# Generalized quaternionic free rotational Dirac equation and spinor solutions
in the electromagnetic field

Shikha Bhatt and B. C. Chanyal Corresponding Author (Email:
<EMAIL_ADDRESS>
###### Abstract

Starting with the quaternionic Minkowski space-time and its four-vector
representation, a rotational analogue of the quaternionic Dirac equation in
the electromagnetic field is developed, which includes not only the energy
solutions but also the angular momentum solutions for rotating Dirac fermions.
The striking feature of the quaternionic approach is that it depicts a unified
representation of energy-momentum in a single framework as the space-time.
Furthermore, we establish the generalized Schrodinger-Pauli-like equation
associated with generalized electric and magnetic dipole energies that
corresponding to their dipole moments. As such, the present analysis deals
with the invariant behavior of the extended quaternionic rotational Dirac
equation under Lorentz-Poincaré, Gauge, duality, and CPT invariance.

Keywords: Minkowski space-time, quaternion, Dirac equation, electromagnetic
field, dipole moments, energy-momentum.

Mathematics Subject Classification 2020:11R52, 81Q05, 20Gxx

Department of Physics,
G. B. Pant University of Agriculture and Technology,
Pantnagar-263145, Uttarakhand, India

##  1 Introduction

The predictions of the Classical theory have a substantial impact on the
complete description of matter, but its failure to characterize subatomic
particle behavior led to the development of quantum mechanics. Quantum
mechanics employs the idea of a wave function, which has mathematical
construct that provides information in the form of probability amplitudes [1].
With the generalization of de Broglie postulates, Erwin Schrodinger proposed
the non relativistic wave equation, however it could not be applied to
particles traveling with relativistic velocity [2]. Furthermore, by combining
quantum mechanics with special theory of relativity, Dirac and Klein-Gordon
discovered the relativistic wave equation, which became the foundation of
‘Relativistic Quantum Mechanics’. The behavior of particles at high energies
is described by relativistic wave equations, which are well invariant under
Lorentz translation. Dirac suggested his renowned differential equation to
correct for the deficiencies of the Klein-Gordon equation, which describes the
motion of spin 1/2 particles [3]. The presence of anti-particles was predicted
by the Dirac equation. His equation was expressed in terms of Dirac matrices,
which were also used to describe the interaction of Dirac particles with an
external electromagnetic field. Generally, researchers used a four-vector
potential with scalar and vector potentials to characterize an external
electromagnetic field. In this case, the Dirac equation can be changed using
minimal coupling when a particle interacts with an electromagnetic field [4].
The hyper-complex algebra is now more commonly used in modern physics to
express many physical quantities. The first hyper-complex algebra that is
commonly used in physics is the quaternion algebra [5]. According to Dickson
[6], there are four forms of division algebra: real numbers, complex numbers,
quaternion, and octonion algebra. Complex numbers (two dimensions),
quaternionic algebra (four dimensions), and octonion algebra (eight
dimensions) are all extensions of real numbers. Many novel results in modern
physics have been explained using hypercomplex algebras [7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20]. Quaternions have a structure that is
extremely similar to that of four vectors, with quaternionic counterparts of
four vectors such as four momentum, four velocity, four potential, and so on
[21]. Apart from that, the quaternionic algebra has been used to describe a
variety of physical theories [22, 23, 24, 25], including Maxwell’s
electromagnetic field equations, Dirac Lagrangian, dual magneto-hydrodynamics
(DMHD) equations, and the Bernoulli and Navier–Stokes equations for dyonic
plasma, and so on. Davies [26] examined the Dirac equation in quaternionic
form, and Rawat et al. [27] discussed the quaternionic Dirac equation in the
bi-spin state, providing positive and negative energy solutions. Rawat and
Negi [28] have proposed a link between the quaternionic form of the Dirac
equation and supersymmetry in presence of electromagnetic field [29]. The non-
conservation of the photon energy-momentum tensor has been investigated [30,
31, 32, 33, 34], which is more in accordance with the terminology used in this
paper. Non-conservation of the photon energy-momentum tensor [34] is also
identified in (i) de Broglie- Proca’s theory (ii) Standard-Model Extension
(iii) Non-Linear Electromagnetism. Chanyal [35, 36] provided a fresh notion
for relativistic quantized electromagnetic fields theory of dyons in terms of
quaternion variable. In addition, the quaternionic algebra has been utilized
to represent the rotational motion of a rigid body [37], where the
quaternionic unit elements are typically associated with matrices of the
rotational group SO(3). Beside, on the $R\times S^{3}$ topology, Carmeli [38,
39, 40] studied the Klein-Gordon equation, the Weyl equation, and the Dirac
equation in rotational form. The rotational analogue of the dynamics of
relativistic spin-1/2 fermions and the rotational Dirac equation in terms of
quaternionic form have recently been constructed [41]. So, in light of the
aforementioned literature on rotating Dirac fermions, we have extended the
quaternionic rotational Dirac equation in the presence of an external
electromagnetic (EM) field.

Starting with the quaternionic Minkowski space-time and its four-vector
representation, we write the rotational analogue of the generalized
quaternionic Dirac equation for a free fermion spinor where the rotational
Dirac matrices are associated with quaternionic basis vectors. We also define
the fermionic spinor field for quaternions. The energy-momentum like solutions
for Dirac rotating particles are expressed in terms of four-component spinors.
Using the quaternionic minimum coupling of four-momentum with the EM-field, a
rotational analogy of the quaternionic Dirac equation in the EM-field is
developed, which contains not only the rotational energy (corresponding to the
quaternionic scalar component) but also the angular momentum (corresponding to
the quaternionic vector component) for Dirac fermions associated with electric
and magnetic fields. Furthermore, the solutions for these quaternionic
rotational energy and momentum equations in the EM-field are obtained in terms
of the four component form of the Dirac spinors with spin up and down state.
We establish the generalized Schrodinger-Pauli like energy equation, which is
associated with unperturbed and perturbed energies due to EM-field
interaction. The generalized electric and magnetic dipole moments are
constructed corresponding to generalized electric and magnetic dipole
energies. Moreover, we also addressed the invariant behavior of the extended
quaternionic rotational Dirac equation for Lorentz, gauge, duality, and CPT
invariance to check the various symmetries.

##  2 The quaternions

The quaternions, denoted by $\mathbb{H}$, are a type of hypercomplex algebra
which is a four-dimensional norm division algebra over a set of real numbers
$(\mathbb{R})$. It is a four-dimensional extension of complex numbers with a
scalar and a vector part that is used to explain dynamic behaviour. A
quaternion variable $Q_{q}\in\mathbb{H}$ can be represented as

$\displaystyle Q_{q}=$
$\displaystyle\,\,e_{0}x_{0}+\left(e_{1}x_{1}+e_{2}x_{2}+e_{3}x_{3}\right)\,,$
(1)

where ($x_{0},x_{1},x_{2},x_{3})\in\mathbb{R}$ and ($e_{0},e_{1},e_{2},e_{3}$)
are the fundamental quaternionic units known as the quaternionic basis
elements. However, the scalar unit is denoted by $e_{0}$, and the vector units
are denoted by $e_{j}$ $(j=1,2,3)$. The quaternion multiplication can be
determined by the following rules:

$\displaystyle e_{0}^{2}=$
$\displaystyle\thinspace\,\,e_{0}=-e_{k}^{2}=1\,\,,$ $\displaystyle
e_{0}e_{k}=$ $\displaystyle\,\,\,e_{k}e_{0}=e_{k}\thinspace\thinspace,$
$\displaystyle e_{k}e_{l}=$
$\displaystyle\,-\delta_{kl}e_{0}+\epsilon_{klm}e_{m}\,\thinspace,\thinspace\,\,\,\,\forall(k,l,m=1,2,3)\,\thinspace.$
(2)

Here $\delta_{kl}$ represents delta symbol and $\epsilon_{klm}$ represents the
Levi-Civita symbol. Moreover, the quaternionic multiplication (2) can also be
written as given Table-1.

| $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$
---|---|---|---|---
$e_{0}$ | $1$ | $e_{1}$ | $e_{2}$ | $e_{3}$
$e_{1}$ | $e_{1}$ | $-1$ | $e_{3}$ | $-e_{2}$
$e_{2}$ | $e_{2}$ | $-e_{3}$ | $-1$ | $e_{1}$
$e_{3}$ | $e_{3}$ | $e_{2}$ | $-e_{1}$ | $-1$
Table 1: Quaternion multiplication table

As such, the relation

$\displaystyle e_{k}e_{l}+e_{l}e_{k}=\,\thinspace$
$\displaystyle-2\delta_{kl}\,\thinspace,$ $\displaystyle
e_{k}e_{l}-e_{l}e_{k}=\,\thinspace$ $\displaystyle 2\epsilon_{klm}e_{m}\,\,,$
(3)

represent the anti-commutation and commutation property of the quaternionic
units. The addition of two quaternions will be expressed as

$\displaystyle Q_{q}+R_{q}=$ $\displaystyle\,\thinspace
e_{0}\left(x_{0}+y_{0}\right)+e_{1}\left(x_{1}+y_{1}\right)+e_{2}\left(x_{2}+y_{2}\right)+e_{3}\left(x_{3}+y_{3}\right)$
$\displaystyle=$ $\displaystyle\,\thinspace
e_{0}z_{0}+\left(e_{1}z_{1}+e_{2}z_{2}+e_{3}z_{3}\right)\equiv
T_{q}\thinspace\,,\thinspace\thinspace\thinspace\>\,\left(Q_{q},R_{q},T_{q}\thinspace\in\thinspace\mathbb{H}\right)\thinspace\thinspace,$
(4)

which satisfies the closure property of addition. As such, the quaternions
show the commutative and associative properties of addition, respectively,
$Q_{q}+R_{q}=R_{q}+Q_{q}$ and
$\left(Q_{q}+R_{q}\right)+T_{q}=Q_{q}+\left(R_{q}+T_{q}\right).$ The
quaternionic multiplication of any two quaternions can be written by using
Table-1 as

$\displaystyle Q_{q}\circ R_{q}=$
$\displaystyle\,\thinspace(e_{0}x_{0}+e_{1}x_{1}+e_{2}x_{2}+e_{3}x_{3})\circ(e_{0}y_{0}+e_{1}y_{1}+e_{2}y_{2}+e_{3}y_{3})$
$\displaystyle=$
$\displaystyle\,\,e_{0}\left(x_{0}y_{0}-\overrightarrow{x}\cdot\overrightarrow{y}\right)+e_{j}\left(x_{0}y_{j}+x_{j}y_{0}+\left(\overrightarrow{x}\times\overrightarrow{y}\right)_{j}\right)\,\,,\,\,\,\left(\forall\,j=1,2,3\right),$
$\displaystyle=$ $\displaystyle\,\,\text{Sr}\\{Q_{q}\circ
R_{q}\\}+\text{Vr}\\{Q_{q}\circ R_{q}\\}$ (5)

where ‘$\circ$’ indicates the quaternion product while $(\cdot)$ and
$(\times)$ are the usual scalar and vector products. The scalar part of the
quaternionic multiplication is ‘Sr’, while the vector part is ‘Vr’. Under
multiplication, quaternions are associative but not commutative since
$\overset{\rightarrow}{x}\times\overset{\rightarrow}{y}\neq\overset{\rightarrow}{y}\times\overset{\rightarrow}{x}.$
The quaternionic conjugate ($Q_{q}^{\ast}$) of $Q_{q}$ for which
$e_{0}^{*}=e_{0},\,e_{1}^{*}=-e_{1},\,e_{2}^{*}=-e_{2},\,e_{3}^{*}=-e_{3}$ is
defined as

$\displaystyle Q_{q}^{\ast}=$ $\displaystyle\,\thinspace
e_{0}x_{0}-\left(e_{1}x_{1}+e_{2}x_{2}+e_{3}x_{3}\right)\,\,.$ (6)

Thus, using the conjugation property of quaternion, the real part (i.e.
scalar) can be distinguished from its imaginary part (i.e. vector), as

$\displaystyle\text{Sr}=\thinspace\frac{1}{2}(\thinspace
Q_{q}+Q_{q}^{*})=\thinspace$ $\displaystyle x_{0}\,\,,$
$\displaystyle\text{Vr}=\thinspace\frac{1}{2}(\thinspace
Q_{q}-Q_{q}^{*})=\thinspace$ $\displaystyle
e_{1}x_{1}+e_{2}x_{2}+e_{3}x_{3}\thinspace\thinspace.$ (7)

Notice that, the quaternions are called real quaternions if the vector part is
zero, and pure quaternions if the scalar part is zero. As such, the modulus
(or norm) of a quaternion is expressed as

$\mid Q_{q}\mid=\,\,\sqrt{Q_{q}^{*}\circ
Q_{q}}=\,\,\sqrt{x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}\,\,,$ (8)

which satisfy

$\mid Q_{q}R_{q}\mid\,\thinspace=\thinspace\,\mid Q_{q}\mid\mid
R_{q}\mid\,\,.$ (9)

The quaternionic inverse becomes

$\displaystyle Q_{q}^{-1}=$ $\displaystyle\,\,\frac{Q_{q}^{*}}{\mid
Q_{q}\mid^{2}}\>\thinspace,\>\thinspace\>\left(\mid
Q_{q}\mid\,\neq\,0\right)\,\thinspace.$ (10)

A scalar product can also be defined for quaternions as

$\displaystyle\left(Q_{q}\cdot R_{q}\right)=$
$\displaystyle\,\thinspace-\frac{1}{2}(Q_{q}\circ R_{q}^{*}+R_{q}\circ
Q_{q}^{*})=\,\thinspace-\frac{1}{2}(Q_{q}^{*}\circ R_{q}+R_{q}^{*}\circ
Q_{q})\,\thinspace.$ (11)

Even so, the quaternionic basis may also be shown as a $2\times 2$ matrix
form, such that $e_{0}=\,1_{2\times 2}$ and $e_{j}=-i\sigma_{j}$ where
$\sigma_{j}$ are the standard Pauli matrices [29]. Furthermore, quaternions
show a remarkable resemblance to rotational tau matrices (or isospin matrices)
and can be utilized to discuss particle rotational motion [37].

##  3 Quaternionic Minkowski space-time

The quaternions can be utilized as an analogous version of the four vectors
because they are regarded comparable to the 4-dimensional Minkowski space-time
with structure $(-,+,+,+)$. Thus,

$\displaystyle X_{q}=\thinspace\,\left(-ict,\overset{\rightarrow}{R}\right)=$
$\displaystyle\,\,\left\\{e_{0}\left(-ict\right)+e_{1}X_{1}+e_{2}X_{2}+e_{3}X_{3}\right\\}\thinspace,$
(12) $\displaystyle
X_{q}^{*}=\thinspace\thinspace\left(-ict,-\overset{\rightarrow}{R}\right)=$
$\displaystyle\,\,\left\\{e_{0}\left(-ict\right)-e_{1}X_{1}-e_{2}X_{2}-e_{3}X_{3}\right\\}\thinspace,$
(13)

where $X_{q}$ is the quaternionic 4-position. As such, the quaternionic
4-displacement vector, 4-velocity, and 4-gradient will be expressed as,
respectively,

$\displaystyle ds_{q}=\thinspace\thinspace\left(-icdt,dx,dy,dz\right)$
$\displaystyle=\,\thinspace
e_{0}\left(-icdt\right)+e_{1}dx+e_{2}dy+e_{3}dz\thinspace,$ (14)
$\displaystyle U_{q}=\,\thinspace\left(-ic,\overset{\rightarrow}{u}\right)$
$\displaystyle=\thinspace\thinspace
e_{0}\left(-ic\right)+e_{1}u_{1}+e_{2}u_{2}+e_{3}u_{3}\,,$ (15) $\displaystyle
D_{q}=\thinspace\thinspace\left(-\frac{1}{ic}\frac{\partial}{\partial
t},\overrightarrow{\nabla}\right)$ $\displaystyle=\thinspace\thinspace
e_{0}\left(-\frac{1}{ic}\frac{\partial}{\partial
t}\right)+e_{1}\frac{\partial}{\partial x}+e_{2}\frac{\partial}{\partial
y}+e_{3}\frac{\partial}{\partial z}\,.$ (16)

The quaternionic 4-momentum can be written as

$\displaystyle
P_{q}=\,\thinspace\left(-i\frac{E}{c},\overset{\rightarrow}{p}\right)=$
$\displaystyle\,\thinspace
e_{0}\left(-i\frac{E}{c}\right)+e_{1}p_{1}+e_{2}p_{2}+e_{3}p_{3}\thinspace.$
(17)

Further, the generalized quaternionic four potential ($V_{q}$), the electric
four potential ($A_{q}$), and the magnetic four potential ($B_{q}$) can all be
expressed in terms of Minkowski structure as

$\displaystyle
V_{q}=\,\thinspace\left(-i\frac{\Phi}{c},\overset{\rightarrow}{V}\right)=$
$\displaystyle\,\thinspace
e_{0}\left(-i\frac{\Phi}{c}\right)+e_{1}V_{1}+e_{2}V_{2}+e_{3}V_{3}\,,$ (18)
$\displaystyle
A_{q}=\,\,\left(-i\frac{\phi_{e}}{c},\overset{\rightarrow}{A}\right)=$
$\displaystyle\,\thinspace
e_{0}\left(-i\frac{\phi_{e}}{c}\right)+e_{1}A_{1}+e_{2}A_{2}+e_{3}A_{3}\,,$
(19) $\displaystyle
B_{q}=\thinspace\,\left(-i\frac{\phi_{m}}{c},\overset{\rightarrow}{B}\right)=$
$\displaystyle\,\thinspace
e_{0}\left(-i\frac{\phi_{m}}{c}\right)+e_{1}B_{1}+e_{2}B_{2}+e_{3}B_{3}\,,$
(20)

where the generalized vector potential ($\overset{\rightarrow}{V}$) and scalar
potential ($\Phi$) are written as
$\overset{\rightarrow}{V}=\,\thinspace\overset{\rightarrow}{A}-ic\overset{\rightarrow}{B}$
and $\Phi=\thinspace\,\phi_{e}-ic\phi_{m}$, respectively. Additionally, the
quaternionic mass can be expressed as,[42]

$\displaystyle M_{q}=$
$\displaystyle\,\,e_{0}m_{0}+\left(e_{1}m_{1}+e_{2}m_{2}+e_{3}m_{3}\right)\,\thinspace,$
(21)

where $m_{0}=\frac{E_{0}}{c^{2}}$ is defined the rest mass while
$m_{j}=\thinspace\left|\frac{p_{j}}{u_{j}}\right|$, $\forall(j=1,2,3)$ is
defined the moving mass. The quaternionic moment of inertia (MOI) can thus be
written as

$\displaystyle I_{q}=$ $\displaystyle\,\thinspace
M_{q}\left(X_{q}\circ\left(X_{q}\right)^{*}\right)=M_{q}\left(X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$
$\displaystyle=$ $\displaystyle\,\thinspace
e_{0}I_{0}+\left(e_{1}I_{1}+e_{2}I_{2}+e_{3}I_{3}\right)\thinspace,$ (22)

where $I_{0}\sim m_{0}|X|^{2}$ is the MOI about the quaternionic $e_{0}$ axis
while $I_{j}\sim m_{j}|X_{j}|^{2}$ are the MOI about the quaternionic $e_{j}$
axes. We can also express the quaternionic rotational 4-momentum as using the
above quaternionic quantities, i.e.,

$\displaystyle L_{q}=$ $\displaystyle\,\,X_{q}\circ
P_{q}\,\thinspace=\,\thinspace
e_{0}E_{0}+\left(e_{1}L_{1}+e_{2}L_{2}+e_{3}L_{3}\right)\thinspace.$ (23)

Here, equation (23) gives the scalar component $(E_{0})$ that can be
represented as quaternionic rest mass energy, while the vector components
$(\overrightarrow{L})$ can represented as pure quaternionic angular momentum.
So, we have

$\displaystyle E_{0}=$ $\displaystyle\,\thinspace
X_{0}p_{0}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{p}\thinspace,$
(24) $\displaystyle\overrightarrow{L}=$
$\displaystyle\,\,X_{0}\overrightarrow{p}+p_{0}\overrightarrow{X}+\left(\overrightarrow{X}\times\overrightarrow{p}\right)\thinspace\,.$
(25)

Interestingly, the quaternionic rest mass energy consisted the scalar terms
while the quaternionic angular momentum consisted all the vector terms. We get
the pure rotational energy
$E_{0}\rightarrow-\overrightarrow{X}\cdot\overrightarrow{p}$ and angular
momentum
$\overrightarrow{L}\rightarrow\left(\overrightarrow{X}\times\overrightarrow{p}\right)$
for a pure quaternion, while for real quaternionic consideration, we only have
the rest mass energy $E_{0}\rightarrow X_{0}p_{0}$ with no rotational motion
$\overrightarrow{L}\rightarrow 0$.

##  4 Quaternionic Rotational Dirac (QRD) equation

A relativistic equation that explains the dynamic behaviour of spin 1/2
particles is the Dirac equation. Let us start with the rotational analogue of
the Dirac equation in order to write the generalized quaternionic Dirac
equation for a free fermion spinor field [41], as

$\displaystyle\left(H_{q}\circ
L_{q}-\mathscr{\mathfrak{B}}\lambda^{2}I_{q}\right)\circ\mathtt{\boldsymbol{\Psi}}=$
$\displaystyle\,\thinspace 0\,\thinspace,$ (26)

where $H_{q},L_{q},\mathscr{\mathfrak{B}},\lambda,I_{q}$ and
$\boldsymbol{\Psi}$ are rotational analogous to the standard Dirac variables
$\overrightarrow{\alpha}$, $\overrightarrow{p}$, $\beta,\,c,\,m$ and $\psi$.
The rotational Dirac matrix elements $H_{q}$ and $\mathscr{\mathfrak{B}}$ can
be represented in terms of $2\times 2$ matrix as

$\displaystyle H_{q}=$
$\displaystyle\,\,\,\left\\{D^{0}\left(H\right),D^{j}\left(H\right)\right\\}\,,\,\,\,\forall\,D^{0}(H)=\thinspace\left(\begin{array}[]{cc}e_{0}&0\\\
0&e_{0}\end{array}\right)\,,\,D^{j}(H)=\thinspace\left(\begin{array}[]{cc}0&ie_{j}\\\
ie_{j}&0\end{array}\right)\,,$ (31) $\displaystyle\mathscr{\mathfrak{B}}=$
$\displaystyle\,\,\left(\begin{array}[]{cc}e_{0}&0\\\
0&-e_{0}\end{array}\right)\thinspace,$ (34)

where $D^{0}\left(H\right)$ and $D^{j}\left(H\right)$ for
$\left(j=1,2,3\right)$ are the quaternionic $D$-matrices. Here the speed of
rotating fermions is denoted by ‘$\lambda$’ i.e.,
$\lambda=\,c\sqrt{\frac{M_{q}}{I_{q}}}=\frac{c}{\mid X\mid}$ (see Ref. [40]).
Therefore, the generalized QRD equation becomes

$\displaystyle\left[e_{0}\right.$
$\displaystyle\left(D^{0}\left(H\right)E_{0}-\lambda\left(\overrightarrow{D}\left(H\right)\cdot\overrightarrow{L}\right)-\mathfrak{B}\lambda^{2}I_{0}\right)$
$\displaystyle+$ $\displaystyle\left.e_{j}\left(\lambda
D^{0}\left(H\right)L_{j}+D^{j}\left(H\right)E_{0}+\lambda\left(\overrightarrow{D}\left(H\right)\times\overrightarrow{L}\right)_{j}-\mathscr{\mathfrak{B}}\lambda^{2}I_{j}\right)\right]\circ\boldsymbol{\Psi}=\,\,0\,\,,$
(35)

The advantage of QRD equation is that it describing not only the rotational
energy but also the rotating momentum for the quaternionic scalar and vector
parts. This is the most significant aspect of quaternionic analysis, as it
allows for dual representation of four vectors in a single framework. Thus,
the energy-momentum solutions of the generalized QRD equation (35) can be
obtained with the quaternion fermionic spinor field $(\boldsymbol{\Psi})$ as

$\displaystyle\boldsymbol{\Psi}=$ $\displaystyle\,\thinspace
e_{0}\Psi_{0}+\left(e_{1}\Psi_{1}+e_{2}\Psi_{2}+e_{3}\Psi_{3}\right)\,\thinspace,$
$\displaystyle\simeq$ $\displaystyle\>\left(\begin{array}[]{c}\Psi_{0}\\\
\Psi_{1}\\\ \Psi_{2}\\\ \Psi_{3}\end{array}\right)\,.$ (40)

Equation (40) indicates the four components representation of quaternionic
Dirac spinor, and it can be simplified as
$\boldsymbol{\Psi}=\left(\Psi_{K}+e_{2}\Psi_{L}\right)$, where
$\Psi_{K}=\,\,\left(\Psi_{0}+e_{1}\Psi_{1}\right)$ and
$\Psi_{L}=\,\thinspace\left(\Psi_{2}-e_{1}\Psi_{3}\right)$. So, the energy
solutions of equation (35) can be examined by equating the scalar coefficient
($e_{0}$) :

$\displaystyle\left[D^{0}\left(H\right)E_{0}-\lambda\left(\overrightarrow{D}\left(H\right)\cdot\overrightarrow{L}\right)-\mathfrak{B}\lambda^{2}I_{0}\right]\boldsymbol{\Psi}=$
$\displaystyle\,\,0\,\,,$ (41)

which gives

$\displaystyle\Psi_{0}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}-\lambda^{2}I_{0}}\Psi_{2}(E_{0},\overset{\rightarrow}{L})\,\,,$
(42) $\displaystyle\Psi_{1}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}-\lambda^{2}I_{0}}\Psi_{3}(E_{0},\overset{\rightarrow}{L})\,\,,$
(43) $\displaystyle\Psi_{2}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}+\lambda^{2}I_{0}}\Psi_{0}(E_{0},\overset{\rightarrow}{L})\,\,,$
(44) $\displaystyle\Psi_{3}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}+\lambda^{2}I_{0}}\Psi_{1}(E_{0},\overset{\rightarrow}{L})\,\,.$
(45)

Equations (42) and (43) give the negative energy solutions for the anti-
fermions while equations (44) and (45) give the positive energy solutions for
fermions. Accordingly, by equating vector coefficient ($e_{j}$) in equation
(29) the momentum like solutions for Dirac rotating particles can be expressed
as

$\displaystyle\Psi_{0}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{\left[e_{j}E_{0}+\lambda\left(\overrightarrow{e}\times\overrightarrow{L}\right)_{j}\right]}{i\lambda\left(L_{j}-\lambda
I_{j}\right)}\Psi_{2}(E_{0},\overset{\rightarrow}{L})\thinspace\thinspace,$
(46) $\displaystyle\Psi_{1}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{\left[e_{j}E_{0}+\lambda\left(\overrightarrow{e}\times\overrightarrow{L}\right)_{j}\right]}{i\lambda\left(L_{j}-\lambda
I_{j}\right)}\Psi_{3}(E_{0},\overset{\rightarrow}{L})\thinspace\thinspace,$
(47) $\displaystyle\Psi_{2}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{\left[e_{j}E_{0}+\lambda\left(\overrightarrow{e}\times\overrightarrow{L}\right)_{j}\right]}{i\lambda\left(L_{j}-\lambda
I_{j}\right)}\Psi_{0}(E_{0},\overset{\rightarrow}{L})\thinspace\thinspace,$
(48) $\displaystyle\Psi_{3}(E_{0},\overset{\rightarrow}{L})=$
$\displaystyle\,\,\frac{\left[e_{j}E_{0}+\lambda\left(\overrightarrow{e}\times\overrightarrow{L}\right)_{j}\right]}{i\lambda\left(L_{j}-\lambda
I_{j}\right)}\Psi_{1}(E_{0},\overset{\rightarrow}{L})\,\thinspace,$ (49)

where the equations (46) and (47) represent the momentum solution in fermionic
spinor field while equations (48) and (49) represent the momentum solution in
anti fermionic spinor field.

##  5 Energy-momentum description of rotational Dirac spinors in presence of
electromagnetic field

Let us start with the quaternionic four-momentum minimum coupling on the
interaction of a charged particle with an electromagnetic (EM) field as

$\displaystyle P^{\mu}\,\,\rightarrow\,\,\,$
$\displaystyle\,\thinspace\thinspace
P^{\mu}-\frac{Q}{c}V^{\mu}\,\thinspace\equiv\thinspace\Pi^{\mu}\,\,,\thinspace\thinspace\thinspace\,(\mu=0,1,2,3)\,\thinspace,$
(50)

where $P^{\mu}$ is the canonical momentum, $Q$ is the generalized charge of
dyons $\left(Q=\mathfrak{e}-ic\left(\mathfrak{m}\right)\right)$ where
$\mathfrak{e}$ is the electric charge and $\mathfrak{m}$ is the magnetic
charge and $\Pi^{\mu}$ is the kinetic momentum. Now, considering the rotating
Dirac fermions, as an analogous, the four angular momentum can be transferred
as
$L^{\mu}:\rightarrow\,\,L^{\mu}-\frac{Q}{\lambda}\left(X^{\mu}V^{\mu}\right)\,\thinspace\equiv\thinspace\Pi_{L}^{\mu}$,
where $\Pi_{L}^{\mu}$ is the kinetic angular momentum. Thus we can express the
quaternionic form of rotational four momentum in presence of EM-field
$(L_{q_{em}})$ as

$\displaystyle L_{q_{em}}=$ $\displaystyle\thinspace\thinspace
L_{q}-\mathscr{L}\thinspace\thinspace,$ (51)

where $\mathscr{L}=\frac{Q}{\lambda}\left(X_{q}\circ V_{q}\right)$ is the
electromagnetic interaction term coupled with quaternionic angular momentum,
such that

$\displaystyle\mathscr{L}=$
$\displaystyle\,\,\frac{Q}{\lambda}\left[\,e_{0}\left(X_{0}\Phi-\overrightarrow{X}\cdot\overrightarrow{V}\right)+e_{j}\left(X_{0}\overrightarrow{V}+\Phi\overrightarrow{X}+\left(\overrightarrow{X}\times\overrightarrow{V}\right)_{j}\right)\right]\,\,.$
(52)

On substituting the values of $V_{q}$ given in (19), we have

$\displaystyle\mathscr{L}=$
$\displaystyle\frac{Q}{\lambda}\left[e_{0}\left\\{\left(X_{0}\phi_{e}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{A}\right)-i\lambda(X_{0}\phi_{m}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{B})\right\\}\right.$
$\displaystyle+$
$\displaystyle\left.e_{j}\left\\{X_{0}\overset{\rightarrow}{A}+\phi_{e}\overset{\rightarrow}{X}+\left(\overset{\rightarrow}{X}\times\overset{\rightarrow}{A}\right)_{j}-i\lambda\left(X_{0}\overset{\rightarrow}{B}+\phi_{m}\overset{\rightarrow}{X}+\left(\overset{\rightarrow}{X}\times\overset{\rightarrow}{B}\right)_{j}\right)\right\\}\right]\,\,.$
(53)

The quaternionic rotational four-momentum can be obtained in presence of EM-
field, i.e.,

$\displaystyle L_{q_{em}}=$ $\displaystyle\thinspace\thinspace\thinspace
e_{0}\left[\left(X_{0}p_{0}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{p}\right)-\frac{Q}{\lambda}\left(X_{0}V_{0}-\overrightarrow{X}\cdot\overrightarrow{V}\right)\right]$
$\displaystyle+$ $\displaystyle
e_{j}\left[\left(X_{0}\overrightarrow{p}+p_{0}\overrightarrow{X}+\left(\overrightarrow{X}\times\overrightarrow{p}\right)_{j}\right)-\frac{Q}{\lambda}\left(X_{0}\overrightarrow{V}+\Phi\overrightarrow{X}+\left(\overrightarrow{X}\times\overrightarrow{V}\right)_{j}\right)\right]\,\thinspace.$
(54)

Now, we examine at the quaternionic scalar portion known as rotational EM-
energy $(E_{em})$ that corresponds to the basis element $e_{0}$ as

$\displaystyle E_{em}=$
$\displaystyle\,\thinspace\left[\left(X_{0}p_{0}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{p}\right)-\frac{Q}{\lambda}\left\\{\left(X_{0}\phi_{e}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{A}\right)-i\lambda\left(X_{0}\phi_{m}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{B}\right)\right\\}\right]$
$\displaystyle=$ $\displaystyle\,\,E_{0}-\left(E_{e}-i\lambda E_{m}\right)\>,$
(55)

where the first term $E_{0}$ is the quaternionic rotational energy given in
equation (24), the second term indicates the rotational electric energy
$(E_{e})$ and the third term indicates the rotational magnetic energy
$(E_{m})$, so that

$\displaystyle E_{e}=$
$\displaystyle\,\thinspace\frac{Q}{\lambda}\left[\left(X_{0}\phi_{e}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{A}\right)\right]\,\,,$
(56) $\displaystyle E_{m}=$
$\displaystyle\thinspace\thinspace\frac{Q}{\lambda}\left[\left(X_{0}\phi_{m}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{B}\right)\right]\,\,.$
(57)

Consider the scalar functions
$\phi_{e}^{X}\mapsto\left(X_{0}\phi_{e}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{A}\right)$
and
$\phi_{m}^{X}\mapsto\left(X_{0}\phi_{m}-\overset{\rightarrow}{X}\cdot\overset{\rightarrow}{B}\right)$
are, respectively, the generalized quaternionic rotational electric and
magnetic potentials. Then, the equation (55) can be written as
$E_{em}=E_{0}-\frac{Q}{\lambda}\Phi^{X}$, where
$\Phi^{X}=\phi_{e}^{X}-i\lambda\phi_{m}^{X}$ is the quaternionic rotational
generalized scalar potential. Similarly, in equation (54) the quaternionic
vector portion denoted the EM-angular momentum ($\overrightarrow{L}_{em}$)
corresponding to basis element $e_{j}$, i.e.,

$\displaystyle\overrightarrow{L}_{em}=$
$\displaystyle\thinspace\thinspace\left[X_{0}\overrightarrow{p}+p_{0}\overrightarrow{X}+\left(\overrightarrow{X}\times\overrightarrow{p}\right)-\frac{Q}{\lambda}\left\\{\left(X_{0}\overset{\rightarrow}{A}+\phi_{e}\overset{\rightarrow}{X}+\left(\overset{\rightarrow}{X}\times\overset{\rightarrow}{A}\right)\right)-i\lambda\left(X_{0}\overset{\rightarrow}{B}+\phi_{m}\overset{\rightarrow}{X}+\left(\overset{\rightarrow}{X}\times\overset{\rightarrow}{B}\right)\right)\right\\}\right]$
$\displaystyle=$
$\displaystyle\,\,\overrightarrow{L}-\left(\overrightarrow{L}_{e}-i\lambda\overrightarrow{L}_{m}\right)\,\thinspace.$
(58)

In equation (58), $L_{e}$ and $L_{m}$ are the electric and magnetic angular
momentum, respectively. As such,

$\displaystyle\overrightarrow{L}_{e}=$
$\displaystyle\thinspace\,\frac{Q}{\lambda}\overrightarrow{A}^{X}\,\,,$ (59)
$\displaystyle\overrightarrow{L}_{m}=$
$\displaystyle\thinspace\thinspace\frac{Q}{\lambda}\overrightarrow{B}^{X}\,\,,$
(60)

where
$\overrightarrow{A}^{X}\mapsto\left(X_{0}\overset{\rightarrow}{A}+\phi_{e}\overset{\rightarrow}{X}+\left(\overset{\rightarrow}{X}\times\overset{\rightarrow}{A}\right)\right)$
can be considered as quaternionic generalized rotational electric vector
potential and
$\overrightarrow{B}^{X}\mapsto\left(X_{0}\overset{\rightarrow}{B}+\phi_{m}\overset{\rightarrow}{X}+\left(\overset{\rightarrow}{X}\times\overset{\rightarrow}{B}\right)\right)$
is considered as quaternionic rotational generalized magnetic vector
potential. Thus, the generalized quaternionic angular momentum in presence of
EM-field becomes
$\overrightarrow{L}_{em}:\longmapsto\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)$
along with vector potential
$\overrightarrow{V}^{X}=\overrightarrow{A}^{X}-i\lambda\overrightarrow{B}^{X}$.

##  6 Generalized QRD solutions in presence of EM-field

Let us rewrite the generalized Dirac equation in the presence of an EM-field
associated with four-momentum minimum coupling (51) for rotating fermions as

$\displaystyle\left[H_{q}\circ\left(L_{q}-\mathscr{L}\right)-\mathfrak{B}\lambda^{2}I_{q}\right]\circ\boldsymbol{\Psi}=$
$\displaystyle\,\thinspace 0\thinspace\,.$ (61)

It is worth noting that equation (61) is rotational analog of the standard
Dirac equation in presence of external EM-field as
$\left(\overrightarrow{\alpha}\cdot\left(\overrightarrow{p}-\frac{Q}{c}\overset{\rightarrow}{V}\right)\right.$$\left.-\mathscr{\beta}mc^{2}\right)\psi=0.$
As a result, using the equations (54), the equation (61) gives the generalized
QRD equation in EM-field,

$\displaystyle\left[e_{0}\left[D^{0}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda\left(\overrightarrow{D}\left(H\right)\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)-\mathfrak{B}\lambda^{2}I_{0}\right]+e_{j}\left[\lambda
D^{0}\left(H\right)\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right.\right.$
$\displaystyle+$
$\displaystyle\left.D^{j}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\left.\lambda\left(\overrightarrow{D}\left(H\right)\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}-\mathfrak{B}\lambda^{2}I_{j}\right]\right]\circ\boldsymbol{\Psi}=\,$
$\displaystyle 0\>,$ (62)

where the components of quaternionic electromagnetic coupling term
$H_{q}\circ\left(L_{q}-\mathscr{L}\right)=H_{q}\circ L_{q_{em}}$ can be
expressed as

$\displaystyle H_{q}\circ L_{q_{em}}=$
$\displaystyle\,\,e_{0}\left[D^{0}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda
D^{1}\left(H\right)\left(L_{1}-\frac{Q}{\lambda}\overrightarrow{V_{1}}^{X}\right)-\lambda
D^{2}\left(H\right)\left(L_{2}-\frac{Q}{\lambda}\overrightarrow{V_{2}}^{X}\right)-\lambda
D^{3}\left(H\right)\left(L_{3}-\frac{Q}{\lambda}\overrightarrow{V_{3}}^{X}\right)\right]$
$\displaystyle+$ $\displaystyle e_{1}\left[\lambda
D^{0}\left(H\right)\left(L_{1}-\frac{Q}{\lambda}\overrightarrow{V_{1}}^{X}\right)+D^{1}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda
D^{2}\left(H\right)\left(L_{3}-\frac{Q}{\lambda}\overrightarrow{V_{3}}^{X}\right)-\lambda
D^{3}\left(H\right)\left(L_{2}-\frac{Q}{\lambda}\overrightarrow{V_{2}}^{X}\right)\right]$
$\displaystyle+$ $\displaystyle e_{2}\left[\lambda
D^{0}\left(H\right)\left(L_{2}-\frac{Q}{\lambda}\overrightarrow{V_{2}}^{X}\right)+D^{2}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda
D^{3}\left(H\right)\left(L_{1}-\frac{Q}{\lambda}\overrightarrow{V_{1}}^{X}\right)-\lambda
D^{1}\left(H\right)\left(L_{3}-\frac{Q}{\lambda}\overrightarrow{V_{3}}^{X}\right)\right]$
$\displaystyle+$ $\displaystyle e_{3}\left[\lambda
D^{0}\left(H\right)\left(L_{3}-\frac{Q}{\lambda}\overrightarrow{V_{3}}^{X}\right)+D^{3}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda
D^{1}\left(H\right)\left(L_{2}-\frac{Q}{\lambda}\overrightarrow{V_{2}}^{X}\right)-\lambda
D^{2}\left(H\right)\left(L_{1}-\frac{Q}{\lambda}\overrightarrow{V_{1}}^{X}\right)\right]\thinspace,$
(63)

It is important to note that in the presence of an electromagnetic field, the
generalized QRD equation has a dual structure that includes both the EM-
rotational energy (scalar component) and the EM-rotational momentum (vector
component) of the Dirac fermions. Now, with adding the two and four-component
spinors field, we will construct the energy and momentum like solutions of
generalized QRD equation (62) in the following subsections.

###  6.1 Energy solutions corresponding to real quaternion

In this case, we must equate the scalar coefficient of $e_{0}$ as to calculate
the energy solution of the generalized QRD equation (62) as

$\displaystyle\left[D^{0}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda\left(\overrightarrow{D}\left(H\right)\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)-\mathfrak{B}\lambda^{2}I_{0}\right]\boldsymbol{\Psi}=$
$\displaystyle\,\,0\,\,.$ (64)

On substituting the values of quaternionic $D$-matrices, we obtain

$\displaystyle\left(\begin{array}[]{cc}\left[\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}\right]&-i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]\\\
-i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]&\left[\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}\right]\end{array}\right)\left(\begin{array}[]{c}\Psi_{K}\\\
\Psi_{L}\end{array}\right)=$ $\displaystyle\,\,0\,\,.$ (69)

If we use a four-component quaternionic spinors field, then we get

$\displaystyle\Psi_{0}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}\Psi_{2}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$
(70) $\displaystyle\Psi_{1}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}\Psi_{3}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$
(71) $\displaystyle\Psi_{2}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}\Psi_{0}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$
(72) $\displaystyle\Psi_{3}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{i\lambda\left(\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}\Psi_{1}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,.$
(73)

Now, the entire spinor solutions for the QRD equation may now be calculated by
introducing the plane wavefunction ($\boldsymbol{\psi}$) as
$\boldsymbol{\Psi}=\chi_{s}\boldsymbol{\psi}$ where $\chi_{s}$ is the Dirac
spinor in presence of EM-field. To calculate the rotational energy solutions
of generalized quaternionic spinors with spin up state [43], we put
$\Psi_{0}=1$, $\Psi_{1}=0$ for positive energy and $\Psi_{2}=1$, $\Psi_{3}=0$
for negative energy Dirac spinors as

$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{+}^{\uparrow}}=\,\,N_{+}$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}\\\
0\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\,\,,$ (78)
$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{-}^{\uparrow}}=\,\,N_{-}$
$\displaystyle\left(\begin{array}[]{c}\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}\\\
0\\\ 1\\\
0\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\thinspace\thinspace,$ (83)

where
$N_{+}=\frac{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}{\sqrt{\left[\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}\right]^{2}+\lambda^{2}\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)^{2}}}$
and
$N_{-}=\frac{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}{\sqrt{\left[\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}\right]^{2}+\lambda^{2}\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)^{2}}}$
are the normalization constants. Similarly, for spin down state we put
$\Psi_{0}=0$ $\Psi_{1}=1$ for positive energy and $\Psi_{2}=0$ $\Psi_{3}=1$
for negative energy as

$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{+}^{\downarrow}}=\,\,N_{+}$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\thinspace\thinspace,$ (88)
$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{-}^{\downarrow}}=\,\,N_{-}$
$\displaystyle\left(\begin{array}[]{c}0\\\
\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}\\\
0\\\
1\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\,\thinspace.$ (93)

###  6.2 Momentum solutions corresponding to pure quaternion

In order to calculate the rotational momentum solutions of generalized QRD
equation in EM-field, we equate the coefficient of $e_{j}$ in equation (62) as

$\displaystyle\left[\lambda
D^{0}\left(H\right)\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+D^{j}\left(H\right)\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{D}\left(H\right)\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}-\mathfrak{B}\lambda^{2}I_{j}\right]\boldsymbol{\Psi}=$
$\displaystyle\,\,0\>,$ (94)

which gives

$\displaystyle\left(\begin{array}[]{cc}\lambda\left[\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right]&i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left[\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]_{j}\right]\\\
i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left[\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]_{j}\right]&\lambda\left[\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right]\end{array}\right)\left(\begin{array}[]{c}\Psi_{K}\\\
\Psi_{L}\end{array}\right)=$ $\displaystyle\,\,0\,\,.$ (99)

On substituting the values of $\Psi_{K}$ and $\Psi_{L},$ we obtain four
component spinors fields as

$\displaystyle\Psi_{0}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right)}\Psi_{2}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$ (100)
$\displaystyle\Psi_{1}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right)}\Psi_{3}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$ (101)
$\displaystyle\Psi_{2}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right)}\Psi_{0}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$ (102)
$\displaystyle\Psi_{3}\left(E_{em},\overrightarrow{L}_{em}\right)=$
$\displaystyle\,\,\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right)}\Psi_{1}\left(E_{em},\overrightarrow{L}_{em}\right)\,\,,$ (103)

where equations (100) and (101) yield the angular momentum solution for the
anti-particle whereas from equations (102) and (103) yield the momentum
solutions for the particle. The term $e_{j}E_{em}$ is due to the
electromagnetic energy of spin $\frac{1}{2}$ particles and
$\lambda\left(\overrightarrow{L}_{em}-\lambda
I_{j}\right),\lambda\left(\overrightarrow{L}_{em}+\lambda I_{j}\right)$ are
the energy terms for the anti-particles whereas
$\left(\overrightarrow{e}\times\overrightarrow{L}_{em}\right)$ can be
interpreted as the directional interaction governed by spin and the orbital
angular momentum. As a result, for angular momentum solutions with rotating
fermions in the spin up state, we put $\Psi_{0}=1$, $\Psi_{1}=0$ for fermionic
angular momentum and $\Psi_{2}=1$, $\Psi_{3}=0$ for anti-fermionic angular
momentum as

$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{+}^{\uparrow}}=\,\,M_{+}$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right)}\\\
0\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\thinspace,$ (108)
$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{-}^{\uparrow}}=\,\,M_{-}$
$\displaystyle\left(\begin{array}[]{c}\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right)}\\\ 0\\\ 1\\\
0\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\,.$ (113)

where the normalization constants are

$\displaystyle M_{+}=$
$\displaystyle\thinspace\thinspace\frac{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right)}{\sqrt{\left[\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right)\right]^{2}-\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]^{2}}}\thinspace\thinspace,$
$\displaystyle M_{-}=$
$\displaystyle\,\,\frac{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right)}{\sqrt{\left[\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right)\right]^{2}-\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left(\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right)_{j}\right]^{2}}}\thinspace\thinspace.$

Correspondingly, to write the angular momentum solutions of rotating fermions
in the spin down state, we put $\Psi_{0}=0$, $\Psi_{1}=1$ for fermionic
angular momentum and $\Psi_{2}=0$, $\Psi_{3}=1$ for anti-fermionic angular
momentum as

$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{+}^{\downarrow}}=\,\,M_{+}$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left[\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)+\lambda
I_{j}\right)}\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\thinspace,$ (118)
$\displaystyle\boldsymbol{\Psi}:\rightarrow\boldsymbol{\Psi_{-}^{\downarrow}}=\,\,M_{-}$
$\displaystyle\left(\begin{array}[]{c}0\\\
\frac{-i\left[e_{j}\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda\left[\overrightarrow{e}\times\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]_{j}\right]}{\lambda\left(\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)-\lambda
I_{j}\right)}\\\ 0\\\
1\end{array}\right)e^{\frac{i}{\hslash}\left(\overset{\rightarrow}{k}\cdot\overset{\rightarrow}{r}-\omega
t\right)}\,\,.$ (123)

Now, using fundamental symmetries and conservation rules, we will examine the
validity of the foregoing QRD equations and their solutions in the EM field.

####  6.2.1 Lorentz-Poincaré Invariance

The Lorentz transformation can be expressed using a $4\times 4$ transformation
matrix $\varLambda$ as

$\displaystyle\left(X^{\mu}\right)^{\prime}=$
$\displaystyle\sum_{\nu=0}^{3}\left(\Lambda_{\nu}^{\mu}\right)X^{\nu}\,\,,\thinspace\thinspace\,(\mu,\nu=0,1,2,3),$
(124)

where

$\displaystyle\varLambda_{\nu}^{\mu}=$
$\displaystyle\left(\begin{array}[]{cccc}\cosh\omega&-i\sinh\omega&0&0\\\
i\sinh\omega&\cosh\omega&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right)\,\thinspace.$ (129)

Here $\omega$ denotes the boost parameter. To check the Lorentz covariance, we
can use the covariant form of the QRD equation as follows:

$\displaystyle\left[\gamma^{\mu}\left(L_{q}^{\mu}-\frac{Q}{\lambda}(X_{q}^{\mu}V_{q}^{\mu})\right)-\lambda
I_{q}\right]\boldsymbol{\Psi}=$ $\displaystyle\,\,0\thinspace\thinspace.$
(130)

Here $\gamma^{\mu}$ are the Dirac gamma matrices which can be represented in
terms of quaternionic D-matrices as

$\displaystyle\gamma^{0}=$
$\displaystyle\thinspace\mathfrak{B}=\thinspace\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\,\thinspace\mbox{and
}\gamma^{j}=\thinspace\mathfrak{B}D^{j}(H)=\thinspace\left(\begin{array}[]{cc}0&ie_{j}\\\
-ie_{j}&0\end{array}\right)\,.$ (135)

Now, we have

$\displaystyle\left[\gamma^{\mu}\left(L_{q}^{\mu^{\prime}}-\frac{Q}{\lambda}(X_{q}^{\mu^{\prime}}V_{q}^{\mu^{\prime}})\right)-\lambda
I_{q}\right]\boldsymbol{\Psi^{\prime}}=$ $\displaystyle\thinspace\thinspace
0\thinspace\thinspace,$ (136)

with

$\displaystyle\boldsymbol{\Psi^{\prime}=}$
$\displaystyle\thinspace\thinspace\boldsymbol{S\,\Psi}\>,$ (137)

where $\boldsymbol{S}$ is a $4\times 4$ matrix. Therefore, we may write

$\displaystyle\boldsymbol{S^{-1}}\left[\gamma^{\mu}\boldsymbol{S}\varLambda_{\nu}^{\mu}\left(L_{q}^{\mu}-\frac{Q}{\lambda}\left(X_{q}^{\mu}V_{q}^{\mu}\right)\right)-\lambda
I_{q}\boldsymbol{S}\right]\boldsymbol{\Psi}=$
$\displaystyle\,\,0\thinspace\thinspace,$ (138)

which shows

$\displaystyle\boldsymbol{S^{-1}}\gamma^{\mu}\boldsymbol{S}\left(L_{q}^{\mu}-\frac{Q}{\lambda}\left(X_{q}^{\mu}V_{q}^{\mu}\right)\right)=$
$\displaystyle\,\thinspace\varLambda_{\nu}^{\mu}\gamma^{\nu}\left(L_{q}^{\mu}-\frac{Q}{\lambda}\left(X_{q}^{\mu}V_{q}^{\mu}\right)\right)\thinspace\thinspace.$
(139)

where the corresponding term $(\lambda I_{q})$ is treated as a constant during
the transformation in order to keep the covariant form of the Dirac equation.
Furthermore, the Lorentz boost can define $\boldsymbol{S}$ for a Lorentz
transformation along the $e_{1}$-axis as

$\displaystyle\boldsymbol{S}=$
$\displaystyle\,\,\cosh\frac{\omega}{2}+iD^{1}(H)\sinh\frac{\omega}{2}\,\thinspace,$
(140)

where $D^{1}(H)\sim\left(\begin{array}[]{cc}0&ie_{1}\\\
ie_{1}&0\end{array}\right)$. As a result, under Lorentz transformation, the
generalized QRD equation in the EM-field is well invariant.

####  6.2.2 Gauge Invariance

Let us consider the analogous gauge transformation of fermionic spinor field
as

$\displaystyle\boldsymbol{\Psi^{\prime}=}$
$\displaystyle\thinspace\thinspace\exp\left(-i\frac{Q}{\hslash\lambda}\xi(\overset{\rightarrow}{r},t)\right)\boldsymbol{\Psi}\,\,,$
(141)

where $\xi(\overset{\rightarrow}{r},t)$ is an arbitrary differentiable gauge
function in Minkowski space-time. The gauge transformation of the two four-
potentials is also introduced as

$\displaystyle\overrightarrow{A}^{\prime}=$
$\displaystyle\thinspace\thinspace\overset{\rightarrow}{A}-\nabla\xi\thinspace\thinspace\thinspace\mbox{and}\thinspace\thinspace\thinspace\phi_{e}^{\prime}=\,\,\phi_{e}+\left(\frac{\partial\xi}{\partial
t}\right)\,\thinspace,$ (142) $\displaystyle\overrightarrow{B}^{\prime}=$
$\displaystyle\thinspace\,\overset{\rightarrow}{B}-\nabla\xi\thinspace\thinspace\thinspace\mbox{and}\thinspace\thinspace\thinspace\phi_{m}^{\prime}=\,\,\phi_{m}+\left(\frac{\partial\xi}{\partial
t}\right)\,\thinspace,$ (143)

along with

$\displaystyle\left[H_{q}\circ\left(L_{q}-\frac{Q}{\lambda}(X_{q}\circ
V_{q}^{\prime})\right)-\mathfrak{B}\lambda^{2}I_{q}\right]\circ\boldsymbol{\Psi^{\prime}}=\,\,$
$\displaystyle 0\thinspace\,,$ (144)

where $V_{q}^{\prime}$ is the generalized two four-potential transforming
under the gauge transformation and can be defined as

$\displaystyle\overrightarrow{V}^{\prime}=$
$\displaystyle\overset{\rightarrow}{V}-\nabla\xi\thinspace\thinspace\thinspace\mbox{and}\thinspace\thinspace\thinspace\Phi^{\prime}=\Phi+\left(\frac{\partial\xi}{\partial
t}\right)\,\thinspace,$ (145)

where $\overset{\rightarrow}{V}$ and $\Phi$ are already defined in equations
(18) and (19).

####  6.2.3 Duality Invariance

To check the duality invariance of generalized QRD equation in EM-field, let
us define the transformation of two field tensors $F_{\mu\nu}$ and
$\mathscr{\mathcal{F_{\mu\nu}}}$ as

$\displaystyle F_{\mu\nu}^{\prime}\longrightarrow$
$\displaystyle\,\,F_{\mu\nu}\cos\alpha-\mathcal{F}_{\mu\nu}\sin\alpha\thinspace\,$
(146) $\displaystyle\mathcal{F_{\mu\nu}^{\prime}}\longrightarrow$
$\displaystyle\thinspace\thinspace\mathcal{F_{\mu\nu}}\sin\alpha+F_{\mu\nu}\cos\alpha\thinspace\thinspace,\thinspace\thinspace\thinspace\left(0\leq\alpha\leq\frac{\pi}{2}\right)\thinspace\thinspace.$
(147)

The electric and magnetic field vectors can be transformed by replacing
$F_{\mu\nu}\longrightarrow\overrightarrow{\mathsf{E}}$ and
$\mathcal{F_{\mu\nu}}\longrightarrow\overrightarrow{\mathsf{H}}$ where
$\overrightarrow{\mathsf{E}}$ and $\overrightarrow{\mathsf{H}}$ are the
electric and magnetic field vectors. Then, the transformation becomes

$\displaystyle\begin{pmatrix}\overrightarrow{\mathsf{E}}\\\
\overrightarrow{\mathsf{H}}\end{pmatrix}:=$
$\displaystyle\,\thinspace\begin{pmatrix}\begin{array}[]{cc}0&-i\lambda\\\
\frac{i}{\lambda}&0\end{array}\end{pmatrix}\begin{pmatrix}\overrightarrow{\mathsf{E}}\\\
\overrightarrow{\mathsf{H}}\end{pmatrix}\thinspace\thinspace.$ (148)

Here we took the general case of $\alpha=\frac{\pi}{2}$. The matrix
$\begin{pmatrix}\begin{array}[]{cc}0&-i\lambda\\\
\frac{i}{\lambda}&0\end{array}\end{pmatrix}$ is the duality transformation
matrix. Accordingly, dual scalar potentials, dual vector potentials and dual
charge can also transform as

$\displaystyle\begin{pmatrix}\phi_{e}\\\ \phi_{m}\end{pmatrix}:=$
$\displaystyle\thinspace\thinspace\begin{pmatrix}\begin{array}[]{cc}0&-i\lambda\\\
\frac{i}{\lambda}&0\end{array}\end{pmatrix}\begin{pmatrix}\phi_{e}\\\
\phi_{m}\end{pmatrix}\thinspace\thinspace.$ (149)
$\displaystyle\begin{pmatrix}\overrightarrow{A}\\\
\overrightarrow{B}\end{pmatrix}:=$
$\displaystyle\thinspace\thinspace\begin{pmatrix}\begin{array}[]{cc}0&-i\lambda\\\
\frac{i}{\lambda}&0\end{array}\end{pmatrix}\begin{pmatrix}\overrightarrow{A}\\\
\overrightarrow{B}\end{pmatrix}\thinspace\thinspace,$ (150)
$\displaystyle\begin{pmatrix}\mathfrak{e}\\\ \mathfrak{m}\end{pmatrix}:=$
$\displaystyle\,\thinspace\begin{pmatrix}\begin{array}[]{cc}0&-i\lambda\\\
\frac{i}{\lambda}&0\end{array}\end{pmatrix}\begin{pmatrix}\mathfrak{e}\\\
\mathfrak{m}\end{pmatrix}\,\,.$ (151)

The generalized QRD equation in the presence of an EM-field is thus well
invariant under duality transformation when applying the above transformation
equations.

####  6.2.4 CPT Invariance

In order to check the CPT invariance of the generalized QRD equation in the
EM-field, Table 2 summarizes the transformations of various physical
parameters under parity, time reversal, and charge conjugation [44, 45].

Physical quantities |  $\mathsf{P}$-symmetry |  $\mathsf{C}$-symmetry |  $\mathsf{T}$-symmetry
---|---|---|---
$\overrightarrow{\nabla}$ | $-\overrightarrow{\nabla}$ | $\overrightarrow{\nabla}$ | $\overrightarrow{\nabla}$
$\partial_{t}$ | $\partial_{t}$ | $\partial_{t}$ | $-\partial_{t}$
$\lambda$ | $-\lambda$ | $\lambda$ | $-\lambda$
$X_{0}$ | $X_{0}$ | $X_{0}$ | $-X_{0}$
$\overset{\rightarrow}{X}$ | $-\overset{\rightarrow}{X}$ | $\overset{\rightarrow}{X}$ | $\overset{\rightarrow}{X}$
$p_{0}$ | $p_{0}$ | $p_{0}$ | $p_{0}$
$\overset{\rightarrow}{p}$ | $-\overset{\rightarrow}{p}$ | $\overset{\rightarrow}{p}$ | $-\overset{\rightarrow}{p}$
$\phi_{e}$ | $\phi_{e}$ | $-\phi_{e}$ | $\phi_{e}$
$\phi_{m}$ | $-\phi_{m}$ | $-\phi_{m}$ | $-\phi_{m}$
$\overset{\rightarrow}{A}$ | $-\overset{\rightarrow}{A}$ | $-\overset{\rightarrow}{A}$ | $-\overset{\rightarrow}{A}$
$\overset{\rightarrow}{B}$ | $\overset{\rightarrow}{B}$ | $-\overset{\rightarrow}{B}$ | $\overset{\rightarrow}{B}$
$\overrightarrow{\mathsf{E}}$ | $\mathsf{-\overrightarrow{\mathsf{E}}}$ | $-\mathsf{\overrightarrow{\mathsf{E}}}$ | $\mathsf{\overrightarrow{\mathsf{E}}}$
$\overrightarrow{\mathsf{H}}$ | $\mathsf{\overrightarrow{\mathsf{H}}}$ | $-\overrightarrow{\mathsf{H}}$ | $-\overrightarrow{\mathsf{H}}$
Table 2: CPT transformation of various quaternionic physical quantities

Apart from the physical quantities, the transformation of the fermionic spinor
fields can be represented as

$\displaystyle\mathsf{P}\boldsymbol{\Psi}(\overset{\rightarrow}{r},t)\mathsf{P^{-1}}=$
$\displaystyle\thinspace\,\gamma_{0}\boldsymbol{\Psi}(-\overset{\rightarrow}{r},t)\,\thinspace,$
(152)
$\displaystyle\mathsf{C}\boldsymbol{\Psi}(\overset{\rightarrow}{r},t)\mathsf{C^{-1}}=$
$\displaystyle\thinspace\thinspace
i\gamma_{2}\boldsymbol{\Psi}^{*}(\overset{\rightarrow}{r},t)\,\thinspace,$
(153)
$\displaystyle\mathsf{T}\boldsymbol{\Psi}(\overset{\rightarrow}{r},t)\mathsf{T^{-1}}=$
$\displaystyle\thinspace\thinspace
i\gamma_{1}\gamma_{3}\boldsymbol{\Psi}(\overset{\rightarrow}{r},-t)\,\,,$
(154)

where $\mathsf{P,C}$ and $\mathsf{T}$ represent parity, charge conjugation and
time reversal operators while
$\gamma_{0}=\mathfrak{B},\gamma_{j}=\mathfrak{B}D^{j}(H)$ are equivalent to
the quaternionic matrices. From equations (152) to (154), we get

$\displaystyle\mathsf{CPT}\left[\left(H_{q}\circ
L_{q_{em}}-\mathfrak{B}\lambda^{2}I_{q}\right)\circ\boldsymbol{\Psi}\right]\mathsf{T^{-1}P^{-1}C^{-1}}=$
$\displaystyle\,\thinspace 0\thinspace\thinspace,$ (155)

Therefore, under CPT transformation, (61) yields an invariant QRD equation in
the EM-field.

##  7 Generalized Quaternionic Electromagnetic Moment

###  7.1 Corresponding to the real quaternion

The Dirac fermions has an electromagnetic moment or electromagnetic energy
that may be determined using the two component form of Pauli’s spinors. To
find the electromagnetic moment corresponding to the scalar coefficient in the
generalized QRD equation, we may start with the given equation (69) with
condition $E_{em}=E_{em}^{\prime}+\lambda^{2}I_{0}$ where $E_{em}$ is
rotational EM-energy, $E_{em}^{\prime}$ is the rotational kinetic energy and
$\lambda^{2}I_{0}$ is rotational analogous of rest mass energy in EM-field,
then one can write

$\displaystyle\left(E_{em}^{\prime}+2\lambda^{2}I_{0}\right)\Psi_{L}-i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}_{em}\right)\Psi_{K}=$
$\displaystyle\,\,0\,\,,$ (156)

Since, in the non-relativistic limit $E_{em}^{\prime}<<\lambda^{2}I_{0},$ we
have

$\displaystyle\Psi_{L}\,\,\simeq$
$\displaystyle\thinspace\,\left(\frac{i\left(\overrightarrow{e}\cdot\overrightarrow{L}_{em}\right)}{2\lambda
I_{0}}\right)\Psi_{K}\,\,.$ (157)

As a result, by substituting equation (157) in second component solution of
equation (69), we obtain

$\displaystyle\left(E_{em}-\lambda^{2}I_{0}\right)\Psi_{K}+$
$\displaystyle.\left(\frac{1}{2I_{0}}\left(L_{em}\right)^{2}-\frac{Q\hslash}{2I_{0}\lambda}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{H}}\right)-\frac{iQ\hslash}{2I_{0}}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{E}}\right)\right)\Psi_{K}=0\,\thinspace.$
(158)

This equation can be referred as the generalized Schrodinger-Pauli like
equation where the unperturbed energy term is
$\left|\frac{1}{2I_{0}}\left(L_{em}\right)^{2}+\lambda^{2}I_{0}\right|$, while
the perturbed energy terms are
$\left|\frac{Q\hslash}{2I_{0}\lambda}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{H}}\right)\right|$
and
$\left|\frac{iQ\hslash}{2I_{0}}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{E}}\right)\right|$
corresponding to magnetic and electric dipole moments
$\frac{Q\hslash\overset{\rightarrow}{e}}{2I_{0}\lambda}$ and
$\frac{iQ\hslash\overset{\rightarrow}{e}}{2I_{0}}$, respectively. Furthermore,
we have found dual perturbed energies, respectively, the generalized electric
dipole energy [46] i.e.
$\left|\frac{\kappa}{\lambda}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{H}}\right)\right|$
and the generalized magnetic dipole energy i.e.
$\left|i\kappa\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{E}}\right)\right|$
due to EM-interaction in fermionic field, where
$\kappa=\frac{Q\hslash}{2I_{0}}$ is a constant.

###  7.2 Corresponding to the pure quaternion

In this case, we choose vector coefficient in the generalized QRD equation and
applying the condition
$\overrightarrow{L}_{em}=\overrightarrow{L}_{em}^{\prime}+\lambda I_{j}$, and
obtain

$\displaystyle\lambda\left(L_{j_{em}}^{\prime}+2\lambda
I_{j}\right)\Psi_{L}+i\left[e_{j}E_{em}+\lambda\left(\overrightarrow{e}\times\overrightarrow{L}_{em}\right)_{j}\right]\Psi_{K}=$
$\displaystyle\,\,0\,\,,$ (159)

For the non-relativistic limit $\lambda
L_{j_{em}}^{\prime}<<2\lambda^{2}I_{j},$ then equation (159) gives

$\displaystyle\Psi_{L}\approx\thinspace\,$
$\displaystyle\frac{-i\left[e_{j}E_{em}+\left(\overrightarrow{e}\times\overrightarrow{L}_{em}\right)_{j}\right]}{2\lambda^{2}I_{j}}\Psi_{K}\thinspace\thinspace.$
(160)

Substituting equation (159) in second component solution of equation (99), and
obtain

$\displaystyle\left[\lambda\overrightarrow{L}_{em}^{\prime}-\frac{1}{2\lambda^{2}I_{j}}\left(E_{em}^{2}\right)+\frac{1}{2I_{j}}\left\\{-2\left(\overrightarrow{L}_{em}^{2}\right)+\frac{Q\hslash}{2I_{j}\lambda}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{H}}\right)+\frac{Q\hslash
i}{2I_{j}}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{E}}\right)\right\\}+\frac{2e_{j}E_{em}\left(\overrightarrow{e}\times\overrightarrow{L}_{em}\right)_{j}}{2\lambda
I_{j}}\right]\Psi_{K}=$ $\displaystyle\,\,0\,\,.$ (161)

As equation (158), the equation (161) can describe the electromagnetic energy
in quaternionic rotational momentum space. The last term denotes the energy
corresponding to the directional spin orbit coupling in the quaternionic
angular momentum-space.

On the other hand, if we consider ‘rotating dyons’ in the generalized
fermionic spinor dual-fields that includes both an electric field due to
electrons and a magnetic field due to magnetic monopoles, then the following
field constituents for generalized Dirac spinors may be used for unified
dyonic fields [14, 35, 36]:

$\displaystyle Q_{\text{Dyon}}\,=$
$\displaystyle\left(\mathfrak{e}-i\lambda\mathfrak{m}\right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Dyonic
Charge})$ $\displaystyle\Phi_{\text{Dyon}}^{X}\,=$
$\displaystyle\left(\phi_{e}^{X}-i\lambda\phi_{m}^{X}\right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Dyonic
Scalar Potential})$ $\displaystyle\overrightarrow{V}_{\text{Dyon}}^{X}\,=$
$\displaystyle\left(\overrightarrow{A}^{X}-i\lambda\overrightarrow{B}^{X}\right)\,\,,\,\,\,\,\,\,\,\,\,\,(\text{Dyonic
Vector Potential})$ $\displaystyle E_{\text{Dyon}}\,=$
$\displaystyle\,\left(E_{e}-i\lambda
E_{m}\right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Dyonic Rotational EM-
Energy})$ $\displaystyle\overrightarrow{L}_{\text{Dyon}}\,=$
$\displaystyle\,\left(\overrightarrow{L}_{e}-i\lambda\overrightarrow{L}_{m}\right)\,\,,\,\,\,\,\,\,\,\,\,\,(\text{Dyonic
EM-Angular Momentum})$ $\displaystyle E_{\text{Dyon}}^{\text{di}}\,=$
$\displaystyle\frac{Q\hslash}{2iI_{0}}\left(\overset{\rightarrow}{e}\cdot\overrightarrow{\mathsf{\Psi}}_{\text{Dyon}}\right)\,\,,\,\,\,\,\,\,\,(\text{Dyonic
Dipole Energy})$
$\displaystyle\overrightarrow{\mathsf{\Psi}}_{\text{Dyon}}\,=$
$\displaystyle\left(\overrightarrow{\mathsf{E}}-\frac{i}{\lambda}\overrightarrow{\mathsf{H}}\right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{Dyonic
EM-Field})$

Hence, we examined a symmetrical form of the generalized QRD equation in the
presence of an external EM-field, in which two four-potentials are recognized
as the gauge potentials associated with a particle (or antiparticle)
containing the simultaneous existence of electric and magnetic charge
(monopole) of dyons. [H]  Physical quantities General Case Quaternionic-
Rotational Case Dirac equation without EM-field:
$\left(\overrightarrow{\alpha}.\overrightarrow{p}-\beta mc^{2}\right)\psi=0$
$\left(H_{q}\circ
L_{q}-\mathscr{\mathfrak{B}}\lambda^{2}I_{q}\right)\circ\mathtt{\boldsymbol{\Psi}}=0$
Minimal Substitution:
$\overset{\rightarrow}{p}\rightarrow\overset{\rightarrow}{p}-\frac{e}{c}\overset{\rightarrow}{A}\thinspace\,,E\rightarrow
E-e\phi$ $L^{\mu}:\rightarrow L^{\mu}-\frac{Q}{\lambda}\left(X^{\mu}\circ
V^{\mu}\right)\,\thinspace\equiv\thinspace\Pi_{L}^{\mu}\thinspace\thinspace(\mu=0,1,2,3)$
Dirac equation with EM-field:
$\left(\overrightarrow{\alpha}\cdot\left(\overrightarrow{p}-\frac{e}{c}\overset{\rightarrow}{A}\right)-\mathscr{\beta}mc^{2}\right)\psi=0$
$\left(H_{q}\circ\left(L_{q}-\frac{Q}{\lambda}(X_{q}\circ
V_{q})\right)-\mathfrak{B}\lambda^{2}I_{q}\right)\circ\boldsymbol{\Psi}=0$
Spinor solutions: $\psi_{+}^{\uparrow}=\,\,N_{+}\left(\begin{array}[]{c}1\\\
0\\\ \frac{c(\overrightarrow{\sigma}.\overset{\rightarrow}{P})}{E+mc^{2}}\\\
0\end{array}\right),\psi^{\downarrow+}=\,\,N_{+}\left(\begin{array}[]{c}0\\\
1\\\ 0\\\
\frac{c\left(\overrightarrow{\sigma}\cdot\overrightarrow{P}\right)}{E+mc^{2}}\end{array}\right)$
$\boldsymbol{\Psi}_{+}^{\uparrow}=\,\,N_{+}\left(\begin{array}[]{c}1\\\ 0\\\
\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}+\lambda^{2}I_{0}}\\\
0\end{array}\right)\boldsymbol{\Psi}^{\downarrow+}=\,\,N_{+}\left(\begin{array}[]{c}0\\\
1\\\ 0\\\
\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}+\lambda^{2}I_{0}}\end{array}\right)$
$\psi^{\uparrow-}=\,\,N_{-}\left(\begin{array}[]{c}\frac{c\left(\overrightarrow{\sigma}\cdot\overrightarrow{P}\right)}{E-mc^{2}}\\\
0\\\ 1\\\
0\end{array}\right),\psi^{\downarrow-}=\,\,N_{-}\left(\begin{array}[]{c}0\\\
\frac{c\left(\overrightarrow{\sigma}\cdot\overrightarrow{P}\right)}{E-mc^{2}}\\\
0\\\ 1\end{array}\right)$
$\boldsymbol{\Psi}^{\uparrow-}=\,\,N_{-}\left(\begin{array}[]{c}\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}-\lambda^{2}I_{0}}\\\
0\\\ 1\\\
0\end{array}\right),\boldsymbol{\Psi}^{\downarrow-}=\,\,N_{-}\left(\begin{array}[]{c}0\\\
\frac{i\lambda\left(\overrightarrow{e}\cdot\overrightarrow{L}\right)}{E_{0}-\lambda^{2}I_{0}}\\\
0\\\ 1\end{array}\right)$ Spinor solutions with EM-field:
$\psi{}_{+}^{\uparrow}=\,\,N_{+}\left(\begin{array}[]{c}1\\\ 0\\\
\frac{c\left[\overrightarrow{\sigma}\cdot\left(\overrightarrow{P}-\frac{\mathfrak{e}}{c}\overrightarrow{A}\right)\right]}{\left(E-\mathfrak{e}\phi\right)+mc^{2}}\\\
0\end{array}\right)$
$\boldsymbol{\Psi_{+}^{\uparrow}}=\,\,N_{+}\left(\begin{array}[]{c}1\\\ 0\\\
\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}\\\
0\end{array}\right)$
$\psi{}_{+}^{\downarrow}=\,\,N_{+}\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\frac{\lambda\left[\overrightarrow{\sigma}\cdot\left(\overrightarrow{P}-\frac{\mathfrak{e}}{c}\overrightarrow{A}\right)\right]}{\left(E-\mathfrak{e}\phi\right)+mc^{2}}\end{array}\right)$
$\boldsymbol{\Psi_{+}^{\downarrow}}=\,\,N_{+}\left(\begin{array}[]{c}0\\\ 1\\\
0\\\
\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)+\lambda^{2}I_{0}}\end{array}\right)$
$\psi{}_{-}^{\uparrow}=\,\,N_{-}\left(\begin{array}[]{c}\frac{c\left[\overrightarrow{\sigma}\cdot\left(\overrightarrow{P}-\frac{\mathfrak{e}}{c}\overrightarrow{A}\right)\right]}{\left(E-\mathfrak{e}\phi\right)-mc^{2}}\\\
0\\\ 1\\\ 0\end{array}\right)$
$\boldsymbol{\Psi_{-}^{\uparrow}}=\,\,N_{-}\left(\begin{array}[]{c}\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}\\\
0\\\ 1\\\ 0\end{array}\right)$
$\psi{}_{-}^{\downarrow}=\,\,N_{-}\left(\begin{array}[]{c}0\\\
\frac{c\left[\overrightarrow{\sigma}\cdot\left(\overrightarrow{P}-\frac{\mathfrak{e}}{c}\overrightarrow{A}\right)\right]}{\left(E-\mathfrak{e}\phi\right)-mc^{2}}\\\
0\\\ 1\end{array}\right)$
$\boldsymbol{\Psi_{-}^{\downarrow}}=\,\,N_{-}\left(\begin{array}[]{c}0\\\
\frac{i\lambda\left[\overrightarrow{e}\cdot\left(\overrightarrow{L}-\frac{Q}{\lambda}\overrightarrow{V}^{X}\right)\right]}{\left(E_{0}-\frac{Q}{\lambda}\Phi^{X}\right)-\lambda^{2}I_{0}}\\\
0\\\ 1\end{array}\right)$

Comparison of some physical quantities

##  8 Conclusion

Hypercomplex division algebra (viz. quaternion) is regarded as a necessity to
study an additional dimension of any system or theory in a single framework.
In this case, quaternonic representation of four-potential, four-momentum,
four-position etc. have been discussed in view of Minkowski structure. The
generalized quaternionic rotational Dirac equation and its solutions for
fermions without EM-field has been discussed in equation (35). The generalized
QRD equation has the benefit of representing both aspects as rotational energy
and rotating momentum corresponding to quaternionic scalar and vector
components. The most important feature of quaternionic analysis is that it
allows for the dual representation of four vectors in a unified structure. As
a result, the EM-field interaction in the extended QRD equation is stated in
equation (54), where equations (55) and (58) define the resulting quaternionic
rotational EM-energy corresponding to the scalar component and quaternionic
rotational angular momentum corresponding to the vector component,
respectively. Accordingly, the electric and magnetic angular momentum are
established in terms of quaternionic generalized rotational electric and
magnetic vector potentials. Furthermore, the four component Dirac spinors with
spin up and down states have been used to examine the solutions for
quaternionic rotational energy and momentum equations in the EM-field. The
generalized Schrodinger-Pauli-like energy equation, which is related with
unperturbed and perturbed energies as a result of EM-field interaction, has
been constructed in equation (158). Due to EM-interaction in a fermionic
field, we have interpreted the dual perturbed energies, respectively, the
generalized electric dipole energy and the generalized magnetic dipole energy
in view of quaternionic analysis. Table-3 shows a comparison of the linear and
rotational behavior of several quaternionic physical quantities. The beauty of
present analysis, to examine the different symmetries the extended
quaternionic rotational Dirac equation in presence of EM-field has been shown
to be well invariant under the Lorentz, gauge, duality, and CPT invariances.
With the interaction of EM-field, the extended QRD equation possesses
symmetrical structure in electric and magnetic fields, with two four-
potentials associated with Dirac fermions (or anti-fermions) including the
simultaneous existence of electric and magnetic charge of rotating dyons.

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