}{\sqrt{2}} \exp (-i \varphi/2) \sin (\theta/2), \hspace{0.1in} & A_4 = \frac{1}{\sqrt{2}} \exp (i \varphi/2) \cos (\theta/2)  \label{Eq_1_56a}
\end{align}
\noindent (Because of the structure of $\Sigma$, this eigenvector has as its first two and last two components equal to the two spin-up and two spin-down components, respectively, from the non-relativistic Pauli spin theory).

Substituting these coefficients into the expressions in Appendix C, the velocity components and spin components corresponding to the spin state $\phi_n (\tau)$ are:
\begin{align}  \nonumber
	u^1 & = c \cos^2 \frac{\theta}{2} \cos(\omega_0 \tau + \varphi) - c \sin^2 \frac{\theta}{2} \cos(\omega_0 \tau - \varphi)  \\  \nonumber
		& = c \cos \theta \cos \varphi \cos (\omega_0 \tau) - c \sin \varphi \sin (\omega_0 \tau) \\  \nonumber
	u^2 & = c \cos^2 \frac{\theta}{2} \sin(\omega_0 \tau + \varphi) + c \sin^2 \frac{\theta}{2} \sin(\omega_0 \tau - \varphi)  \\  \nonumber
		& = c \cos \varphi \sin (\omega_0 \tau) + c \cos \theta \sin \varphi \cos (\omega_0 \tau) \\
	u^3 & =  - c \sin \theta  \cos (\omega_0 \tau)  \label{Eq_1_57a}
\end{align}
\begin{align}  \nonumber
	s^1 & =  \frac{\hbar}{2} \sin \theta \cos \varphi = \frac{\hbar}{2} n^1  \\  \nonumber
	s^2 & =  \frac{\hbar}{2} \sin \theta \sin \varphi  = \frac{\hbar}{2} n^2  \\
	s^3 & =  \frac{\hbar}{2} \cos \theta = \frac{\hbar}{2} n^3  \label{Eq_1_57b}
\end{align}
As expected, the $s^j$ are the projections of the spin vector $\mathbf{s}_n =  \frac{\hbar}{2} \mathbf{n}$ onto the axes defined by the orthonormal triad of 3-vectors $\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \}$.

For the special cases $\theta = 0$ and $\theta = \pi$, Eqs.~\eqref{Eq_1_57a} and ~\eqref{Eq_1_57b} give agreement with the velocity and spin components for the spin-up and spin-down states
in Eqs.~\eqref{Eq_1_50a} and~\eqref{Eq_1_51a}, respectively, as expected.
The corresponding state functions $\phi_{up}$ and $\phi_{dn}$ are given by substituting $\theta = 0$ and $\theta = \pi$ in the coefficients in Eqs.~\eqref{Eq_1_56a} and using them in Eq.~\eqref{Eq_1_48a}:
\begin{align}  \nonumber
	\phi_{up} & = \left[ \frac{1}{\sqrt{2}} \exp (-i (\omega_0 \tau + \varphi)/2), 0, 0, \frac{1}{\sqrt{2}} \exp (i (\omega_0 \tau + \varphi)/2) \right]^T \\
	\phi_{dn} & = \left[ 0, \frac{1}{\sqrt{2}} \exp (-i (\omega_0 \tau - \varphi)/2), - \frac{1}{\sqrt{2}} \exp (i (\omega_0 \tau - \varphi)/2), 0 \right]^T \label{Eq_1_58a}
\end{align}
The state function $\phi_n (\tau)$ for spin about direction $\mathbf{n}$ can therefore be written as a superposition of $\phi_{up}$ and $\phi_{dn}$, which are orthogonal unit vectors under the usual Hermitian inner product on ${\cal C}^4$.
Furthermore, using Euler's formula for $\exp (i \varphi/2)$, it is readily seen that $\phi_{up}$ and $\phi_{dn}$ are themselves linear superpositions of ``positive energy" and ``negative energy" solutions, which are
the orthogonal unit vectors given by setting $\varphi = 0$ and $\varphi = \pi$ in each of Eqs.~\eqref{Eq_1_58a}. Therefore, the spin state for direction $\mathbf{n}$ is a linear superposition of spin-up and spin-down states as follows:
\begin{align}  \nonumber
	    \phi_{n} = & \cos \frac{\theta}{2} \phi_{up} + \sin \frac{\theta}{2} \phi_{dn}  \\ \nonumber
	    		= & \cos \frac{\theta}{2} \cos \frac{\varphi}{2} \left[ \frac{1}{\sqrt{2}} \exp (-i \omega_1 \tau), 0, 0,  \frac{1}{\sqrt{2}} \exp (i \omega_1 \tau) \right]^T \\ \nonumber
			 - i & \cos \frac{\theta}{2} \sin \frac{\varphi}{2} \left[ \frac{1}{\sqrt{2}} \exp (-i \omega_1 \tau), 0, 0, - \frac{1}{\sqrt{2}} \exp (i \omega_1 \tau) \right]^T \\ \nonumber
			 + & \sin \frac{\theta}{2} \cos \frac{\varphi}{2} \left[ 0, \frac{1}{\sqrt{2}} \exp (-i \omega_1 \tau), - \frac{1}{\sqrt{2}} \exp (i \omega_1 \tau), 0 \right]^T \\
			 + i & \sin \frac{\theta}{2} \sin \frac{\varphi}{2} \left[ 0, \frac{1}{\sqrt{2}} \exp (-i \omega_1 \tau),  \frac{1}{\sqrt{2}} \exp (i \omega_1 \tau), 0 \right]^T \label{Eq_1_59a}
\end{align}

Consider suppressing the contribution to the superposition for $\phi_n (\tau)$ that comes from the ``negative energy" solution by removing the third and fourth components from each of the four column vectors
in Eqs.~\eqref{Eq_1_59a}, then the common factor $\exp (-i \omega_1 \tau)$ has no effect when calculating dynamic variables and so can be removed. This changes $\phi_n$ to the superposition
\begin{equation}  \nonumber
	\widetilde{\phi}_n = \frac{1}{\sqrt{2}} \cos \frac{\theta}{2}  \exp (-i \frac{\varphi}{2}) [1,0]^T + \frac{1}{\sqrt{2}} \sin \frac{\theta}{2}  \exp (i \frac{\varphi}{2}) [0,1]^T
\end{equation}
\noindent as in Pauli spin theory.

It is readily shown from Eqs.~\eqref{Eq_1_57a} that $\mathbf{u} \cdot \mathbf{n} = 0$ so that the electron motion produced by this superposition of spin-up and spin-down states does indeed lie in a plane orthogonal
to the spin direction $\mathbf{n}$. The superposition spin state is therefore physically distinct from each of the four component spin states, which on their own produce motion confined to the $x^1-x^2$ plane.
Furthermore, from Eqs.~\eqref{Eq_1_57a}, $\mathbf{u} \cdot \mathbf{u} = c^2$ and $\dot{\mathbf{u}} \cdot \dot{\mathbf{u}} = c^2 \omega_0^2$, so the constraints $C1$ and $C2$ in Eqs.~\eqref{C1_C2b} are indeed satisfied.
Constraint $C3$ is trivially satisfied because here it is just a statement of the rest energy: $\pi_0 u_0 = E = mc^2$, the kinetic energy of the spin motion.
As a final remark, the electron cannot be spinning simultaneously about two distinct spin axes whose directions correspond to $\mathbf{n}_1$ and  $\mathbf{n}_2$, say, so the Hermitian matrix operators
$\Sigma_1$ and  $\Sigma_2$ corresponding to the spin vectors according to Eq.~\eqref{Eq_1_55a} cannot have common eigenvectors, implying that they are not commutative.

\subsection{Neo-Classical Wave Function and Derivation of Dirac's Wave Equation}

Consider first a \emph{free} electron, then the Lorentz transformation in Eq.~\eqref{Eq_1_10b} gives proper time $\tau(x)$ as a function of an observer's 4-vector position $x$ for the electron.
Define the neo-classical \emph{wave function} $\psi(x) = \phi(\tau(x))$ in terms of the state function  $\phi(\tau)$.
Consider now the operator $i \hbar \partial_{\mu} = i \hbar \frac{\partial} {\partial x^{\mu}}$ acting on this wave function:
\begin{equation} \label{Eq_1_35}
	i \hbar \partial_{\mu} \psi(x) = i \hbar \dot{\phi}(\tau(x)) \partial_{\mu} \tau = \partial_{\mu} \tau \hat{H} \psi(x)
\end{equation}
\noindent using Eq.~\eqref{Eq_1_23}. We deduce from Eq.~\eqref{Eq_1_6} that $mc^2 \partial_{\mu} \tau = \pi_{\mu}$ since $d\tau = \partial_{\mu} \tau~dx^{\mu}$ and $dx^{\mu}$ $(\mu = 0,1,2,3)$ is arbitrary. This implies:
\begin{equation} \label{Eq_1_36}
	i \hbar \partial_{\mu} \psi(x) =  \frac{\pi_{\mu}} {mc^2} \hat{H} \psi(x)
\end{equation}
\noindent and so:
\begin{equation} \label{Eq_1_37}
	\bar{\psi}(x)~(i \hbar \partial_{\mu}) \psi(x) = \frac{\pi_{\mu}} {mc^2} \bar{\psi} \hat{H} \psi = \pi_{\mu}
\end{equation}
\noindent using Eq.~\eqref{Eq_1_30}. Thus, the operator $\hat{\pi}_{\mu} = i \hbar \partial_{\mu}$ acting on the neo-classical wave function gives the kinetic momentum, as in QM theory.

From Eq.~\eqref{Eq_1_36}:
\begin{equation} \label{Eq_1_38}
	c \gamma^{\mu} \hat{\pi}_{\mu} \psi(x) = \frac{1} {mc^2} \hat{H}^2 \psi(x)
\end{equation}
\noindent Applying the identity in Eq.~\eqref{Eq_1_39}, we get the covariant form of \emph{Dirac's wave equation} ({\color{Green}Bjorken and Drell (1964)}):
\begin{equation} \label{Eq_1_40}
	\widetilde{H} \psi(x) = \hat{u}^{\mu} \hat{\pi}_{\mu} \psi(x) = mc^2 \psi(x)
\end{equation}
\noindent showing that it corresponds to an eigenvalue equation for the Hamiltonian energy operator $\widetilde{H} =  \hat{u}^{\mu} \hat{\pi}_{\mu}$.
Dirac's equation can also be interpreted as an operator form of the constraint $C3$ in Eq.~\eqref{Eq_1_7}.
It is a linear partial differential equation for the neo-classical wave function $\psi(x)$ whereas the set of ordinary differential equations in Eq.~\eqref{Eq_1_31} involving the state function $\phi(\tau)$ is nonlinear in general.
However, since the partial differential equation is for the field $\psi(x)$ over the space of the observer's 4-coordinates, this comes at the cost of no longer tracking the space-time path of the electron.
Therefore, information present in the original equations of motion is lost in Dirac's wave equation, implying that it no longer gives a complete description of the electron.

From Eq.~\eqref{Eq_1_40}:
\begin{equation} \label{Eq_1_40b}
	\widetilde{H}^2 \psi = \widetilde{H} (mc^2 \psi) = (mc^2)^2 \psi
\end{equation}
\noindent where
\begin{equation} \label{Eq_1_40c}
	\widetilde{H}^2 =  \hat{u}^{\mu}  \hat{u}^{\nu} \hat{\pi}_{\mu} \hat{\pi}_{\nu} = - (c \hbar)^2 \partial_\mu \partial_\nu (\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu)/2
				= - (c \hbar)^2 \partial_\mu \partial_\nu g^{\mu\nu} I_4
\end{equation}
\noindent so
\begin{equation} \label{Eq_1_40d}
	 \left[ \partial_\mu \partial^\mu + \left( \frac{mc}{\hbar} \right)^2 \right] \psi = 0
\end{equation}
Thus, each component of the neo-classical wave function $\psi$ for the free electron satisfies the Klein-Gordon equation, the QM theory version of Eq.~\eqref{Eq_1_41}.

We can get Dirac's equation for the case of an electron in an em-field by using the usual minimal coupling from QM theory, as did {\color{Green}Dirac (1928)} after developing his equation for the free electron.
Thus, the kinetic momentum operator in Eq.~\eqref{Eq_1_40} becomes $\hat{\pi}_{\mu} = i \hbar \partial_{\mu} - qA_{\mu}$, so that now $\hat{p}_{\mu} = i \hbar \partial_{\mu}$ is the canonical momentum operator.
However, minimal coupling in the neo-classical Lagrangian function, which `is given in Eq.~\eqref{Eq_1_11}, does not produce the same Dirac equation as this QM minimal coupling prescription.
Indeed, the derivation presented for Dirac's wave equation does not work in the presence of an em-field by setting the operator $i \hbar \partial_{\mu}$ equal to $\hat{p}_{\mu}$ rather than $\hat{\pi}_{\mu}$.
Furthermore, the neo-classical Dirac-Schr\"{o}dinger equation~\eqref{Eq_1_23} for the state function needs to be supplemented by the Lorentz force law, as in Eqs.~\eqref{Eq_1_31},
because the Hamiltonian $\hat{H}(\tau)$ in Eq.~\eqref{Eq_1_23} depends on the momentum $\pi(\tau)$, whereas the Dirac Hamiltonian $\widetilde{H}$ does not.
The Lorentz force law is implicit in Dirac's wave equation under the minimal coupling prescription; it appears, for example, in the Heisenberg equations from Dirac's equation in proper time
({\color{Green}Barut (1987)}) and is identical to the last equation in  Eqs.~\eqref{Eq_1_34}.

From the Lorentz transformation in Eq.~\eqref{Eq_1_10b}, setting $\omega_1 = \frac{mc^2}{\hbar} = \frac{1}{2} \omega_0$,
\begin{equation} \label{Eq_1_53}
	\omega_1 \tau(x) = (\pi*x) / \hbar = \left( Et - \mathbf{P} \cdot \mathbf{x} \right) / \hbar \equiv \theta(x)
\end{equation}
\noindent Substituting this in Eq.~\eqref{Eq_1_42}, the solution for the neo-classical wave function for the free electron with global momentum $\pi$ is:
\begin{equation} \label{Eq_1_54}
	\psi(x) = \phi(\tau(x)) = \left[ \cos \theta(x) I_4 - \frac{i}{mc^2} \sin \theta(x) \hat{H} \right] A
\end{equation}
\noindent for arbitrary energy-normalized $A \in {\cal C}^4$.
This equation shows that the phase $\theta(x)$ defined in Eq.~\eqref{Eq_1_53} gives the wave function $\psi(x)$ an apparent plane-wave characteristic with the well-known De Broglie frequency-energy and
wavenumber-momentum relations, as well as an apparent wave propagation speed of $\frac{c^2}{V} > c$ when viewed by an observer fixed with respect to the reference frame $X_o$.
Also, $2 \theta(x)$ gives the position of the free electron in its circular motion about the spin center (to within an unknown additive constant) when it has inertial observer coordinates $x = (ct, \mathbf{x})$.
Furthermore, $S(x) = \theta(x) \hbar = \pi*x$ is the action function from classical electron mechanics.

Since Dirac's wave equation is linear, a superposition of solutions of the form of Eq.~\eqref{Eq_1_54},  each with a distinct  global momentum $\pi$, is also a solution.
Each such solution has a distinct proper time $\tau(x)$ according to Eq.~\eqref{Eq_1_53} when the electron has space-time coordinates $x$ relative to $X_o$.
Such a superposition of neo-classical wave functions has no clear interpretation in the present theory whereas in QM theory,  a superposition of wave functions over all possible global momenta
is postulated to provide a probability distribution (at each observer time $t$) over the electron's spatial coordinates in a region known to contain the electron.

Expressing the sine and cosine terms as complex exponentials:
\begin{align}  \nonumber
	\psi(x) & = \phi(\tau(x)) = \left[ \frac{1}{2} \left(I_4 + \frac{1}{mc^2} \hat{H} \right) \exp (-i \theta(x)) + \frac{1}{2} \left(I_4 - \frac{1}{mc^2} \hat{H} \right) \exp (i \theta(x)) \right] A \\
		  & =  \psi^+(x) + \psi^-(x)  \label{Eq_1_55}
\end{align}
\noindent where from the results of the previous subsection,
\begin{equation} \label{Eq_1_56}
	\psi^+ (x) = \exp (- i\theta(x)) A^+, \hspace{0.2 in} \psi^- (x) = \exp ( i\theta(x)) A^-
\end{equation}
\noindent are the so-called \emph{positive energy} and \emph{negative energy} wave function solutions, although they both have positive energy
since they are just the previous neo-classical state functions $\phi^+$ and $\phi^-$ in Eq.~\eqref{Eq_1_46a} with $\tau$ transformed using Eq.~\eqref{Eq_1_53}.
Eq.~\eqref{Eq_1_55} agrees with the wave function solution of Dirac's wave equation given in textbooks (e.g. {\color{Green}Bjorken and Drell (1964)}).

Recall that $\hat{H}$ incorporates the specified global kinetic momentum $\pi$ and it is replaced by the operator $\hat{\pi}$ in $\widetilde{H}$.
For the free electron case considered here, Eqs.~\eqref{Eq_1_56} show that $\hat{\pi}_\mu \psi^+ (x) = \pi_\mu \psi^+ (x)$ whereas $\hat{\pi}_\mu \psi^- (x) = - \pi_\mu \psi^- (x)$,
so $\widetilde{H} \psi^+(x) = \hat{H} \psi^+(x)$ and $\widetilde{H} \psi^-(x) = - \hat{H} \psi^-(x)$. Dirac's equation therefore gives $\hat{H} \psi^+(x) = mc^2 \psi^+(x)$ and $\hat{H} \psi^-(x) = - mc^2 \psi^-(x)$,
which is consistent with the result in the previous subsection  that $A^+$ and $A^-$ are eigenvectors of $\hat{H}$ belonging to the two-dimensional eigenspaces corresponding to eigenvalues $mc^2$ and $-mc^2$, respectively.

When the free electron is at rest globally relative to the observer's inertial reference frame at $O$, then the global 3-vector momentum $\mathbf{P} = \bf{0}$, the kinetic energy is just the spin energy $E=mc^2$ and
from Eq.~\eqref{Eq_1_53}, proper time $\tau = t$, the observer time. The wave function $\psi_n$ for the electron spinning about direction $\mathbf{n}$ is then identical to the state function $\phi_n$ in Eq.~\eqref{Eq_1_59a}.

\section{On Introducing Probability into the Neo-classical Theory}

The neo-classical spin model presented here has a complete set of physical variables describing the electron but that does not mean that their values will always be known with certainty.
Any variables whose values are uncertain can be described by an appropriate probability distribution (e.g. {\color{Green}Beck (2018, 2019)}).
Although a detailed study of what probability distributions should be added to the present neo-classical theory to bring it closer to QM theory is left for future work,
a few connections are made in this subsection.

\subsection{Uncertainty in Initial Conditions and Liouville's Equation}

The most general case for the electron spin model requires a joint PDF (probability density function) over 16 physical variables and it is a function of proper time $\tau$.
Consider, for example, the PDF $p(x, u, y, \pi | \tau)$ for the state-space form of the equations of motion of the electron given in Eqs.~\eqref{Eq_1_17}.
Only the joint PDF for the initial conditions $x(0), u(0), y(0), \pi(0)$ at $\tau = 0$ needs to be specified since $p(x, u, y, \pi | \tau)$ satisfies Liouville's equation from stochastic dynamics theory
(e.g. {\color{Green}Gardiner (1985)}), a PDE to be solved using as the initial condition $p(x, u, y, \pi | 0)$.
This PDE applies when the evolution of the state is described by deterministic equations, as in the neo-classical model here, but the initial conditions are uncertain.

Consider now the state-space form of the equations of motion of the electron given in Eqs.~\eqref{Eq_1_31}. For the free electron, if $\phi(0)$ and $\pi(0)$ are specified, then $\phi(\tau)$ and $\pi(\tau) = \pi(0)$ are known.
There remains the possibility, however, that the initial space-time coordinate $x(0)$ of the electron is uncertain. In this case, the first of Eqs.~\eqref{Eq_1_31} implies that
$\dot{x}^{\mu}(\tau)  = \bar{\phi}(\tau) \hat{u}^{\mu} \phi(\tau) = u^{\mu}(\tau)$ is known and so $p(x | \tau, \phi(0), \pi(0))$ satisfies Liouville's equation:
\begin{equation} \label{Eq_1_60}
	\partial_\tau p(x |\tau, \phi(0), \pi(0)) + \partial_\mu \left[ p(x |\tau, \phi(0), \pi(0)) \, u^\mu (\tau(x)) \right] = 0
\end{equation}
\noindent where $\tau(x)$ is the proper time for the electron when it is at the space-time point $x$, which for a free electron with global momentum $\pi$ is given by  $\tau(x) = \tau_0 + (\pi * x)/ (mc^2)$.
Here $\tau_0$ is an arbitrary constant that would be zero if $\tau=0$ is chosen to occur when $x=0$, that is, at the origin $\mathbf{x}=0$ at observer time $t=0$.

If the uncertainty in $x$ does not change with proper time $\tau$, then the PDF $p(x | \phi(0), \pi(0))$ is a solution of  the ``stationary''  Liouville's equation:
\begin{equation} \label{Eq_1_61}
	\partial_\mu \left[ p(x | \phi(0), \pi(0)) \, u^\mu (\tau(x)) \right] = 0
\end{equation}
\noindent implying that only an initial distribution $p(x | \phi(0), \pi(0))$ satisfying this PDE gives a stationary PDF.  Define
\begin{equation} \label{Eq_1_62}
	 U^\mu (x) = c \frac{u^\mu (\tau(x))}{u^0 (\tau(x))} = \frac{\dot{x}^\mu (\tau(x))}{\dot{t} (\tau(x))} = \frac{dx^\mu}{dt} \left(\tau(x) \right)
\end{equation}
\noindent implying that $U^0(x) = c$, then the stationary Liouville equation becomes:
\begin{equation} \label{Eq_1_63}
	\partial_\mu \left[ p(x | \phi(0), \pi(0)) \, \dot{t} (\tau(x)) \, U^\mu(x) \right] = 0
\end{equation}
\noindent where $U^\mu (x)$ can be expressed using the total velocity operator $\hat{u}^\mu = c \gamma^\mu$ and Dirac's wave function $\psi(x) = \phi(\tau(x))$ for the free electron:
\begin{equation} \label{Eq_1_64}
	 U^\mu (x) =  \frac {\bar{\psi}(x) \hat{u}^{\mu} \psi(x)}{\psi^*(x) \psi(x)}
\end{equation}

Consider now the continuity equation derived from Dirac's wave equation:
\begin{equation} \label{Eq_1_65}
	0 = \partial_\mu \left[ j^\mu (x) \right] = \partial_\mu \left[ \psi^*(x) \psi(x) \, U^\mu (x) \right]
\end{equation}
\noindent where the probability current is defined by:
\begin{equation} \label{Eq_1_66}
	 j^\mu (x) =  \bar{\psi}(x) \hat{u}^{\mu} \psi(x) = \psi^*(x) \psi(x) \, U^\mu (x)
\end{equation}
\noindent In QM, $ \psi^*(x) \psi(x)$ is identified with $p(x | \psi)$, viewed as the PDF $p(\mathbf{x}, t | \psi)$ for the spatial position $\mathbf{x}$ of the electron at time $t$,
conditional on knowing the wave function $\psi$, so the continuity equation becomes with this identification:
\begin{equation} \label{Eq_1_67}
	\partial_\mu \left[ p(x | \psi) \, U^\mu (x) \right] = 0
\end{equation}
\noindent Notice that if $\dot{t} (\tau(x)) = 1$ for all $x$, as in the non-relativistic case, the continuity equation and the stationary Liouville equation are identical with the common solution
$p(x | \phi(0), \pi(0)) = p(x | \psi)$ that is stationary with respect to $\tau = t$ and so is just $p(\mathbf{x} | \psi)$.
In general, however, $\dot{t} (\tau(x))$ is highly oscillatory and $p(x | \phi(0), \pi(0)) \, \dot{t} (\tau(x))$ cannot be identified with the PDF $p(\mathbf{x}, t | \psi)$ to give agreement between
Liouville's equation and the continuity equation with the stated QM identification of $ \psi^*(x) \psi(x)$ as a PDF.

For the free electron, Liouville's equation in Eq.~\eqref{Eq_1_61} has the solution $p(x)$ constant over a finite space-time region ${\cal V}$ where it can be normalized so that
$\int_{\cal V} p(x) dx = 1$. This is evident from the fact that the continuity equation derived from Dirac's wave equation for $\psi(x) = \phi(\tau(x))$ is:
\begin{equation} \label{Eq_1_68}
	\partial_\mu \left[ u^\mu (\tau(x)) \right] = \partial_\mu \left[ \bar{\psi}(x)  \, \hat{u}^\mu \psi(x)  \right] = 0
\end{equation}
\noindent Thus, for any proper time $\tau$, the uncertainty in the electron's space-time coordinates corresponds to a uniform distribution over ${\cal V}$;
in particular, the PDF for the free electron's spatial coordinates, $p(\mathbf{x})$, is constant, as in QM theory.

\subsection{Uncertainty in Spin Measurement Outcomes and Malus's Law}

As noted at the end of the previous section, the  wave function $\psi_n(x) $ for the free electron spinning about direction $\mathbf{n}$ is identical to the state function $\phi_n(\tau)$ given by the superposition in Eq.~\eqref{Eq_1_59a}.
Thus, applying Born's rule from QM, the coefficients $\cos \frac{\theta}{2}$ and $\sin \frac{\theta}{2}$ when squared give the probabilities $P_{up}$ and $P_{dn}$ of getting spin-up and spin-down states, respectively,
upon exit from a Stern-Gerlach device, given that the electron is in a state of spin about the direction $\mathbf{n}$ upon entry:
\begin{equation} \label{Eq_1_57}
	P_{up} = \cos^2 \frac{\theta}{2} = \frac{1}{2} \left( 1 + \mathbf{n} \cdot {\bf e}_3 \right),  \hspace{0.1in} P_{dn} = \sin^2 \frac{\theta}{2} = \frac{1}{2} \left( 1 - \mathbf{n} \cdot {\bf e}_3 \right)
\end{equation}
\noindent This is Malus's Law for spin-1/2 particles (e.g. {\color{Green}W\'odikiewicz (1985)}).

It is interesting to note that the expressions for $u^i (i=1,2)$ in Eqs.~\eqref{Eq_1_57a} for the electron spinning about direction $\mathbf{n}$ upon entering the Stern-Gerlach device
are equal to the predicted means of these velocities for the electron emerging from the device, since the expressions can be interpreted as the sum $P_{up} u^i_{up} + P_{dn} u^i_{dn}$
where $u^i_{up}$ is given by Eqs.~\eqref{Eq_1_50a} and $u^i_{dn}$ is given by Eqs.~\eqref{Eq_1_51a}. However, $u^3$ in Eqs.~\eqref{Eq_1_57a} does not agree with the mean velocity in the ${\bf e}_3$ direction,
which is obviously zero because $u^3_{up}$ and $u^3_{dn}$ are both zero.

Born's rule could be incorporated into the neo-classical spin theory to extend it to spin ``measurement" outcomes.
This gives the state function a role beyond being just a mathematical device for conveniently solving Eqs.~\eqref{Eq_1_18}, or their equivalent in Eqs.~\eqref{Eq_1_1}, suggesting that it captures some additional physics.
The term ``measurement" of spin here, although commonly used, is misleading because it suggests that a pre-existing spin direction is being measured; however, as inferred from Malus's Law,
the incoming spin direction is rotated by the magnetic field of the Stern-Gerlach device into either the up or the down direction, albeit in a deterministically unpredictable way.

There has been a long interest in examining the possibility that this indeterminacy in spin outcomes is due to ``hidden variables" (e.g. {\color{Green}Bell (1987), Brunner et al. (2014)}).
In the context of the neo-classical model presented here, since the spin direction is an input in Malus's Law, this leaves the spin phase, which specifies where the electron is in its spin cycle,
as the only possible hidden variable that could influence the outcome.
An analogous setting that comes to mind is the flipping of a coin. If the initial conditions for the center of mass and angular velocity of the coin where known to a very high precision, then the outcome of a head or a tail
could be predicted deterministically; however, the  angular velocity is so high that the outcome is very sensitive to any uncertainty in the values of the initial conditions, so that in practice, it is uncertain.
The spin frequency $\omega_0$ of the electron is ultra-high but in contrast to the flipping of the coin, it is not clear if the spin outcome for the electron emerging from the Stern-Gerlach device could be dependent on the spin phase.

\section{Concluding Remarks}

The neo-classical spin model presented here gives a  \emph{complete} relativistic mechanics description of the electron where its total motion is described by a fourth-order differential equation in proper time
that is equivalent to a second-order equation for the electron's luminal motion about its spin center coupled to an equation for the sub-luminal motion of this center that corresponds to Newton's second law.
As a result, the model exhibits spin as part of the electron's dynamic motion that explains Schr\"{o}dinger's zitterbewegung, the magnetic moment produced by the Lorentz force acting on this spin,
the wave function that satisfies Dirac's relativistic wave equation, and wave function superposition to give its free spin dynamic motion (but not measurement outcome probabilities).
The wave function is simply the neo-classical state function satisfying a Dirac-Schr\"{o}dinger spinor equation after using the Lorentz transformation to express proper time
in the state function in terms of an observer's space-time coordinates.
Although the theory exhibits a magnetic moment that is a half of Dirac's theory, it does so by deriving an energy equation with a spin-field interaction term that is the sum of of two energy terms
due to electric and magnetic moments produced by the Lorentz force, rather than invoking the usual minimal coupling prescription assumed in quantum mechanics, as Dirac did to extend his free-electron theory.

The presented theory provides a basis for gaining a deeper understanding of the foundations of QM because its equations of motion incorporate dynamical variables that are at a deeper level than in Dirac's wave equation.
As in classical mechanics, however, the initial values of these dynamic variables may be uncertain and so a probability model would be appropriate for the "hidden variables"; in particular, those variables associated with
the ultra-high frequency spin motion, such as the initial phase angle for its circular motion.