Advances in High Energy Physics

Volume 2018, Article ID 7843730, 13 pages

https://doi.org/10.1155/2018/7843730

## Quaternionic Approach to Dual Magnetohydrodynamics of Dyonic Cold Plasma

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

Correspondence should be addressed to B. C. Chanyal; moc.liamg@laynahccb

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 SCOAP^{3}.

#### Abstract

The dual magnetohydrodynamics of dyonic plasma describes the study of electrodynamics equations along with the transport equations in the presence of electrons and magnetic 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 universe was composed almost entirely of the baryonic matter (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 collective behavior of plasma. In 1942, Alfven [3] gave the theory of magnetohydrodynamics (MHD) and suggested that electrically 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 them as 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 fluctuation (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 monopole 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 maintained complete symmetry between electric and magnetic fields. Therefore, the name of particles carrying simultaneously the electric and magnetic charges is dyons. Further, the theoretical approach of Schwinger [9, 10] and 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 theoretical 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 quaternion 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 electrodynamics 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 quaternionic 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, magnetic monopoles, and their counter partners, namely, ions and magnetoions. In this study, we clarify that the dominating 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 equations 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 which may 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 is defined by the set of all real linear combinations of the unit elements , such thatwhere the real number is called the real part and is called the imaginary part of a complex number . If the real part , then we can say that is purely imaginary. As such, the Euclidean scalar product as is then defined bywhere and are two complex numbers. The modulus of any complex number is also defined by .

However, a complex field is a finite dimensional real vector space, so that we can easily extend the complex number into the quaternionic field 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 to mechanics 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 over the field of real numbers aswhere the Hamilton vector space () has the quaternionic elements (, , , and ), which are called quaternion basis elements while , , , and are the real quarterate of a quaternion. As such the addition of two quaternions and is given byHere, the quaternionic addition is clearly associative and commutative. The additive identity element is defined by the zero element; i.e.,and the additive inverse of is given byCorrespondingly, the product of two quaternions, i.e., , can be expressed byWe may notice that this quaternionic product is associative, but not commutative. The quaternionic unit elements followed the given relations,where is the delta symbol and is the Levi Civita three-index symbol having value for cyclic permutation, for anticyclic permutation, and for any two repeated indices. Further, we also may write the following relations to quaternion basis elementswhere 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.,where we take for noncommutative product of quaternion. The quaternionic product with the scalar quantity is given byAs such, the multiplication identity element can expressed by the unit elements,Moreover, a quaternion can also be decomposed in terms of scalar and vector parts aswhere the quaternionic conjugate is expressed byThe real and imaginary parts of can be written asIf and , then is said to be purely imaginary quaternions. Therefore, all quaternions with zero real are simplified as imaginary space of , where the imaginary space is a three-dimensional real vector space,Interestingly, we may write the following form of quaternion asThe quaternionic Euclidean scalar product can also be expressed asLike complex numbers, the modulus of quaternion is then defined asSince there exists the norm of a quaternion, we have a division; i.e., every has an inverse of a quaternion and is expressed asThe quaternion conjugation satisfies the following property:The norm of the quaternion is positive definite and obeys the composition lawThe 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.

#### 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 () is expressed bywhere is the velocity of particles, (, ) are the effective charge and mass of the species particles, and (, ) 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:In (25) and (26), we may consider that the two dominating particles (i.e., electrons and ions both) constitute the dynamics 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 becomewhere , 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 asand the current density becomesThe continuity equations can be written asAs such, the momentum equation for plasma fluid is expressed as [35]where is the pressure force introduced due to the inhomogeneity of the plasma and is a Lorentz force per unit volume element. Now, we introduce an acceleration to the conducting fluid,where the term is used for the convective acceleration of fluid. Furthermore, the generalized Ohm’s law becomes [35]where denotes the conductivity of fluid. One can define Maxwell’s equations with natural unit () asInterestingly, 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 () (equivalent to masses), and velocities (). 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.,We consider that the effected behavior of electrons are comparable 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 asAs such, the Navier-Stokes and Ohm’s equations becomewhere to the case if the current is small compared to . The ideal MHD equations () for cold plasma may then be expressed asTo consider wave behavior of cold particles, the cold plasma wave has temperature independent dispersion relation. If is Alfven velocity, then the dispersion relation for cold plasma waves become [35] 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.,In the above generalized Dirac-Maxwell’s equations, and are the electric and magnetic charge densities while and 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 aswhere , and 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 aswhereupon the dual-current densities (electric and magnetic) are defined byThe conservation laws for the dynamics of dyonic cold plasma can be written asThe generalized Navier-Stokes force equation can also be exhibited in presence of magnetic monopole; i.e.,where the duality invariant Lorentz force equation for dyons isand 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 asIn the same way, Ohm’s law for the dyonic cold plasma is expressed aswhere is the magnetic conductivity. Therefore, from (63) to (64), we can conclude that for infinite conductivity of dyons () the electric and magnetic field vectors constitute from the rotation of each other, i.e., , and . The above classical field equations given by (49) to (64) of dyons are referred to dual MHD field equations of dyonic cold plasma.

#### 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 () and two-enthalpy () of dyons for plasma fluid dynamics aswhere () are quaternionic variables associated with two four-velocities of electrons and magnetic monopoles of dyons and denoted the speed of particles (dyons) moving in conducting 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 () of dyons aswhere the vector components 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.