B. C. Chanyal and Mayank Pathak
Department of Physics, G. B. Pant University of Agriculture &
Technology, Pantnagar, Uttarakhand 263145, India
Correspondence should be addressed to B. C. Chanyal;
[email protected]
Received 23 June 2018; Accepted 7 August 2018; Published 14 August
2018
Academic Editor: Antonio J. Accioly
Copyright © 2018 B. C. Chanyal and Mayank Pathak. This is an open
access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited. The publication of this article was funded by SCOAP3.
The dual magnetohydrodynamics of dyonic plasma describes the study
of electrodynamics equations along with the transport equations in
the presence of electrons andmagnetic monopoles. In this paper, we
formulate the quaternionic dual fields equations, namely, the
hydroelectric and hydromagnetic fields equations which are an
analogous to the generalized Lamb vector field and vorticity field
equations of dyonic cold plasma fluid. Further, we derive the
quaternionic Dirac-Maxwell equations for dual magnetohydrodynamics
of dyonic cold plasma. We also obtain the quaternionic dual
continuity equations that describe the transport of dyonic fluid.
Finally, we establish an analogy of Alfven wave equation which may
generate from the flow of magnetic monopoles in the dyonic field of
cold plasma.The present quaternionic formulation for dyonic cold
plasma is well invariant under the duality, Lorentz, and CPT
transformations.
1. Introduction
In the past few decades, astronomers predicted that the universewas
composed almost entirely of the baryonicmatter (ordinary matter).
According to Bachynski [1], more than 99% of the matter in the
universe is in plasma state. This type of matter may consist of
baryonic and nonbaryonic matter. The first experimental evidence of
the existence of plasma was given by American Physicists [2]. In
plasma, consisting of charged and neutral particles, the interionic
force between particles shows electromagnetic in nature. Therefore,
due to the long range order of Coulomb force charged particles
interact with all other charged particles resulting in a collec-
tive behavior of plasma. In 1942, Alfven [3] gave the theory of
magnetohydrodynamics (MHD) and suggested that electri- cally
conducting fluid can support the propagation of shear waves called
the Alfven waves. Basically, MHD describes the behavior of
electrically conducting fluid in the presence of magnetic field
[4]. It is macroscopic theory that assumes the electrons, ions, and
charged particles move together and treated themas a single fluid
component known as single fluid theory. The plasma along with MHD
is simply described by a single temperature, velocity, and density.
However, when
the MHD wave propagates faster than plasma thermal speed then the
effect of temperature can be neglected [5]. This is called a cold
plasma approximation (i.e., in cold plasma approximation,
temperature does not take into account). In this approximation,
there is no wave related to pressure fluc- tuation (e.g., sound
waves). On the other hand, the hot and warm plasmas are another
sates of plasma where the collision between electrons and gas
molecules are so frequent that there is a thermal equilibrium
between electron and the gas molecules.
Meyer-Vernet [6] discussed the role of magnetic monop- ole in
conducting fluid (plasma). The magnetic monopole proposed by Dirac
[7], is a hypothetical elementary particle having only one magnetic
pole. Dirac also pointed out that if there exists any monopole in
the universe then all the electric charge in the universe will be
quantized [8]. Schwinger [9, 10], an exception to the argument
against the existence of monopole, formulated relativistically
covariant quantum field theory of magnetic monopoles which main-
tained complete symmetry between electric and magnetic fields.
Therefore, the name of particles carrying simulta- neously the
electric and magnetic charges is dyons. Fur- ther, the theoretical
approach of Schwinger [9, 10] and
Hindawi Advances in High Energy Physics Volume 2018, Article ID
7843730, 13 pages https://doi.org/10.1155/2018/7843730
2 Advances in High Energy Physics
Zwanziger [11] describes the theory of dyonic particles. Peres [12]
pointed out the controversial nature [13] of the singular lines of
magnetic monopoles and established the charged quantization
condition in purely group theoreti- cal manner without using them.
In view of mathematical physics, the study of four-dimensional
particles (dyons) in distinguish mediums can be explained by
division algebras. There are four types of divisions algebras [14],
namely, the real, complex, quaternion, and octonion algebras. The
complex algebra is an extension of real numbers; the quater- nion
is an extension of complex numbers while the octonion is an
extension of quaternions. Quaternionic algebra [15] can also
express by the four-dimensional Euclidean spaces [16, 17], and it
has vast applications in the multiple branches of physics.
Further, Rajput [18] pointed out an effective unified theory for
quaternionic generalized electromagnetic and gravitational fields
of dyons by using the quaternion algebra. The quaternionic form of
classical and quantum electrody- namics has been already discussed
[19–22]. Many authors [23–29] have studied the role of hypercomplex
algebras in various branches of physics. Recently, Chanyal [30, 31]
independently proposed a novel approach on the quater- nionic
covariant theory for relativistic quantum mechanics and established
the quantized Dirac-Maxwell equations for dyons. Besides, in
literature [32–34], the reformulation of incompressible plasma
fluids and MHD equations has been discussed in terms of
hypercomplex numbers. Keeping in view the importance of
quaternionic algebras, we establish the MHD field equations for
dyonic cold plasma. Starting with the definitions of one-fluid and
two-fluid theory of plasma, we identify the cold plasma
approximation where the thermal effects (or pressure effects) of
conducting fluid will be neglected. Further, we introduce the dual
MHD equations of dyonic plasma consisted with electrons, mag- netic
monopoles, and their counter partners, namely, ions and
magnetoions. In this study, we clarify that the domi- nating aspect
for the dyonic cold plasma approximation is the dynamics of
electrons along with magnetic monopoles. As we know the generalized
Dirac-Maxwell like equations are primary equations to explain the
dynamics of dyonic cold plasma. Therefore, undertaking the
quaternionic dual- velocity and dual-enthalpy of dyonic cold
plasma, we have made an attempt to formulate the quaternionic
hydroelectric and hydromagnetic fields equations, which are an
analogous to the generalized Lamb vector field and vorticity field
of conducting dyonic fluid. The Lorenz gauge conditions for dyonic
cold plasma fluid are also obtained. Further, we derive the
generalized quaternionic Dirac-Maxwell equa- tions to the case of
dual magnetohydrodynamics of dyonic cold plasma. We have discussed
that these Dirac-Maxwell equations for dyonic cold plasma are well
invariant under the duality, Lorentz, and CPT transformations.
Finally, the Alfven wave like equation is established whichmay
propagate from the flow of magnetic monopoles in the dyonic cold
plasma.
2. The Quaternions
Through the extension of the set of natural numbers to the
integers, a complex number C is defined by the set of all real
linear combinations of the unit elements (1, ), such that
C → { = 1 + 2 | (1, 2 ∈ R)} , (1)
where the real number1 is called the real part and2 is called the
imaginary part of a complex number . If the real part Re() = 0,
then we can say that is purely imaginary. As such, the Euclidean
scalar product as C × C → R is then defined by , = Re ( ⋅ ) = 11 +
22, (2)
where = 1+2 and = 1+2 are two complex numbers. The modulus of any
complex number is also defined by || =√ ⋅ = √21 + 22.
However, a complex field C is a finite dimensional real vector
space, so thatwe can easily extend the complex number into the
quaternionic field H by losing the commutativity of multiplication.
Thus, the quaternion represents the natural extension of complex
numbers and forms an algebra under addition and multiplication.
Hamilton [15] described a four- dimensional quaternionic algebra
and applied it tomechanics in three-dimensional space. A striking
feature of quaternions is that the product of two quaternions is
noncommutative, meaning that the product of two quaternions depends
on which factor is to the left of the multiplication sign and which
factor is to the right.
Thus the allowed four-dimensional Hamilton vector space is defined
by quaternion algebra H over the field of real numbers R as
H → {{{ = 3∑=0 = 00 + 11 + 22 + 33 | ∀ ∈ R
}}} , (3)
where the Hamilton vector space (H) has the quaternionic elements
(0, 1, 2, and 3), which are called quaternion basis elements while
0, 1, 2, and 3 are the real quarterate of a quaternion. As such the
addition of two quaternions =00 + 11 + 22 + 33 and = 00 + 11 + 22 +
33 is given by + = 0 (0 + 0) + 1 (1 + 1) + 2 (2 + 2)+ 3 (3 + 3) , ∀
(, ) ∈ H. (4)
Here, the quaternionic addition is clearly associative and
commutative. The additive identity element is defined by the zero
element; i.e., 0 = 00 + 10 + 20 + 30, (5)
Advances in High Energy Physics 3
and the additive inverse of ∈ H is given by− = 0 (−0) + 1 (−1) + 2
(−2) + 3 (−3) . (6)
Correspondingly, the product of two quaternions, i.e., ( ) ∈ H, can
be expressed by = 0 (00 − 11 − 22 − 33)+ 1 (01 + 10 + 23 − 32)+ 2
(02 − 13 + 20 + 31)+ 3 (03 + 12 − 21 + 30) . (7)
We may notice that this quaternionic product is associa- tive, but
not commutative. The quaternionic unit elements(0, 1, 2, 3)
followed the given relations,20 = 1,2 = −1,0 = 0 = , = −0 + , (∀, ,
= 1, 2, 3)
(8)
where is the delta symbol and is the Levi Civita three-index symbol
having value = +1 for cyclic permutation, = −1 for anticyclic
permutation, and = 0 for any two repeated indices. Further, we also
may write the following relations to quaternion basis elements[, ]
= 2,{, } = −20, () = () , (9)
where the brackets [ , ] and { , } are used, respectively, for
commutation and the anticommutation relations. Thus the above
multiplication rules governed the ordinary dot and cross product;
i.e., = (00 − ⋅ , 0 + 0 + ( × )) , (10)
where we take × = 0 for noncommutative product of quaternion. The
quaternionic product with the scalar quantity is given by = 0 (0) +
1 (1) + 2 (2) + 3 (3) . (11)
As such, the multiplication identity element can expressed by the
unit elements,1 = 01 + 10 + 20 + 30. (12)
Moreover, a quaternion can also be decomposed in terms of scalar
(()) and vector (V()) parts as () = 12 ( + ) , (13)
V () = 12 ( − ) , (14)
where the quaternionic conjugate is expressed by = 00 − (11 + 22 +
33) . (15)
The real and imaginary parts of can be written as
Re (H) = 0, (16)
Im (H) = {11 + 22 + 33 | ∀=1,2,3 ∈ R} ⊆ H. (17)
If Re(H) = 0 and = 0, then is said to be purely imaginary
quaternions. Therefore, all quaternions with zero real are
simplified as imaginary space of H, where the imaginary space Im(H)
∈ R3 is a three-dimensional real vector space,
Im () = (1, 2, 3) Im ()† = (123). (18)
Interestingly, we may write the following form of quaternion
as
= Re () + 3∑ =1
Im () . (19)
The quaternionic Euclidean scalar product H × H → R can also be
expressed as, = Re ( ) = 00 + 11 + 22 + 33. (20)
Like complex numbers, the modulus of quaternion is then defined as
|| = √20 + 21 + 22 + 23 . (21)
Since there exists the norm() = of a quaternion, we have a
division; i.e., every has an inverse of a quaternion and is
expressed as −1 = || . (22)
The quaternion conjugation satisfies the following property:1 2 = 1
2. (23)
The norm of the quaternion is positive definite and obeys the
composition law(1 2) = (1) (2) . (24)
The quaternion elements are non-Abelian in nature and thus
represent a noncommutative division ring. Quaternion is an
important fundamental mathematical tool appropriate for
four-dimensional world.
4 Advances in High Energy Physics
3. Magnetohydrodynamics of Cold Plasma
Let us start with the basic parameters of the plasma. As we know
that the plasma exists in many more forms in nature which has a
wide spread use in the science and technology. The theory of plasma
is divided into three categories [35], namely, the microscopic
theory, kinetic theory, and the fluid theory. In briefly, the
microscopic theory is based on the motion of all the individual
particles (e.g., electrons, ions, atoms, molecules, and radicals).
According to Klimontovich [36], the time evolution of the particle
density ( →(r, k, )) is expressed by + k ⋅ ∇ + (E + k × B) ⋅ ∇ = 0,
(25)
where k is the velocity of particles, (, ) are the effective charge
and mass of the −species particles, and (E, B) are the electric and
magnetic field produced by the microscopic particles. Besides, the
collisionless kinetic theory of plasma proposed by Vlasov [37],
which has included the Boltzmann distribution function as [35], is
as follows: + k.∇ + (E + k × B) .∇ = 0. (26)
In (25) and (26), we may consider that the two dominating particles
(i.e., electrons and ions both) constitute the dynam- ics of
plasma, called the two-fluid theory of plasma [35–38]. For the
two-fluid theory of plasma, at a given position (), the mass and
charge densities become () = () + () , (27) () = () + () ,
(28)
where , , and are defined as the mass, total number, and charge of
electrons while , , and are defined as the mass, total number, and
charge of ions, respectively. The center of mass fluid velocity can
be expressed as
k = 1 () (k () + k ()) , (29)
and the current density becomes
J = k + k. (30)
The continuity equations can be written as + ∇ ⋅ (k) = 0, (mass
conservation law) (31) + ∇ ⋅ J = 0, (charge conservation law)
(32)
As such, the momentum equation for plasma fluid is expressed as
[35]( + k ⋅ ∇) k = (J × B) + E − ∇, (33)
where ∇ is the pressure force introduced due to the inho- mogeneity
of the plasma and (J × B) is a Lorentz force per unit volume
element. Now, we introduce an acceleration to the conducting
fluid,k → ( + k ⋅ ∇) k, (34)
where the term (k ⋅∇)k is used for the convective acceleration of
fluid. Furthermore, the generalized Ohm’s law becomes [35]2 J = 2∇
+ E + (k × B) − (J × B)
− J , (35)
where denotes the conductivity of fluid. One can define Maxwell’s
equations with natural unit ( = = 1) as
∇ ⋅ E = , (36)
∇ × B = E + J. (39)
Interestingly, if we combine together the conducting fluidic field
and electromagnetic field then the relevant theory comes out which
is called MHD. The MHD of cold plasma is an approximation theory of
fluid dynamics where we neglect temperature effect and combine the
electron equation with ionic equation to form a one-fluid model
[39]. For the cold plasma model, many researchers [40, 41]
suggested that, at a given position, all particle-species (mostly
ions and electrons) have comparable temperatures (), energies (E)
(equivalent to masses), and velocities (k). It follows that the
fluid velocity is identical for particle velocity. Now, we may
summarize the following conditions for the cold plasma
approximation, i.e., ∼ (neglected)
E ∼ E
(40)
We consider that the effected behavior of electrons are com-
parable to the ions, while their temperatures and pressure-
gradients are taken negligible in case of homogeneous cold plasmas.
Thus, using approximation (40), the average mass and charge
densities to cold plasma are expressed as → () () ≡ () , (41) → ()
() ≡ () . (42)
Advances in High Energy Physics 5
As such, the Navier-Stokes and Ohm’s equations become ( + k ⋅ ∇) k
= E, (43)
J = (E + k × B) , (44)
where (J×B) ∼ 0 to the case if the current is small compared to (k
× B). The ideal MHD equations ( ∼ 0) for cold plasma may then be
expressed as + ∇ ⋅ (k) = 0, (45)
( + k ⋅ ∇) k = 0, (46)
∇ × (k × B) = B , (47)
∇ × B = E + J. (48)
To consider wave behavior of cold particles, the cold plasma wave
has temperature independent dispersion relation. If k is Alfven
velocity, then the dispersion relation for cold plasma waves become
[35] 2 = 2k2/(1 + k2). Interestingly, the cold plasma waves
propagate like as Alfven waves which are independent of
temperature.
4. Dual MHD Equations for Dyonic Cold Plasma
The dual MHD field not only consists of electrons and ions but also
has the magnetic monopole and their ionic partners magnetoions
[42]. Generally, the composition of an electron and a magnetic
monopole referred a dyon [25]. In this study, we may neglect the
magnetoionic contribution like ions to continue the dyonic cold
plasma approximations. Dirac [8] proposed the symmetrized field
equations by postulating the existence of magnetic monopoles;
i.e.,
∇ ⋅ E = , (49)
∇ × B = E + J. (52)
In the above generalized Dirac-Maxwell’s equations, andm are the
electric and magnetic charge densities while J and Jm are the
corresponding current densities. To study the dyonic cold plasma
field, there are a couple of masses and charges species in presence
of dyons. Thus, the generalized dual densities (mass and charge
densities) may be expressed for one-fluid theory of dyonic cold
plasma as (, m) → ( + mm) , (53) (, m) → ( + mm) , (54)
wherem, m, and m are defined as the mass, total number, and charge
of magnetic monopoles, respectively. As such, we can express the
center of mass velocity of dyonic fluid in cold plasma as
k 1 (k () + kmmm ()) , (55)
whereupon the dual-current densities (electric andmagnetic) are
defined by
J = k, and Jm = mmkm. (56)
The conservation laws for the dynamics of dyonic cold plasma can be
written as + ∇ ⋅ (k) = 0,(dyons mass conservation law) (57)
+ ∇ ⋅ J = 0,(electric charge conservation law) (58)
m + ∇ ⋅ Jm = 0,(magnetic charge conservation law) . (59)
The generalized Navier-Stokes force equation can also be exhibited
in presence of magnetic monopole; i.e.,( + k ⋅ ∇) k = (J × B) − (Jm
× E) + E+ mB − (∇) , (60)
where the duality invariant Lorentz force equation for dyons
is
F = E + (J × B) + mB − (Jm × E) (61)
and the dyonic pressure gradient term (∇) is negligible to the case
of cold plasma approximation. Conditionally, if the influence of
dyonic current is small then the force equation can be written as (
+ k ⋅ ∇) k = E + mB. (62)
In the same way, Ohm’s law for the dyonic cold plasma is expressed
as
J = (E + k × B) , (63)
Jm = m (B − k × E) . (64)
where m is the magnetic conductivity. Therefore, from (63) to (64),
we can conclude that for infinite conductivity of dyons (,m → ∞)
the electric and magnetic field vectors constitute from the
rotation of each other, i.e., E = −(k × B), and B = (k × E). The
above classical field equations given by (49) to (64) of dyons are
referred to dualMHDfield equations of dyonic cold plasma.
6 Advances in High Energy Physics
Table 1: Analogies between electrodynamics and hydrodynamics in
presence of dyons.
Electrodynamics case Hydrodynamics case
fluid) B (magnetic vector potential) → (magnetic velocity of
the
fluid) (electric scalar potential) → (electric enthalpy of
the
fluid)m (magnetic scalar potential) → (magnetic enthalpy of
the
fluid)
5. Quaternionic Formulation to Dual Fields of Dyonic Cold
Plasma
In order to write the dual MHD field equations for dyonic cold
plasma, we may start with quaternionic two-velocity (u, ) and
two-enthalpy (, ) of dyons for plasma fluid dynamics as
U (1, 2, 3, 0) = {, , , − 0 } , (65)
V (1, 2, 3, 0) = {, , , −0} , (66)
where (U,V) are quaternionic variables associated with two
four-velocities of electrons andmagnetic monopoles of dyons and 0
denoted the speed of particles (dyons) moving in con- ducting cold
plasma. Here, we have taken the two-enthalpy of dyons, i.e., the
internal energy of dyons associated with electrons and magnetic
monopoles. Like many physicists [32, 43, 44], there is an analogy
between the electromagnetic and hydrodynamic. Thus, we may write
the analogy of two four- potentials (A,B) of dyons as
A(A, − ) → U(u, − 0 ) , (67)
B (B, −m) → V (, −0) , (68)
where the vector components u → (, , ), →(, , ) are analogous to
electric and magnetic vector potentials of dyons while the scalar
components (, ) are analogous to their scalar potentials. It should
be notice that the role of quaternionic two four-velocities of
dyonic fluid in generalized hydrodynamics of cold plasma is the
same as the quaternionic two four-potentials of dyons in
generalized electrodynamics. Now, we may summarize the dyonic
potentials corresponding to its fluid behavior in Table 1.
The unified structure of quaternionic two four-velocities (W ∈ H)
for the generalized fields of dyonic cold plasma can be written
as
W = (U − 0 V)= 1 ( − 0 ) + 2 ( − 0 )+ 3 ( − 0 ) − 0 0 ( − 0) ,
(69)
and it reduces to
W = 3∑ =1
− 0 0Ω0 = 3∑ =1
( − 0 ) − 0 0 ( − 0) , (70)
where w → (u − (/0)) and Ω0 → ( − 0) are dyonic fluid-velocity and
dyonic enthalpy in cold plasma, respectively. Here, the scalar
component (Ω0) represents the amount of dyonic internal energy
required to move one kilogram of the fluid element. Now, to
formulate the quaternionic dual MHD field equations for dyonic cold
plasma, it is necessary to define quaternionic space-time
differential operator as
D = (∇, − 0 ) 1 + 2 + 3 − 0 , (71)
and its quaternionic conjugate is
D = (−∇, − 0 ) −1 − 2 − 3 − 0 . (72)
The quaternionic product of D D will be
D D = 22 + 22 + 22 − 120 22 = ∇2 − 120 22= D D, (73)
where D D or D D is defined by the D’ Alembert operator. In order
to emphasize the variation of quaternionic space-time to two
four-velocities of dyonic fluid plasma, we may operate the
quaternionic differential operator (D) on generalized two
four-velocities (W) as
D W = 1 {( − − 120 − )+ 0 (− + − − )}
Advances in High Energy Physics 7
+ 2 {( − − 120 − )+ 0 (− + − − )}+ 3 {( − − 120 − )+ 0 (− + − −
z)}− 0 {( + + + 120 )− 0 ( + + + )} . (74)
Equation (74) governed the following quaternionic hydrody- namics
field equation for dyonic cold plasma, i.e.,
D W = Ψ 11 + 22 + 33 + 0, (75)
where → (1, 2, 3) and are the vector and scalar fields connected to
the hydrodynamics of dyonic cold plasma, respectively. Further, the
unified structure of quaternionic hydrodynamics field components
can be expressed as
1 = {(∇ × u) − 120 − }+ 0 {− (∇ × ) − − } , (76)
2 = {(∇ × u) − 120 − }+ 0 {− (∇ × ) − − } , (77)
3 = {(∇ × u) − 120 − }+ 0 {− (∇ × ) − − } , (78)
= −{(∇ ⋅ u + 120 ) − 0 (∇ ⋅ + )} . (79)
We may consider the generalized dual hydrodynamics fields, namely,
the hydroelectric and hydromagnetic fields of dyonic fluid
associated with the dynamics of electrons and magnetic
monopoles in dyonic cold plasma.Thus, the unified fields can be
rewrite as 1 ←→ ( + 0) , (80)
2 ←→ ( + 0) , (81)
2 ←→ ( + 0) , (82)
←→ −(L − 0 L) . (83)
The hydroelectric field vector (E) plays as the generalized Lamb
vector field and the hydromagnetic field vector (B) plays as the
generalized vorticity field [45–47] to the case of dual MHD. The
generalized Lamb vector field may be used to accelerate the dyonic
fluid flow while the vorticity field is its counterpart. Thus, the
generalized dual fields (E,B) for dyonic fluid become
E = −∇ × − u − ∇, (84)
B = ∇ × u − 120 − ∇, (85)
and the dual Lorenz gauge conditions (L, L) for the contin- uous
flow of incompressible dyonic fluid plasma are
L :→ ∇ ⋅ u + 120 = 0, (86)
L :→ ∇ ⋅ + = 0. (87)
The unified quaternionic Lamb-vorticity field vector Ψ (or
generalized hydroelectromagnetic field vector) for dyons can be
expressed as
Ψ = 1 ( + 0) + 2 ( + 0)+ 3 ( + 0) . (88)
Now, applying the quaternionic conjugate of differential operator D
to (88), we obtain
D Ψ = −1 [{(∇ × B) − 120 }+ 0 {(∇ × E) + }]− 2 [{(∇ × B) − 120 }+ 0
{(∇ × E) + }]
8 Advances in High Energy Physics
− 3 [{(∇ × B) − 120 }+ 0 {(∇ × E) + }] + 0 [∇ ⋅ B + 0∇ ⋅ E] .
(89)
Equation (89) shows the quaternionic space-time evaluation of
generalized Lamb-vorticity fields in the incompressible fluid of
dyonic cold plasma. The dynamics of dyonic cold plasma fluid can be
expressed by following equation:
D Ψ = −S (S, ℘) − (11 + 22 + 33 + 0℘) , (90)
whereS is the quaternionic source for the dyonic cold plasma.
Moreover, the quaternionic vector and scalar components of dyonic
sources, i.e., (S, ℘), can be written as
1 ←→ ( − 0 m ) , (91)
2 ←→ ( − 0 m ) , (92)
3 ←→ ( − 0 m ) , (93)
℘ ←→ (m − 0 ) , (94)
where (J, ) are the quaternionic electric source current and source
density associated with the dynamics of hydroelectric field while
(Jm, m) are corresponding magnetic sources associated with the
dynamics of hydromagnetic field of dyonic fluid. Therefore, the
quaternionic unified hydroelec- tromagnetic source for dyonic cold
plasma can be expressed by
S = (1 + 2 + 3 − 0m)− 0 (1 m + 2 m + 3 m + 0 )= (J − 00 ) − 0 ( Jm
− 00m) = (J − 0K) .
(95)
Here, J(, 0) → (J − (/0)(/)),K(, 0) →((1/)Jm−0m) are quaternionic
two four-fluid sources of dyons and (, ) considering the
permittivity and permeabil- ity satisfy 0 = 1/√. Now, equate
quaternionic imaginary and real coefficients in (90) and
obtain
∇ ⋅ E = , (Imaginary part of 0) (96)
∇ ⋅ B = m, (Real part of 0) (97)
(∇ × E) = − − m , (Imaginary part of 1) (98)
(∇ × E) = − − m , (Imaginary part of 2) (99)
(∇ × E) = − − m , (Imaginary part of 3) (100)
(∇ × B) = 120 + , (Real part of 1) (101)
(∇ × B) = 120 + , (Real part of 2) (102)
(∇ × B) = 120 + , (Real part of 3) . (103)
The above eight equations represent the quaternionic field
equations for hydrodynamics of dyonic cold plasma. These obtained
equations are primary equations for dual MHD of dyonic cold plasma,
which are exactly the same as the generalized Dirac-Maxwell
equations given by (49)-(52). As such, we also may write the
unified dual MHDfield equations for dyonic cold plasma as
∇ ⋅Ψ = ℘, (104)
∇ ×Ψ = − 0 Ψ + S. (105)
The present quaternionic formulation describes the macro- scopic
cold plasmabehavior.The solution of differential equa- tions
(104)-(105) provides the evolution of generalized lamb vector field
and generalized vorticity field to the presence of dyonic cold
plasma. Now, we may check the validity of dual MHD field equations
for dyonic cold plasma in given subsections.
5.1. Duality Invariant. Let us check the duality invariant symmetry
for generalized hydroelectric and hydromagnetic fields of dyonic
cold plasma. The duality transformation defines the rotation of
hydroelectric and hydromagnetic field components in the
quaternionic space such that the physics behind the quantity
remains the same after the transformation is performed. Suppose,
and F are the field and dual field tensor, then the duality
transformation becomes [48] :→ cos +F
sin , F :→ −F
sin + cos ,(0 ≤ ≤ 2 ) . (106)
Correspondingly, the quaternionic hydroelectric and hydro- magnetic
fields can also be transformed as(E
B ) → D2×2 (EB) , (107)
Advances in High Energy Physics 9
whereD2×2 = ( cos 0 sin −(1/0) sin cos ) is an unitary matrix
called
the duality transformation matrix (or simply D-matrix). For general
case = /2, the generalized dual fields will be transformed as
(E B ) = ( 0 0− 10 0)(E
B ) :⇒ {{{{{
E → 0B, B → − 10 E. (108)
Here, the D-matrix D2×2 = ( 0 0−1/0 0 ). For quaternionic
dual-velocity and dual-enthalpy of dyons fluid, the following
duality transformation relations governed the streamline flow,
i.e.,
(u ) = ( 0 0− 10 0)(u
) :⇒ {{{{{
u → 0, → − 10u, (109)
( ) = ( 0 0− 10 0)( ) :⇒ {{{{{ → 0, → − 10 . (110)
Accordingly, the dual-current and dual-density of dyonic plasma
will be transformed as
( J
Jm ) :
(111)
→ 0m,m → − 10 . (112)
Interestingly, from relations (108) to (112), we can conclude that
the generalizedDirac-Maxwell equations for dyonic fluid of cold
plasma are invariant under the duality transforma- tions and
showing the highly symmetric nature in presence of dyonic
fluid.
5.2. Lorentz Invariant. Let us start with the most usual trans-
formation [49, 50] that preserves the quaternionic intervals2 = 2 +
2 + 2 − 202; i.e., = Λ, (113)
where is any four-vector and the Lorentz transformation matrix
element Λ is
Λ →( cosh 0 0 − sinh 0 1 0 00 0 1 0 sinh 0 0 cosh ). (114)
Here is the boost parameter. Using the above Lorentz transformation
matrix, we may obtain the following trans- formation equations for
quaternionic four-velocity (W) of dyonic cold plasma which are an
analogous to quaternionic potentials of dyons; i.e., = ( − 0Ω0) , =
, = ,Ω0 = (Ω0 − 0) ,
(115)
where
cosh = 1√1 − tanh2 = 1√1 − 20 = , sinh = 0. (116)
If we consider the massive dyonic particles [51], then the
transformation relations (115) lead to the energy-momentum
transformations for dyonic cold plasma,
P = (P − 0E) ,
P = P,
P = P,
E = (E − 0P) ,
(117)
where the quaternionic four-momentum is defined by P(1, 2, 3, 0) =
(P,P,P,E). It should be notice that the obtained relations (117)
are similar to the usual rela- tivistic Lorentz energy-momentum
transformation relations [49, 50], where we assume that the speed
of dyons (0) is comparable to the speed of light ( ∼ 1). As such,
we also may establish the following transformation relations for
quaternionic source current and source density, i.e., = ( − 0℘) , =
, = ,℘ = (℘ − 0) .
(118)
10 Advances in High Energy Physics
Correspondingly, we obtain the Lorentz transformation rela- tions
for unified hydroelectromagnetic field of dyonic cold plasma, so
that = , = ( − 0) , = ( + 0) , (119)
along with = ( + ) , = ( + 20 ) . (120)
The beauty of the transformation relations (118)-(120) is that the
generalized Dirac-Maxwell equations for dyonic fluid of cold plasma
are well invariant under these Lorentz transformation.
5.3. CPT Invariant. In order to check the CPT invariance [52] for
the dual MHD field equations of dyonic cold plasma, we may write
the charge conjugation matrix (C) to the case of quaternionic
dual-current sources and hydroelectromagnetic fields of dyonic
fluid as C → ( −1 00 −1 ), where the charge conjugation
transformation plays as
C : ( J
Jm ) , (121)
C : (E B ) → (−1 00 −1)(EB) . (122)
Correspondingly, the parity matrix → ( −1 00 1 ) can govern the
following transformations for the dyonic fluid: : ( J
Jm ) → (−1 00 1)( J
Jm ) , (123)
: (E B ) → (−1 00 1)(EB) . (124)
As such, we can write the time reversal matrix, i.e., →( 1 00 −1 ),
and the transformation performs as : (Jm J ) → (1 00 −1)(JmJ ) ,
(125)
: (E B ) → (1 00 −1)(EB) . (126)
The forth component of quaternionic sources can also be transformed
for charge conjugation, parity, and time reversal as the following
ways:
C : ( m) → (−1 00 1)(m) , (127)
: ( m) → (1 00 −1)(m) , (128)
: ( m) → (1 00 1)( m) . (129)
We can summarize the quaternionic physical quantities of
dualMHDfields and their changes under charge conjugation, parity
inversion, and time reversal given by Table 2 [53, 54].
Now, we may apply the CPT transformation relations on generalized
Dirac-Maxwell equations for dyonic fluid of cold plasma as
[54]
C (∇ ⋅ E) −1−1C−1 = C( ) −1−1C−1, C (∇ ⋅ B) −1−1C−1 = C (m)
−1−1C−1, C (∇ × B) −1−1C−1= C( 120 E + J)−1−1C−1+C (J) −1−1C−1, C
(∇ × E) −1−1C−1= C(−B )−1−1C−1+C(−1 Jm)−1−1C−1.
(130)
Therefore, it may conclude that the generalized Dirac- Maxwell
equations for dyonic cold plasma are invariant under CPT
transformations.
6. Quaternionic Hydroelectromagnetic Wave Propagation
To establish the dual hydrodynamics wave equations for dyonic cold
plasma, we can start with the following quater- nionic
relation:
D (D Ψ) = −D S, (131)
where the left hand part of (131) can be written as
D (D Ψ) = 1 {(22 − 120 22 )+ 0 (22 − 120 22 )}+ 2 {(22 − 120 22 )+
0 (22 − 120 22 )}
Advances in High Energy Physics 11
Table 2: Quaternionic physical quantities and their CPT
transformations.
Physical quantities Charge conjugation (C) Parity inversion () Time
reversal () − ∇ ∇ −∇ ∇0 0 −0 −0 J −J −J −J Jm −Jm Jm Jm
E −E −E E B −B B −B − m m −m m
+ 3 {(22 − 120 22 )+ 0 (22 − 120 22 )} . (132)
Accordingly, the right hand part of (131) can be expressed as
D S = 1 {( − − 120 m − m )− 0 (m − m + + )} + 2 {( − − 120 m − m )
− 0 (m − m + + )} + 3 {( − − 120 m − m ) − 0 (m − m + + )} − 0 {( +
+ + 120 )− 0 (m + m + m + m )} .
(133)
Now, equate the real and imaginary parts of quaternionic basis
vectors in (131) and obtain the following relations:
∇ ⋅ J + = 0, (134)
∇ ⋅ Jm + 120 m = 0, (135)
∇ 2B − 120 2B2 − (∇m) − 120 Jm + (∇ × J)= 0, (136)
∇2E − 120 2E2 − 1 (∇) − J − 1 (∇ × Jm) = 0. (137)
Equations (134) and (135) defined the well-known dual continuity
equations while (136) and (137) represented the generalized
hydromagnetic and hydroelectric wave equations for dyonic cold
plasma in presence of electrons and magnetic monopoles. The beauty
of (136) is that it is an analogous to Alfven wave propagation [55,
56] associated with magnetic monopoles, and the same way (137)
describes the counterpart of Alfven wave propagation associated
with the electrons. Thus, the unified hydroelectromagnetic wave
equations for dyonic fluid of cold plasma can also be expressed
as
∇2Ψ − 120 2Ψ2 − (∇℘) − 0 S + (∇ × S) = 0. (138)
Interestingly, the generalized wave equation (138) is invariant
under the duality, Lorentz, and CPT transformations.
7. Conclusion
The dyons are high energetic soliton particles existing in the cold
plasma. The cold plasma model is the simplest model where we assume
negligible plasma temperature, and the corresponding distribution
function shows the Dirac delta function centered at the macroscopic
flow of linearised velocity. Dyonic cold plasma model can be used
in the study of small amplitude electromagnetic waves propagating
in the conducting plasma. In this study, we have applied the
four-dimensional space-time algebra (quaternionic algebra) to
elaborate the dynamics of dyonic fluid in cold plasma field. In
Section 2, we have explained in detail the prop- erties of
quaternionic algebra. However, the quaternion is an important and
appropriate fundamental mathematical tool to understand the
four-dimension space-time world. In Sections 3 and 4, the
fundamental equations for MHD field and their cold plasma
approximation have been defined. The interesting part we have
mentioned here that the dual MHD equations for massive dyons
consisted of electrons and
12 Advances in High Energy Physics
magnetic monopoles. The generalized equations involving the mass
and charge densities are expressed in terms of one- fluid theory of
dyonic cold plasma. Accordingly, we have discussed the dual-current
densities given by (56). The mass conservation law, dual-charge
conservation law, Lorentz force equation, and Ohm’s law for dyonic
cold plasma have been defined. In Section 5, we have described the
quaternionic for- mulation for moving massive dyonic fluid of
incompressible cold plasma. The advantage of the quaternionic
formulation is that, it is better to explain two four-velocities,
hydroelectric (Lamb vector), and hydromagnetic (vorticity) fields
and the dual Lorenz gauge conditions for dyonic cold plasma. It has
been emphasized that the dual hydrodynamics field of dyons (i.e.,
hydroelectric and hydromagnetic fields) deal with both
electrohydrodynamic and magnetic-hydrodynamics. In present study,
the existence of magnetic monopoles has been visualized to MHD
field. It has been shown that the two current sources are also
associated with the quaternionic hydroelectric and hydromagnetic
fields of dyonic plasma fluid. We have established the eight
primary equations of dual MHD field in presence of dyonic fluid.
Interestingly, the unified macroscopic Dirac-Maxwell equations
(104) and (105) have been obtained in the case of dyonic dual MHD.
It has been noticed that like electrodynamics, the Dirac- Maxwell
fluid equations are mandatory to describe the dynamics of MHD
plasma. The beauty of cold plasma field equations is that these
equations are well invariant under the duality, Lorentz, and CPT
transformations. In Section 6, we have obtained the quaternionic
dual continuity equations for incompressible dyonic fluid. The
generalized hydroelectric and hydromagnetic wave equations have
been established for dyonic cold plasma in presence of electrons
and magnetic monopoles. It has been emphasized that the obtained
Alfven wave like equation is associated with magnetic monopoles,
while the counterpart of Alfven wave equation plays as
electric-plasma waves in presence of electrons.
Data Availability
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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