B.C.Chanyal

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

1 Introduction

The paper describes how highest norm division algebra may be applied to chiral electromagnetism in massive fields of dyons.A chiral medium is characterized by either a left-handedness or a right-handedness in its micro structure.As a result,in a chiral medium left and rightcircularly polarized fields propagate with different phase velocities:the field with the latter polarization traveling through a right-handed medium faster than the leftcircularly polarized field,and vice versa.In this paper,we de fine the important role of hyper-complex algebra in electromagnetic fields of chiral media.Replacing the complex numbers by octonions,the physicists Jordan,von Neumann and Wigner[1]investigated a new finite Hilbert space.According to the Cayley Dickson process over the if elds of real numbers,the octonion numbers are widely used in modern theoretical physics.Rather,there has been a resurgence in the formulation of natural laws so that there exist[2]four-division algebras,namely the algebras of real numbers(R),complex numbers(C),quaternions numbers(H),and octonions numbers(O).These four set of algebras(R,C,H,O)are alternative with totally anti-symmetric associators.[3]

During recent decades,the role of octonion algebra has been used in various branch of physics,viz electromagnetism,gravi-electromagnetism,relativistic quantum mechanics,high energy particle physics,etc.Octonions share with complex numbers and its analysis has been discussed by Baez.[4]In view of higher dimensional algebra,the sedenions,like quaternions or even octonions,form a 16-dimensional algebra over the real numbers,has been used for the uni fied theory of gravi-electromagnetism.[5]The role of octonion for generalized electromagnetic field equations for bi-isotropic media have been discussed in simple manner.[6]Several physical quantities to exert an in fluence on the spatial parameters of complex-octonion curved space has been proposed by Weng.[7]One of the most popular species of octonions is split-octonions,[8]expressed by 2×2 Zorn’s vector matrix realization containing both scalars and vectors using a modi fied version of matrix multiplication.In view of quantum formulation of octonion variables,Mironov and Mironov[9−10]have discussed a new approach of octonic representation to establish the first as well as second order equations of relativistic quantum mechanics.Latterly,Chanyalet al.[11−21]have studied the various classical and quantum fields equations in terms of hyper-complex algebra.

Since,the electromagnetic characterization of materials is a fundamental problem in many research areas of electromagnetism.During recent decades,in electromagnetism,a great variety of novel and complex materials have been designed with promising practical applications.However,it has been found that in many cases,the relations between electric and magnetic fields cannot be described by standard constitutive equations.Thus,bi-isotropic media are the most general linear,homogeneous and isotropic materials,and they respond to electromagnetic excitation.In general,the optically active media mostly organic materials are naturally chiral at optical frequencies,where the chirality has an important role in mirror-asymmetry of the constituent micro-structure.Keeping in mind the important fact of electromagnetic if elds in chiral media(CM),in present paper,we derive the electromagnetic chiral field equations of massive dyons in terms of split octonionic formulation.Starting with octonion number,multiplication rules,and its 2×2 Zorn’s vector matrix representation,we represent the electric and magnetic induction vectors and chiral fields components in consistent manner of 2×2 Zorn’s vector matrix realization.As such,we describe the chiral parameter and pairing constant in terms of split octonionic representation of the Drude–Born–Fedorov constitutive relations.We have expressed the split octonionic generalized chiral field vector,that is the uni fied field of electric and magnetic chiral fields.The beauty of octonion matrix representation is that,the every scalar and vector components have its own meaning in generalized chiral electromagnetism.Correspondingly,we obtained the alternative form of generalized Proca–Maxwell’s equations of massive dyons in presence of chiral media.Furthermore,the split octonion form of the continuity equations,wave equations,workenergy theorem(Poynting theorem)and Poynting vector for generalized electromagnetic chiral- fields of massive dyons have been established in simple,compact and consistent manner.In this formulation we also have shown that in the absence of chiral parameter the theory of massive dyons reduces to that for homogeneous or isotropic medium.

2 Field Equations of Dyons for Isotropic and Chiral Media

An isotropic medium,or some time called homogeneous medium,is one such that the permittivity(ϵ)and permeability(µ)of the given medium are uniform in all for the electromagnetic fields,the inductions vectors are expressed as

whereϵ0andµ0are respectively de fined the permittivity and permeability in the free space,whileϵrandµrare relative permittivity and permeability associated with electric and magnetic fields.The constitutive equations in mavectors,respectively.Further we de fined

wherecis the velocity of light in free space(vacuum)while the other velocity variablevshows the speed of electromagnetic wave in isotropic medium.Therefore we get the following four differential equations which referred as the generalized field equations(Generalized Dirac-Maxwell equations)with the existence of magnetic monopole in isotropic media,[23−24]

Here we may express the electric and magnetic fields of virtual particle dyons in isotropic medium in presence of two-four potentials

Rather,the chirality was first observed as optical activity,which is the rotation of the plane of polarization in certain linear isotropic media.The rotation of the plane of polarfore,in order to express the chiral medium the electric and magnetic fields are paired with each other and can be written as[25]

whereβis chiral parameter andϵ′,µ′are pairing constants,and these relation are known as Drude–Born–Fedorov constitutive relations.All three are macroscopic quantities,andβresults directly from any chirality in the nanostructure of the medium.Ifβ→0,then the chirality field equations(9)reduce to the well known constitutive relations given by(1).It should be noted that for the case of local and global conservation of energy the chirality parameters become equal.[26]In other words,the generalized chiral constitutive relations

show that energy is not conserved unlessβd=βb.Furthermore,the generalized chiral constitutive relations(9)also can be expressed in terms of frequency dependent,i.e.,

The above relation(11)has also been employed for the study of chiral optical media.Moreover,the chiral constitutive relations and Maxwell equations are invariant under the electric-magnetic symmetry of simply called the duality transformation,[27]

whereϑis any pesudoscalar variable.However,ifχeandχmare de fined as electric and magnetic susceptibility then the chiral constitutive relations governed by

3 Octonion and Zorn Vector Matrix Realization

The octonions,[4]eight-dimensional space-times algebra,are the most convenient algebra in hypercomplex numbers,which exhibits normed alternative division algebra unlike vectors.In mathematical form,the octonion parameterξ∈Ois the form of

where the basise0andejare the octonions unit element,and satisfy the following multiplication rules

The structure constants,de fined byfjklare completely antisymmetric and take the value 1 for the following seven permutation of(j kl),i.e.

On the other hand,the split-octonions are the another specie of octonions which shows the non associative extension of split quaternions.Chanyalet al.[12]have studied the generalized electrodynamics of dyons in terms of splitoctonions algebra.Thus,in case of split-octonions algebra the basis elements{u0,u∗0,uj,u∗j}are expressed in terms of octonions basis as:

We may now summarize the multiplication rule of split octonionic basis elements in Table 1.

Table 1 Split-octonion multiplication Table.

In order to de fine the split-octonion representation of matrix visualization,we may introduce a convenient realization for the basis elements in term of well known Pauli’s spin matrices as where the quaternionic basis{e0ej(j=1,2,3)}can be expressed as

We know that the split-octonions are nonassociative in nature and they cannot be represented by ordinary matrices.Therefore we can use the Zorn’s vector-matrix algebra because it shows the non-associative property in nature.[28−29]Thus the Zorn’s vector-matrix,Z∈O,can be written by

where(r,s)and(→p,→q)are the scalar and vector coeffi-cients,respectively.The determinant of Eq.(19)can be expressed as

Further,the determinant shows a quadratic form on the Zorn’s vector-matrix and satis fies the following rule

IfZ1andZ2are two Zorn’s vector matrices then their product will be,

where(·)and(×)are dot and cross product of three-vectors.Since the Zorn’s vector-matrix algebra is isomorphic to the split-octonions algebra,then we can express the 2×2 Zorn’s vector matrix of any octonion functionO,

Thus an arbitrary octonion-matrix ʊ∈Ocan be expressed in terms of following 2×2 Zorn’s vector matrix,

whereµ,νare scalars while→xand→yare three-vectors of any octonionic variable.However,the octonion-conjugate of Eq.(24)is de fined by

Furthermore,the octonion-matrix algebra shows the following properties

where P,Q,and S are three octonion-matrices written in terms of Zorn vector-matrix realizations.Besides,the octonions have some different species as like real octonion,complex number coefficient octonion,split octonion,hyperbolic octonion etc.Recently,Chanyalet al.[30−32]proposed the dual octonion algebra and expressed its role in electrodynamics for the massive field of dyons.

4 Dual Euclidean Split-Octonion Space-Time

In view of the interpretation of Eq.(30),Chanyalet al.[14]discussed theU(1)abelian gauge structure forµ=0 and SU(2)non-Abelian gauge structure for the conditionµ=j(∀j=1,2,3).As such for the bi linear termηµνˆ14×4,the dual space time metric

describes the inner product.Furthermore,the split-octonion conjugation is accordingly expressed by following form

The Hermitian conjugation is described[8]by

The multiplication of two Zorn matrices having dual metric space-time,where

Moreover,the split-octonionAµ(x)with a space-time index,[33]may be written in terms of the split generators,i.e.

where we can say that the split-octonion coefficients ofAµ(x)transform like vectors under space-time transformations.The space-time covariant derivative ofAµ(x)can be written as the following form:

where Ωµρα→Ωµρα·12×2is the affinity of the nonsymmetric theory[33]and 12×2≃(u∗0+u0)is the unit element of the split octonion algebra.Similarly the space-time curvature is given byRσµνρ·12×2,whereRσµνρis the curvature of nonsymmetric theory.If octonion affinity become zero,the space-time covariant derivative ofAµ(x)shows ordinary derivative.

5 Split Octonion Electromagnetism in Chiral Media

Keeping in mind the advantage of split octonion and its isomorphism with Zorn vector matrix algebra,in the case of chiral electrodynamics of dyons,any generalized vector field(X)may be written in terms of following split octonion form,[14]

It should be noted that forUe1×Um1field theory of chiral electrodynamics,we have taken the degenerate fields component in both Euclidean spaces.Now we may introduce the electric and magnetic induction vectors(→D,→B)in following vector matrix form of split octonion,

Thus the electric and magnetic chiral induction vectors,respectively associated with two separate vector fields along with chiral coefficientβ,[16]can be expressed as the following split octonion form:

The generalized electromagnetic chiral(EMC) field,denoted by ΨCEM,associated with the uni fied field of both electric and magnetic induction vectors.Now,the generalized EMC- field can be written in terms of 2×2 Zorn vector matrix of split octonionic,

are the vector components of uni fied EMC- fields.In order to show the important role of octonion-matrix representation in chiral fields,let us start with the Euclidean or Minkowski spaces in terms of 2×2 Zorn’s vector matrix realizations of split octonion.The octonion differential operator D with dual degenerate Euclidean spaces may then be written by the following way

Here we have used the operator□ as the D’Alembertian operator,and also take natural unit(c=ħ=1)through out the text.Now,applying the octonion differential operator D to the generalized EMC- field ΨCEMgiven by the Eq.(42),then we obtain the following compact form of 2×2 Zorn vector matrix realization:

Now,we introduce the current source of dyons in generalized EMC- field,denoted byℑ.According to the de finition of massive chiral field for dyons,the current source equation become,[17]

where the scalar and vector matrix components are

The above equation(58)describes the octonionic generalized Dirac-Maxwell equations of massive dyons in presence of Chiral medium.Now,applying the following relations for generalized EMC- field,

which are an alternative,well known form of generalized Proca-Maxwell’s(GPM)equations in presence of massive electromagnetic field of dyons in chiral media.By substituting the above GPM equations,the generalized chiral constitutive relation for massive dyons becomes

Here we should be noted that the chirality factor not only connected with induction field vectors but also connected with current sources as well as the vector potentials of massive dyons.

6 Continuity Equation,Poynting Theorem and Wave Propagation in Chiral Media

In this section,we shall describe some useful conserved quantity for dyons in chiral electromagnetism.Let us start with the chiral field equation(52),and operate D both side from the left,i.e.

By equating the octonionic scalar and vector components,we may write the two sets of differential equations from the 2×2 Zorn vector matrix realization of split octonion,

these equations are governed as the generalized EMC- field equations,which may further reduce to

where the scalar components of Zorn vector matrix realization given by Eq.(69)show the continuity equations(in presence of electric and magnetic charge),and the vector components(70)represent the generalized chiral wave equations for electric and magnetic fields of massive dyons.Moreover,we also may express the wave equations of dyons in free space governed the following generalized EMC- field,i.e.,

along with the Lorentz gauge conditions,respectively for

The advantage of having applied the Lorentz condition is that Eq.(72)for the potentials are uncoupled and each one depends on only one type of source.Furthermore,from the Zorn vector matrix realization of split octonion,we may also express the“work-energy theorem”or“Poynting Theorem”to the case of generalized EMC- fields of dyons.We know that the Poynting theorem which is the analogous to the work-energy theorem of classical mechanics reproducing the continuity equation,so that it relates the energy stored in generalized electromagnetic chiral field to the work done on a charge distribution,through energy flux.Thus,we can relate the conservation of energy for generalized EMC- fields of massive dyons by the relation:the dynamics of electric and magnetic charges of massive dyons as

For massless particles,the above equation leads to

After that,the energy density(energy per unit time,per unit area)transported by the EMC- fields can be proposed by

7 Discussion and Conclusion

We know that the optically active media(mostly organic materials)are naturally chiral at optical frequencies,interesting examples include the famous Watson-Crick double-helix representation of the DNA molecule.Optical activity is usually very weak in the chiral material found in nature,but it can be enhanced in an artificial chiral material,called chiral metamaterial.Thus,in a chiral metamaterial,aligned electric and magneticdipole pairs are induced.Moreover,the strong coupling between electric and magnetic dipoles results in large optical activity so that the chiral metamaterial can be used for circular polarizer design.[37]But the important principle behind chirality is that the mirror-asymmetry of the constituent micro-structure.Handedness is manifest in the micro-structure of homogeneous chiral materials.For example,an isotropic chiral material comprises a random dispersion of handed molecules or inclusions.Therefore,the chirality shows the property of asymmetry important in several branches of physics,e.g.chirality may be found in the spin of the particle,where the handness of the object is determined by the direction in which the particle spins;and in electromagnetic wave propagation,handedness is wave polarization and described in terms of helicity.In experimental point of view,the electromagnetic characterization of chiral media has already been described by Marginedaet al.[38]In the dimension of nanoscale,the chirality and the chiral electromagnetic fields generated by arrays of nano-slits have also been observed.[39−40]Furthermore,the experimental prediction of the chiral waves in graphene medium[41]has been a novel challenged in material science.On the other hand,in study of dyons,the magnetic monopoles are hypothetical particles with a single magnetic charge,either a north pole or a south pole.Some speculative theories suggest that,if they do exist,magnetic monopoles could cause protons to decay.These theories also say that such monopoles would be too heavy to be produced at the Large Hadron Collider(LHC).Nevertheless,if the magnetic monopoles were light enough to appear at the LHC,cosmic rays striking the Earth’s atmosphere would already be making them,and the Earth would very effectively stop and trap them.Recently,the experimental study of magnetic monopole[42](which created dyons),and the interaction of this approximate magnetic monopole field with a beam of electrons has been demonstrated.[43]Besides,the role of complex-octonion space in various branch of physics and its corresponding experimental proposal for laboratory has been discussed by Weng.[44−46]In split octonion space,the in fluence of two chiral constituents of EMC- fields of dyons given by(43)is comparatively tough to be validated within the laboratory experiments at present.In other words,we can say that the higher dimensional uni fied EMC- fields of dyons is very complicated aspect to visualize in the laboratory.But,it may be possible to validate separately the in fluence of the chiral electric and magnetic constitutive fields vector of metamaterial in the laboratory.Theoretically,we can visualize the split-octonionic mathematical modeling of chiral media of dyons by the modi fication of the constitutive relations for normal dielectrics.The physical quantities of massive dyons and their corresponding split-octonionic de finitions are summarized by Table 2.

Table 2 Generalized split octonionic chiral fields of dyons.

From the forgoing analysis of split-octonionic formulation,we have de fined split-octonion algebra that gives an isomorphic representation of 2×2 Zorn’s vector matrix.Since the split-octonionic matrix multiplication is nonassociative,thus any four-vector has written by bi-valued representation of octonionic Zorn’s vector matrices that included scalar component along principle diagonal and vector component as off-diagonal elements.The electric and magnetic induction vectors associated with octonion basis vector and chiral parameterβin terms of split octonionic form of 2×2 Zorn’s matrix.It has been discussed that the split octonionic representation of electromagnetic chiral fields of dyons,which shows isotropic birefringent substance that respond to either electric or magnetic excitation with both electric and magnetic polarizations.The generalized form of octonionic electromagnetic chiral field of dyons(uni fied EMC- field of electric and magnetic field)has been obtained manifestly.Furthermore,we have established the generalized Dirac-Maxwell equations and generalized Proca-Maxwell equations of massive dyons in presence of chiral media by means of split-octonionic form of Zorn’s vector matrix representation.We have derived the continuity equation given by Eq.(69),which identify the law of the conservation of dyonic charge.Correspondingly,we have obtained the chiral wave equations given by Eq.(70)for generalized EMC- field of massive dyons.From this 2×2 Zorn’s vector matrix realization,we also have obtained the work-energy theorem by Eq.(74)and Poynting vector given by Eq.(76)for electromagnetic field of dyons in chiral media.The advantage of reformulating the theory of massive dyons in Zorn’s vector matrix realization is that,in this form one can extend the theory of massive dyons into the massive black-hole,massive dark energy and dark matter,and uni fied theory of fundamental forces.

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