# Zitterbewegung, internal momentum and spin of the circular travelling wave
electromagnetic electron

Malik Mohammad Asif<EMAIL_ADDRESS>Salman Khan<EMAIL_ADDRESS>COMSATS Institute of Information Technology, Park Road, Tarlai Kalan,
Islamabad 45550, Pakistan

(December 14, 2015)

###### Abstract

The study of this paper demonstrates that electron has Dirac delta like
internal momentum ($u,\overrightarrow{p}_{\theta}$), going round in a circle
of radius equal to half the reduced Compton wavelength of electron with
tangential velocity $c$. The circular momentum $\overrightarrow{p}_{\theta}$
and energy $u$ emanate from circular Dirac delta type rotating monochromatic
electromagnetic (EM) wave that itself travels in another circle having radius
equal to the reduced Compton wavelength of electron. The phenomenon of
Zitterbewegung and the spin of electron are the natural consequences of the
model. The spin is associated with the internal circulating momentum of
electron in terms of four component spinor, which leads to the Dirac equation
linking the EM electron model with quantum mechanical theory. Our model
accurately explains the experimental results of electron channelling
experiment, [P. Catillon et al., Found.Phys. $38$, $659$ ($2008$)], in which
the momentum resonance is observed at $161.784MeV/c$ corresponding to
Zitterbewegung frequency of $80.874MeV/c$ electron beam.

Classical field theories; Models beyond standard model; Miscellaneous
theoretical ideas and models; Electrons (including positrons); Theory of
quantized fields

###### pacs:

03.50; 12.60.-i; 12.90.+b; 14.60.Cd; 03.70.+k

##  I Introduction

The electron is elementary particle, embodied with intrinsic properties such
as charge, mass, spin and magnetic dipole moment. Historically, there are
number of proposed extended classical models of electron Jansen ; Barut ;
Dirac2 ; Dirac3 ; Burinskii ; Qiu Hong  encompassing classical properties
effectively. However, these models not only suffer from the electron
instability problem but also are unable to explain the quantum mechanical (QM)
behavior of electron. On the other hand, the celebrated QM model of Dirac
Dirac 1  is an abstract mathematical model, which incorporates the intrinsic
properties of electron in terms of Dirac matrices and $4$-component
wavefunction. Dirac’s later works on extended model of electron Dirac2 ;
Dirac3 ; Dirac 4  attempts to expound its classical properties. However, on
the conceptual background the comprehension of the abstract point QM model and
its possible association to electrons’s properties is still not well clear.

While working on Dirac electron in an external electromagnetic field, Foldy-
Wouthuysen noted that the particle spreads out over a region of dimension of
the order of its Compton wavelength in space with its mean position
$X^{{}^{\prime}}=x$. This behavior of the particle lead to the proposal of
extended model of electron Foldy L. L ; Franz Schwabl . Due to some reasons,
the extended models or their interpretation such as the one given in Foldy L.
L  didn’t get much attention. The detail studies on Dirac model by Schrödinger
and others E. Schrodinger ; Barut and A. J  show that the electron undergoes
an internal oscillatory motion having dimension of the order of Compton
wavelength, called Zitterbewegung (Ztg). From the Dirac equation of motion for
a free electron, Schrödinger showed that even in linear motion the electron
executes an oscillatory motion with a frequency $\nu_{o}=2m_{o}c/h$, where all
the parameters bear their traditional meaning. He further concluded that the
instantaneous velocity of a slowly moving electron is $c$.

Recently the electron model proposed by Alexander Burinskii Burinskii ;
Burinskii 1 ; Burinskii 2  considers electron as a closed gravitational
singular string, based on Kerr-Newman theory. The model is essentially in
curved space time geometry. Similarly, the electron model Atiyah  uses ($4+1$)
Ricci flat space time geometry to obtain exact solutions of the Maxwell’s and
gauged Dirac equations. These solutions are interpreted in terms of a
geometric model of the electron with its spin. Hence these models are
inherently geometric ones, based on curved or ($4+1$) space-time, instead of a
flat spacetime geometry. Again these models do not institute some physical
mechanism(s) at the core level. Using the Maxwell’s theory with the boundary
condition of spherical conducting surface, Dirac presented an electron model
Dirac 4 , which explains the existence of muon as the excited state of
electron, however, it does not assign spin to the electron.

In a very recent study, a traveling wave electromagnetic (TWEM) model for
electron is presented M.M.Asif . The model is based on rotating EM wave
confined in a circle with radius equal to the reduced Compton wavelength (RCW)
($\hslash/mc$) of electron using Maxwell’s theory as a physical mechanism at
the core level. In TWEM model the electric field vector $\vec{E}_{r}$ (a Dirac
delta vector) rotates in the $xy$ plane and the corresponding magnetic field
vector $\vec{H}_{z}$ is pseudo static Dirac delta vector along the
$z$-direction. The EM wave traverses a circular trajectory with RCW of the
electron in half space. The radius and energy of the modeled electron, matches
precisely to the classical counterparts. The stability of EM electron is
ensured by showing zero divergence of source-free EM energy-momentum stress
tensor for the model. The charge generation mechanism has been discussed at
length using gauge of EM fields. The time reversal in travelling wave
corresponds to the reversal of $\vec{E}$ and $\vec{B}$ fields, which in turn
gives rise to antiparticle (the positron) rotating EM wave. With four possible
time-space ($\pm t,\pm\theta$) combinations we obtain four types of EM
rotating waves. The overall picture at this level conforms the local $U(1)$
gauge symmetry. The monochromatic circulating EM wave, carry the corresponding
energy-momentum in half the RCW of electron. The Ztg phenomenon arises
naturally in this model as a consequence of circular flow of EM energy-
momentum.

In this paper we use the TWEM model to demonstrate the electron’s internal
structure in terms of circular flow of energy-momentum ($u_{em,}\
\vec{p}_{\theta}$) at velocity $c$. The two component wavefunction for energy-
momentum, corresponding to left and right circular polarized wave(s) is shown
to be related to the spin of electron, conforming to $SU(2)$ group symmetry.
Furthermore, the negative and positive time circulating energy-momentum are
shown to correspond to four component Dirac like spinor. In the rest frame,
the four components of spinor satisfy massless Klein Gordon equation, which in
turn leads to a massless Dirac or Weyl equation, governed by transformation
rule under covering group $SL(2,C)$ of proper Lorentz group $SO^{+}(3,1)$.
Finally, the massive Dirac equation for TWEM electron has been obtained in
boosted frame linking EM electron model with QM.

The internal dynamical structure of electron based on Ztg can be
experimentally verified in line with experiments of electron channeling in
silicon crystals Catillon ; Gouan . In electron channeling experiment, the
momentum resonance is observed at twice the de Broglie frequency, that is at
$161.784MeV/c$ instead of $80.874MeV/c$. The authors describe it as a
manifestation of de Broglie internal clock if squared amplitude is considered,
that is, the Ztg frequency. In our proposed model the frequency of momentum is
twice the de Broglie frequency of EM circulating wave. This is in good
agreement with the phenomena observed in the electron channelling experiment.

##  II Zitterbewegung and the velocity of electron

In this section, for the paper to be self-contained, we present a brief review
of the basic concepts of Ztg in electron. Although a number of different
interpretations of Ztg of Dirac model are given Foldy L. L ; David Hestenes 1
; David Hestenes 2 ; Sasabe ; E. Romera ; G. Cavalleri ; David Hestenes 3 ; D.
Hestenes ; K. Huang ; Zhi-Yong , however, we will briefly review the
interpretation given in E. Schrodinger ; Barut and A. J . The Dirac equation
for spin-$1/2$ particle is

$i\hbar\frac{\partial}{\partial t}\psi(r,t)=H\psi(r,t),$ (1)

where $H$ is the Hamiltonian for a free particle and can be written as

$H=c\vec{\alpha}\cdot\vec{p}+\beta mc^{2}.$ (2)

In Eq. (2) $\vec{\alpha}$ and $\beta$ are the well known $4\times 4$ Dirac
matrices which satisfy following conditions,

$\beta^{2}=I;\quad\alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=2\delta_{ij}I;\quad\alpha_{i}\beta+\beta\alpha_{i}=0.$
(3)

For a free electron, the velocity determined by Schrödinger E. Schrodinger ,
using Eq. (2) is

$\vec{v}=c\vec{\alpha},$

with $c$ being the expectation value of $\overrightarrow{v}$, which means the
electron is moving with an aggregate constant velocity $c$. It was further
illustrated that $\vec{v}$ can be expressed as.

$\vec{v}=c^{2}H^{-1}\vec{p}\mathbf{+}c\vec{\eta}_{o}e^{-2iHt/\hbar},$ (4)

where $\vec{p}$ is the linear momentum and $\vec{\eta}_{o}$ is a constant
operator given by

$\vec{\eta}_{o}=\vec{\eta}_{o}\mathbf{(}0\mathbf{)=}\vec{\alpha}\mathbf{(}0\mathbf{)-}cH^{-1}\vec{p}\mathbf{.}$

The velocity of electron in Eq. (4) consists of two parts, the regular part
$c^{2}H^{-1}\vec{p}$ and the oscillatory part
$c\vec{\eta}_{o}e^{-2iHt/\hbar}$. The later part represents Ztg having
amplitude equal to $\hbar/2m_{o}c$ (half the RCW of electron) and angular
frequency $\omega_{z}=2m_{o}c^{2}/\hbar$. The oscillatory part, having
extremely small amplitude, was believed unobservable directly P.A.M. Dirac 1 .
However, a recent simulated work R. Gerritsma  on one dimensional Dirac
dynamics for free particle shows that the Ztg is an observable phenomenon.
Also, J. Y. Vaishnav  proposes an experiment with ultracold atoms in a tripod
level scheme on an optical lattice, to observe Ztg, at experimentally
measurable frequencies. The internal dynamics of 4 momentum associated with
the oscillatory part of electron is the source of Ztg of electron and is at
the core of QM theory embedded in $\alpha_{i}$ and $\beta$ of Dirac matrices.

##  III Internal 4-momentum of Circular EM travelling wave electron

The relativistic momentum $\vec{p}^{\ \prime}$ of a free electron is given as

$\left|\vec{p}\ ^{\prime}\right|=\pm\sqrt{\vec{p}^{\ 2}+\vec{p}_{o}^{\ 2}},$
(5)

where $\vec{p}$ is the linear momentum in $x,y,z$ coordinates and
$p_{o}=m_{o}c$ is the magnitude of momentum associated with the rest mass of
electron. The momentum $p_{o}$ is Lorentz invariant and behaves as a constant
in the frame of the center of mass of the particle. However, there is dynamism
of energy-momentum when electron is at rest, as noted by Hestenes David
Hestenes 3  and others H. Torres-Silva . In this article we will consider the
internal momentum dynamism (spin) in the center of mass frame of electron,
which naturally arises in the TWEM model of electron M.M.Asif . The $EM$
momentum of electron in the particle’s rest frame $K_{p}(0)$ is given by

$\vec{p}_{o}\equiv\vec{p}_{\theta}=\frac{1}{c}\vec{S}_{\theta}.$ (6)

The angular momentum $\vec{p}_{\theta}$ is invariant momentum associated with
the spin of electron. In the rest frame, we have $\vec{p}=0$ in Eq. (5) and
$p^{\prime}=\left|\pm\vec{p}_{\theta}\right|=$
$\left|\frac{1}{c}\vec{S}_{\theta}\right|$, which leads to the massless -light
like case.

Using the internal momentum $\vec{p}_{\theta}$ and energy
$u_{em}=(\varepsilon_{o}\vec{E}_{r}^{2}+\mu_{o}\vec{H}_{z}^{2})/2$ of electron
in the rest state, we can define internal momentum of electron as follows

$p_{\theta}^{\mu}=(p_{\theta}^{o},\vec{p}_{\theta}),$ (7)

where $p_{\theta}^{o}=u_{em}/c$. The $p_{{}_{\theta}}^{\mu}$ apparently forms
$4$-vector with $p_{\theta}^{o}$, and $\vec{p}_{\theta}$ the time like and
spatial component(s), respectively, in the rest frame of electron. It,
however, does not transform as $4$-vector, since $p_{\theta}^{\mu}$ is the
internal degree of freedom of electron associated with the spin and subsist as
a $2D$ object. In the rest frame of electron, $p_{\theta}^{\mu}$ can be
considered to dwell in wrapped coordinates within the particle Barut and A. J
; GIOVANNI .

Figure 1: The vector $\vec{E}_{r}$ is circulating in a circle of redius equal
to RCW and the corresponding momentum $\vec{p}_{\theta}$ and energy
$cp_{\theta}$ are circulating in a circle of radius equal to $\frac{1}{2}$RCW.
The vectors $\vec{E}_{r}$ and $\vec{p}_{\theta}$ have angular frequency
$\omega_{o}$ and $2\omega_{o}$, respectively, but the same tangential velocity
$c$.

Using the circular EM travelling wave expressions for TWEM model of electron,
the $\mathbf{p}_{\theta}$ can be expressed as,

$\vec{p}_{\theta}=\frac{1}{c}p_{o}^{{}^{\prime}}e^{\pm i/\hslash(-2u_{em}t\pm
2p_{\theta}r\theta)}\hat{\theta},$ (8)

where $p_{o}^{\prime}=\frac{1}{c}[E_{r}^{o}H_{z}^{o}]$. We have a similar
expression for the energy (time like) component $p_{\theta}^{o}=cp_{\theta}$.
There are four EM rotating (circular polarized like) waves with circulating
momentum given in Eq. (8). Two of these are associated with electron in
positive ($+ve$) time (or $+ve$ energy), right ($\uparrow$)$\equiv\psi^{1}$
and left circular ($\downarrow$) $\equiv\psi^{2}$ and other two associated
with positron in negative ($-ve)$ time (or $-ve$ energy), right ($\uparrow$)
$\equiv\psi^{3}$ and left circular ($\downarrow$) $\equiv\psi^{4}$. Therefore,
we can define four spinors $\psi^{i}$ ($i=1,2,3,4$) corresponding to four
circulating polarized waves with four time-space ($\pm t,\pm\theta$)
combinations.

Initially, we consider the case of $\psi^{1}$ (in $+t,+\theta$ coordinates)
and will extend the idea at the end to encompass four component
$\vec{\psi}\cong\psi^{\alpha};$ $\alpha=1,2,3,4$, to arrive at Dirac equation.
The momentum in coordinate combination ($+t,+\theta$ ) is

$\vec{p}_{\theta}=p_{o}^{\prime}[e^{-i(\omega_{o}^{\prime}t-k_{\theta}^{\prime}\mathbf{\theta})}]\hat{\theta},$
(9)

with $\omega_{o}^{\prime}=2\omega_{o}$ and $k_{\theta}^{\prime}=$
$2\omega_{o}r/c$. In this case the electric field vector $\vec{E}$
($+t,+\theta$ ) rotates with angular velocity $\omega_{o}$, in a circle of
radius $r_{o}\equiv$RCW. In Eq. (9), we note that the momentum
$\vec{p}_{\theta}$ rotates with twice the angular velocity of $\vec{E}_{r}$
vector, in a circle of radius $r_{o}/2$. From Eq. (9), we determine the
angular velocity (or frequency) of energy-momentum as follows

$\omega_{\theta}^{p}=\omega_{\theta}^{u}=\frac{\omega_{o}^{\prime}}{k_{\theta}^{\prime}}=\frac{c}{r_{o}},$
(10)

where $\omega_{\theta}^{p}$ and $\omega_{\theta}^{u}$ are the angular
velocities of momentum and energy, respectively. The corresponding tangential
velocities are $\left|\vec{v}^{p}\right|=\left|\vec{v}^{u}\right|=c$, (see
Fig. (1)). The angular frequency $\omega_{o}^{\prime}$ of energy-momentum is
in conformity with Ztg angular frequency E. Schrodinger ; Barut and A. J .

As both the momentum $\vec{p}_{\theta}$ and the energy $p_{\theta}^{o}c$ are
circulating with the velocity of light, hence, we will first consider massless
Dirac or Weyl spinor within $(1/2)$ RCW of electron. In fact, the Schrödinger
result for the expectation value of $\overrightarrow{v}$ equals $c$ for Ztg of
electron is valid only if the electron (or its constituent spinors) behave as
massless object(s) in the relevant region Qiu Hong . We will however see that
with boost, the left and the right spinors combine through the mass term. This
leads to the actual velocity of the particle in the moving frame and the
spinors still circulate at $c$, the velocity of light.

The extended TWEM model of electron exhibits internal geometry in the form of
rotating transverse EM wave in a circle of radius $r_{o}$ (RCW). That is the
source free, Dirac delta type EM fields exist in a sphere of radius $r_{o}$.
The energy-momentum of EM travelling wave flows in a circle of radius
$(1/2)r_{o}$ synchronously with EM wave in the $xy$ plane with speed $c$, as
shown in Fig.(1). The circulating energy-momentum subsist in $SU(2)$ symmetry
group, bestows QM spin satisfying all $SU(2)$ commutation relations. As we
show below, the magnitude of $z$-component of spin turns out to be $\hslash/2$
and the helicity corresponding to right circular or left circular states
determine its spin up or down state, respectively. A similar geometry of
extended model of electron based on Ztg, is described in Bruno .

##  IV Electron in the rest state

For electron in the rest state, Eq. (9) may be written as

$\vec{p}_{\theta}=p_{\theta}e^{i\theta^{\prime}}\hat{\theta},$ (11)

where $\theta^{\prime}=(2m_{o}r/\hslash)\theta$ and time exponential part is
implicit. Using cylindrical coordinates, we can write

$\vec{p}_{\theta}\equiv p_{x}\pm ip_{y}$ (12)

This is the relation for momentum (and the corresponding energy), rotating,
counterclockwise (ccw) and clockwise (cw) in $xy$ plane, for $+$ and $-$ signs
in Eq. (12), respectively. Then using the analogy of circular polarized waves
as the combination of two orthogonal linear polarized waves with a phase
difference of $\pm\pi/2$, we get

$p_{\theta}e^{i\theta^{\prime}}=p_{o}(\hat{x}\pm
i\hat{y})e^{i\theta^{\prime}}.$ (13)

Similar expression can be obtained for the energy $u_{em}$. The energy-
momentum taken together form four vector like object $p_{\theta}^{\mu}$. Now,
we can write two component wave function in momentum space
$\psi^{\nu}(\vec{p})$, for $\nu=1,2,$

$\psi(\vec{p})=\left[\begin{array}[]{c}\xi_{R}\\\
\xi_{L}\end{array}\right]e^{i\theta^{\prime}}=\left[\begin{array}[]{c}\psi^{1}\\\
\psi^{2}\end{array}\right],$ (14)

where the right and the left eigenfunctions are, respectively, given by
$\xi_{R}\equiv(\xi_{R}^{u},\xi_{R}^{p})$ and
$\xi_{L}\equiv(\xi_{L}^{u},\xi_{L}^{p})$. The superscripts $u$ and $p$ denote
the energy and momentum parts, respectively. Moreover, the $4$-component
spinor $\psi^{\alpha}(\vec{p})$; $\alpha=1,2,3,4$ is obtained by including the
$-ve$ time (or $-ve$ energy).

The four component wavefunction is the solution of the following second order
wave equation in cylindrical coordinates M.M.Asif ,

$\left[\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}-\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}\right]\psi^{\alpha}(\vec{p})=0.$
(15)

This equation is analogous to massless Klein-Gordon equation in momentum
space. Using the standard procedure for obtaining first order differential
equation from second order, we get the massless Dirac equation for the spinor
$\psi^{\alpha}(\vec{p})$

$i\hslash\left[\gamma^{o}\frac{\partial}{\partial
t}+\gamma^{i}\frac{1}{r}\frac{\partial}{\partial\theta}\right]\psi^{\alpha}(\vec{p})=0;\quad
i=1,2,3.$

Writing $\gamma^{\mu}$ in Weyl representation

$\gamma^{o}=\left[\begin{array}[]{cc}0&I\\\
I&0\end{array}\right],\quad\gamma=\left[\begin{array}[]{cc}0&\vec{\sigma}\\\
\mathbf{-}\vec{\sigma}&0\end{array}\right],$

where $I$ is $2\times 2$ identity matrix and $\vec{\sigma}$ are the Pauli spin
matrices. Using the units where $\hslash=c=1$, we arrive at Weyl equations or
massless Dirac equation in chiral representation

$\left[\begin{array}[]{c}(E-\vec{p}_{\theta}\cdot\vec{\sigma}\mathbf{)}\\\
(E+\vec{p}_{\theta}\cdot\vec{\sigma}\mathbf{)}\end{array}\right]\left[\begin{array}[]{c}\vec{\psi}_{R}\\\
\vec{\psi}_{L}\end{array}\right]=\left[\begin{array}[]{c}0\\\
0\end{array}\right]\mathbf{.}$ (16)

Here wave factor $e^{i\theta^{\prime}}$ is implicit and $\vec{\psi}_{R}$ and
$\vec{\psi}_{L}$ are $\times 2$ block wavefunctions

$\vec{\psi}_{R}\equiv\left[\begin{array}[]{c}\psi^{1}\\\
\psi^{2}\end{array}\right],\quad\text{and}\quad\vec{\psi}_{L}\equiv\left[\begin{array}[]{c}\psi^{3}\\\
\psi^{4}\end{array}\right].$

We notice that $\vec{\psi}_{R}$ and $\vec{\psi}_{L}$ are independent and there
is no interference from one to the other in the rest state of electron.

Now, we define helicity operator for our case as

$\hat{h}_{\theta}=\frac{\vec{\sigma}\cdot\vec{p}_{\theta}}{\left|\vec{p}_{\theta}\right|}.$
(17)

The Weyl spinors are eigenstates of helicity operator $h=$
$\vec{\sigma}\mathbf{\cdot}\hat{n}$, with
$\vec{p}/\left|\vec{p}\right|=\hat{n}$. The helicity is well defined for the
massless Dirac fermion or Weyl fermion with eigenvalues equal to
$\mathbf{\pm}1$. In this case we are considering only internal momentum
$\vec{p}_{\theta}$ ($2D$ object in the $xy$ plane), therefore, eigenvalues of
$h_{\theta}$ are $+1$ and $-1$ for ccw and cw rotation, respectively. Dirac
massless case is also equivalent to ultra relativistic helicity (energy
$E>>m$) of particle (moving at velocity near to velocity of light $c$), again
giving $h_{\theta}$ the eigenvalues $\pm 1$. The evidence for helicity ($\pm
1$) of massless Dirac electron is experimentally confirmed at Dirac point,
such as in graphene A. H. Castro Neto ; A. K. GEIM .

The intrinsic spin of electron is associated with the circular flow of energy-
momentum ($p_{\theta}^{o},\vec{p}_{\theta}$) of TWEM model. The spin angular
momentum $S$ is linked to the energy and angular frequency of the circularly
polarized wavepacket as follows Hans

$S\equiv u_{em}/\omega^{\prime}.$

With $p_{\theta}$ circulating in the $xy$-plane, it is straightforward to show
by using $u_{em}=\hslash\omega_{o}$ and $\omega^{\prime}=2\omega_{o}$ that the
$z-$component of spin is given by

$S_{z}=\pm\frac{1}{2}\hslash.$ (18)

where $\pm$ stand for ccw and cw rotations, respectively. Alternatively, with
the geometry of Fig. (1), we can determine the spin using the internal orbital
angular momentum generated due to circular flow of energy-momentum

$\vec{L}_{int}=\vec{r}\times(\vec{E}_{r}\mathbf{\times}\vec{H}_{z}\mathbf{)=}\vec{r}\mathbf{\times}\vec{p}_{\theta}=\left|\vec{r}\right|\left|\vec{p}_{\theta}\right|\hat{\theta}.$

Using all the known values, this leads to $L_{int}=\frac{1}{2}\hslash$. For
free massless spinors, the spin eigenstates can also be taken as the
eigenstates of helicity with eigenvalues $\mathbf{\pm}1$. Correspondingly, the
$z$-components of spin have eigenvalues $\mathbf{\pm}1/2$, which results in
Eq. (18). Similar results for spin emerging from Ztg, have been obtained in
Bruno . Also a topological structure of electron has been hypothesized in Qiu
Hong  and coinciding results for spin have been obtained for postulated Hubius
Helix model of electron.

##  V Electron in motion

To see the behavior of electron wavefunction in motion, we apply the Lorentz
boost to Weyl spinors. The massless Weyl spinors seem to exist on the light
cone, moving at the speed of light. Furthermore, the velocity derivative in
proper time is zero, i.e. $\partial v/\partial\tau=0$ at the speed of light.
Applying boost to an object, already moving with speed of light is pointless.
However in the TWEM electron energy-momentum spinor(s) is circulating around
the origin (that is around the center of energy-momentum) at the speed of
light, Eqs. (9, 10). Then, we can apply boost to center of energy-momentum
frame, which we consider, initially at rest. The general Lorentz boost for
spinors is given by Andrew

$S(\Lambda)=I\cosh\frac{\rho}{2}-\vec{\sigma}\cdot\hat{n}\sinh\frac{\rho}{2},$
(19)

where $\rho$ is rapidity such that $\tanh\rho=\beta$ and $\hat{n}$ is a unit
vector in the direction of motion. Using the relations
$\cosh\frac{\rho}{2}=\left[\frac{1}{2}(1+\gamma)\right]^{\frac{1}{2}}$ and
$\sinh\frac{\rho}{2}=\left[\frac{1}{2}(\gamma-1)\right]^{\frac{1}{2}}$ along
with the relativistic relations of energy $E=\gamma m$ and momentum
$\vec{p}=\gamma m\vec{v}=E$ $\vec{v}$, the Lorentz boost of Eq. (19) in terms
of energy-momentum turns out to be

$S(\Lambda)=\sqrt{\frac{E+m}{2m}}\left[\begin{array}[]{cc}I+\frac{\vec{\sigma}\mathbf{\cdot}\vec{p}}{E+m}&0\\\
0&I-\frac{\vec{\sigma}\mathbf{\cdot}\vec{p}}{E+m}\end{array}\right].$ (20)

Then we can write spinors $\psi_{R}(\vec{p})$ and $\psi_{L}(\vec{p})$ in the
moving frame as

$\left[\begin{array}[]{c}\psi_{R}(\vec{p})\\\
\psi_{L}(\vec{p})\end{array}\right]=S(\Lambda)\left[\begin{array}[]{c}\psi_{R}(0)\\\
\psi_{L}(0)\end{array}\right],$ (21)

where $\psi_{R}(0)$ and $\psi_{L}(0)$ are spinors in the rest frame. For
$\psi_{R}(p)$ and $\psi_{L}(p)$, we get

$\left[\begin{array}[]{c}\psi_{R}(\vec{p})\\\
\psi_{L}(\vec{p})\end{array}\right]=\left[\begin{array}[]{c}\sqrt{\frac{E+m}{2m}}\left[I+\frac{\vec{\sigma}\mathbf{\cdot}\vec{p}}{E+m}\right]\\\
\sqrt{\frac{E+m}{2m}}\left[I-\frac{\vec{\sigma}\mathbf{\cdot}\vec{p}}{E+m}\right]\end{array}\right]\left[\begin{array}[]{c}\psi_{R}(0)\\\
\psi_{L}(0)\end{array}\right].$

Assuming $\psi_{L}(0)=\psi_{R}(0)=\psi(0)$, we obtain massive Dirac equation

$\left[\begin{array}[]{c}E-\vec{p}_{\theta}\cdot\vec{\sigma}\\\
E+\vec{p}_{\theta}\cdot\vec{\sigma}\end{array}\right]\left[\begin{array}[]{c}\vec{\psi}_{R}\\\
\vec{\psi}_{L}\end{array}\right]=\left[\begin{array}[]{c}m\vec{\psi}_{L}\\\
m\vec{\psi}_{R}\end{array}\right].$ (22)

In the moving frame the left and the right spinors are now mixed through the
mass term. Alternatively, we can mix up the left and the right spinors with an
appropriate weightage and can obtain the mass property of the massless
spinning (momentum) waves.

Now, for four component spinor
$\vec{\psi}(\vec{p})\equiv\psi^{\alpha}(\vec{p})$, we can write the massive
Dirac electron equation in the momentum space

$(\gamma^{\mu}p_{\mu}-m)\vec{\psi}(\vec{p})=0.$ (23)

In Minkowski space, replacing $p_{\mu}\longrightarrow i\partial/\partial
x^{\mu}=\partial_{\mu}$, $\mu=0,1,2,3$, the Dirac equation takes the form

$(i\gamma^{\mu}\partial_{\mu}-m)\vec{\psi}(x)=0.$ (24)

Once the relation between energy-momentum of spinning EM fields with Dirac
equation is established,we can associate all attributes of Dirac theory to the
TWEM model of electron. The Dirac fields describing the model are found on the
divergence free symmetric energy-momentum tensor $\Theta^{\alpha\beta}$of the
circulating EM wave which obeys conservation laws
$\partial_{\alpha}\Theta^{\alpha 0}=\partial_{\alpha}\Theta^{\alpha i}=0$
M.M.Asif . There are number of ways to determine spin from Dirac equations,
however, we will follow Hans  to show it for circular flow of energy-momentum
of Dirac field. Using momentum density in Dirac fields, the angular momentum
$\vec{J}$ of electron (wave packet) becomes

$\vec{J}=\frac{\hslash}{2i}\int\vec{x}\times[\psi^{{\dagger}}\vec{\nabla}\psi-(\vec{\nabla}\psi^{{\dagger}})\psi]d^{3}x+\frac{\hslash}{2}\int(\psi^{{\dagger}}\vec{\Sigma}\psi)d^{3}x,$
(25)

where

$\vec{\Sigma}=\left(\begin{array}[]{cc}\vec{\sigma}&0\\\
0&\vec{\sigma}\end{array}\right),$

are Dirac spin matrices. In Eq. (25), the first part represents orbital
angular momentum and the second part represents the spin. From the spin part
of Eq. (25), the Dirac spin operator is obtained as

$\vec{S}\mathbf{=}\frac{\hslash}{2}\vec{\Sigma}.$

The helicity operator $\hat{h}$ of Dirac electron is projection of spin on
linear momentum of electron, then

$\hat{h}=\frac{\vec{\Sigma}\cdot\vec{p}}{2\left|\vec{p}\right|}.$

Alexander Burinskii Burinskii  has obtained similar results by considering
Kerr singular ring (a closed gravitational string) for the electron model. He
treats electron as electromagnetic excitation of the ring, forming, travelling
wave along a string as the Kerr-Newman solution of Einstein-Maxwell’s theory.
The ordinary Dirac theory has been obtained from the solutions of the massless
underlying theory, which regards Ztg as a corpuscular analogue of the
travelling waves. The author attempts to establish relation between gravity
and QM leading to Quantum Gravity. The Kerr singular ring model and TWEM model
are vastly different in terms of their modeling background theories. The Kerr
singular ring model is based on Kerr-Newman gravitational singular string
whereas TWEM model is based on linear Maxwell’s theory and quantization.

In TWEM electron model, the dynamism (due to circular motion) of energy-
momentum gives rise to Ztg and associated properties, such as electron spin
and helicity etc. This model pictures, the internal structure of electron (and
positron) and interprets its underlying physical mechanism. In perspective of
the above deliberations, we may infer that TWEM electron exhibits symmetry
group $U(1)$ at traveling electromagnetic wave level M.M.Asif . The $2D$
(circulating momentum) object belongs to symmetry group $SU(2)$ in the rest
frame as a massless Dirac spinor. Finally under boost and/or rotation,
$SL(2,C)$ symmetry group is pertinent to spinors leading to massive Dirac
theory. Hence electron is a complex object and exhibits blend of different
symmetry groups at different levels of structure and with reference to frame
of observation.

##  VI Conclusion

In summary, we have used the TWEM model to explore the internal structure of
electron. The circular traveling EM wave of the model carries, synchronous
rotating energy-momentum ($u,\vec{p}_{\theta}$), in a circle of radius of half
the reduced Compton wavelength of electron. It is demonstrated that in TWEM
model of electron, the Zitterbewegung and spin arise naturally as a result of
circular motion of the energy-momentum of EM wave. Furthermore, the spinning
energy-momentum wave function forms four component Dirac spinor like object
corresponding to the four possible circular polarized like EM waves. For the
four component energy-momentum spinor, the massless and massive Dirac
equations follow directly in the rest frame and in the boosted frame,
respectively. Alternatively, the four component spinor of the TWEM model is a
solution to the Dirac equation. In this model, we show that the Dirac spinors
are not just mathematical objects but are real objects with a physical
mechanism(s) in the background. The TWEM electron model is blend of symmetry
groups $U(1)$, $SU(2)$ and $SL(2,C)$, that is, different symmetries are
relevant at different levels of internal structure and with reference to frame
of observation.

##  VII Acknowledgement

M.M. Asif pays thanks to Robert L. Shuler Jr., NASA Johnson Space Center, USA
for his constructive suggestions regarding this work. The valuable discussions
with him, provided insight that greatly helped in carrying out the present
work.

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