Generalized Octonion Electrodynamics

TL;DR

This paper reformulates the equations governing dyons using octonion algebra, addressing non-associativity issues to develop a covariant and consistent generalized electrodynamics framework.

Contribution

It introduces a novel octonion-based formulation of dyon electrodynamics, highlighting necessary constraints due to octonion non-associativity for consistency.

Findings

Octonion formulation of dyon equations developed

Constraints imposed to handle non-associativity

Achieves covariant and consistent generalized electrodynamics

Abstract

We have made an attempt to reformulate the generalized field equation of dyons in terms of octonion variables. Octonion forms of generalized potential and current equations are discussed in consistent manner. It has been shown that due to the non associativity of octonion variables it is necessary to impose certain constraints to describe generalized octonion electrodynamics in manifestly covariant and consistent manner.

Equations

$$x=(x_0,x_1,....,x_7)$$

$$=$$

$$e_0=1,$$

$$\left[e_A,e_B\right]$$

$$\bar{x}$$

$$=$$

$$Sc(x)$$

$$(\overline{xy})$$

$$⟨x,y⟩$$

$$⟨e_A,e_B⟩$$

$$N(x)=\overline{x}x$$

$$N(xy)$$

$$x^{-1}$$

$$x^{-1}x$$

$$D$$

$$\mathcal{F}(X)$$

$$D\mathcal{F}(X)$$

$$G(X)D$$

$$D\mathcal{F}$$

$$I_0$$

$$I_1$$

$$I_2$$

$$I_3$$

$$I_4$$

$$I_5$$

$$I_6$$

$$I_7$$

$$\mathbb{V}$$

$$(V_0,V_1,V_2,V_3,V_4,V_5,V_6,V_7)⟹(\phi ,A_x,A_y,A_z,i{B}_x,i{B}_y,i{B}_z,i\varphi )(i=\sqrt{-1})$$

$$\mathbb{V}=e_1(A_x+i{e}_7{B}_x)+e_2(A_y+i{e}_7{B}_y)+e_3(A_z+i{e}_7{B}_z)+(\phi +i{e}_7\varphi )=e_1{\mathrm{V}}_{\mathrm{x}}+e_2{\mathrm{V}}_{\mathrm{y}}+e_3{\mathrm{V}}_{\mathrm{z}}+i{e}_7\varnothing$$

$$D⟼⊡$$

$$\overline{⊡}$$

$$\overline{⊡}\mathbb{V}=$$

$$+$$

Full text

Department of Physics

Kumaun University

S. S. J. Campus

Almora -263601 (U.K.) India

Email:- bcchanyal@gmail.com

ps_bisht123@rediffmail.com

ops_negi@yahoo.co.in

1 Introduction

There has been a revival in the formulation of natural laws so that there exists [1] four-division algebras consisting the algebra of real numbers ($\mathbb{R}$), complex numbers ($\mathbb{C}$), quaternions ($\mathbb{H}$) and Octonions ($\mathcal{O}$). All four algebra’s are alternative with totally anti symmetric associators. Quaternions [2, 3] were very first example of hyper complex numbers have been widely used [4, 5, 6, 7, 8, 9, 10] to the various applications of mathematics and physics. Since octonions [11] share with complex numbers and quaternions, many attractive mathematical properties, one might except that they would be equally as useful as others. Octonion [11] analysis has been widely discussed by Baez [12]. It has also played an important role in the context of various physical problems [13, 14, 15, 16] of higher dimensional supersymmetry, super gravity and super strings etc. In recent years, it has also drawn interests of many [17, 18, 19, 20] towards the developments of wave equation and octonion form of Maxwell’s equations. We [21, 22] have also studied octonion electrodynamics, dyonic field equation and octonion gauge analyticity of dyons consistently and obtained the corresponding field equations (Maxwell’s equations) and equation of motion in compact and simpler formulation. Extending our previous results [22], in this paper, we have made an attempt to reformulate the generalized field equation of dyons in terms of octonion variables. Octonion forms of generalized potential and current equations are discussed in consistent manner. It has been shown that due to the non associativity of octonion variables it is necessary to impose certain constraints to describe generalized octonion electrodynamics in manifestly covariant and consistent manner. We have obtained the generalized Dirac-Maxwell’s (GDM) equations of dyons from octonion wave equation in simple, compact and consistent manner. It has been shown that the octonion variable of dyons reproduces the dynamics of electric (magnetic) charge in the absence of magnetic (electric) charge.

2 Octonion Definition

An octonion $x$ is expressed [21, 22] as a set of eight real numbers

[TABLE]

where $e_{A}(A=1,2,.....,7)$ are imaginary octonion units and $e_{0}$ is the multiplicative unit element. The octet $(e_{0},e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7})$ is known as the octonion basis and its elements satisfy the following multiplication rules

[TABLE]

The structure constants $f_{ABC}$ are completely antisymmetric and take the value $1$ i.e. $f_{ABC}=+1=(123),\,(471),\,(257),\,(165),\,(624),\,(543),\,(736)$. Here the octonion algebra $\mathcal{O}$ is described over the algebra of rational numbers having the vector space of dimension $8$. Octonion algebra is non associative and multiplication rules for its basis elements given by equations (2,3) are then generalized in the following table:

[TABLE]

**              Table1-**

Octonion Multiplication table

Hence, we get the following relations among octonion basis elements i.e.

[TABLE]

where brackets $[\,\,]$ and $\{\,\,\}$ are used respectively for commutation and the anti commutation relations while $\delta_{AB}$ is the usual Kronecker delta-Dirac symbol.Octonion conjugate is thus defined as,

[TABLE]

An Octonion can be decomposed in terms of its scalar $(Sc(x))$ and vector $(Vec(x))$ parts as

[TABLE]

Conjugates of product of two octonions and its own are described as

[TABLE]

while the scalar product of two octonions is defined as

[TABLE]

which can be written in terms of octonion units as

[TABLE]

The norm of the octonion $N(x)$ is defined as

[TABLE]

which is zero if $x=0$, and is always positive otherwise. It also satisfies the following property of normed algebra

[TABLE]

As such, for a nonzero octonion $x$ , we define its inverse as

[TABLE]

which shows that

[TABLE]

3 Octonion Wave Equation

A lot of literature [17, 18, 19, 20] has already been available on octonion wave equation. Accordingly, let us define the octonion differential operator $D$ as

[TABLE]

where $D_{\mu}$ are described as the components of a differential operator in an eight dimensional representation. We describe a function of an octonion variable as

[TABLE]

where $f_{\mu}$ are scalar functions. Since octonions are neither commutative nor associative, one has to be very careful to multiply the octonion either from left or from right in terms of regularity conditions [17]. As such, a function $\mathcal{F}(X)$ of an octonion variable $X={\displaystyle\sum_{\mu=0}^{7}e_{\mu}X_{\mu}}$ is left regular at $X$ if and only if $\mathcal{F}(X)$ satisfies the condition

[TABLE]

Similarly, a function $G(X)$ is a right regular if and only if

[TABLE]

where $G(X)=g_{0}+g_{1}e_{1}+g_{2}e_{2}+.....+g_{7}e_{7}.$ Then we get

[TABLE]

where

[TABLE]

The regularity condition (15) may now be considered as a homogeneous octonion wave equation for octonion variables without sources. On the other hand, equation (17) is considered as the inhomogeneous wave equation $D\mathcal{F}=I$.

4 Octonion Formulation for Generalized Fields of Dyons

In order to write the various quantum equations of dyons in octonion formulation, let us start with potential octonion

[TABLE]

and we identify its components as

[TABLE]

where $(\phi,\,A_{x},\,A_{y},\,A_{z})=(\phi,\overrightarrow{A}\,)=\left\{A_{\mu}\right\}$ and $(\varphi,\,B_{x},\,B_{y},\,B_{z})=(\varphi,\overrightarrow{B}\,)=\left\{B_{\mu}\right\}$ are respectively described as the components of electric $\left\{A_{\mu}\right\}$ and magnetic $\left\{B_{\mu}\right\}$ four potentials of dyons (particles carrying simultaneously the electric and magnetic charges). Equation (19) may then be written as

[TABLE]

where $(\emptyset,\mathrm{V_{x},}\mathrm{V_{y},\mathrm{V_{z}}})=(\emptyset,\overrightarrow{\mathrm{V}}\,)=\left\{\mathrm{V_{\mu}}\right\}$ are then be described as the components of generalized four potential $\left\{\mathrm{V_{\mu}}\right\}$ associated with generalized charge $(q=e+i\,g$) (where $e$ and $g$ are respectively known as electric and magnetic charges) of dyons [6]. In order to obtain the generalized field equations of dyons in four dimensional space time, we identify differential operator (13) to be four dimensional and hence we may write equation (13) as

[TABLE]

where we have taken other components like $\partial_{0},\partial_{4},\partial_{5},\partial_{6}$of equation (13) vanishing. Octonion conjugate of equation (22) may then be written as

[TABLE]

Now operating $\overline{\boxdot}$ given by equation (23) to octonion potential $\mathtt{\mathrm{\mathcal{\mathsf{\mathbf{\mathcal{\mathfrak{\mathbb{V}}}}}}}}$of equation (21), we get

[TABLE]

We are using S.I. system of natural units $(c=\hbar=1)$. On applying the Lorentz gauge conditions, respectively for the dynamics of electric and magnetic charges of dyons as

[TABLE]

we find the following octonion form of equation (24) i.e.

[TABLE]

where $\mathbb{F}$ is again an octonion reproduces the generalized electromagnetic fields of dyons. It is thus described by

[TABLE]

where

[TABLE]

Let us define [6, 21] the generalized electric $(\overrightarrow{E})$ and magnetic $(\overrightarrow{H})$ fields of dyons in terms of components of electric and magnetic four potentials as

[TABLE]

Thus equation (27) reduces to

[TABLE]

where $\overrightarrow{\Psi}=\overrightarrow{H}+i\,e_{7}\overrightarrow{E}$ is the generalized vector field [6, 21] of dyons. Now applying the differential operator (22) to equation (30), we get

[TABLE]

or

[TABLE]

These equations (31,32) may then be written in following compact notation in terms of an octonion i.e.

[TABLE]

where $\mathbb{J}$ is again an octonion and is re-described as the Octonion form of generalized current given by

[TABLE]

where $\mathrm{(\rho,\,\overrightarrow{j})=\{\mathrm{j_{\mu}}\}}$, $(\varrho,\,\overrightarrow{j})=\{\mathrm{k_{\mu}}\}$ and $(\mathrm{J_{0},\overrightarrow{J})=\{\mathrm{J_{\mu}}\}}$ are respectively the four currents associated with electric charge, magnetic monopole and generalized fields of dyons. Equations (31-34) thus lead to following differential equations

[TABLE]

Equation (35) may then be written as

[TABLE]

which are the generalized Dirac-Maxwell’s equations of generalized fields of dyons [21, 22].

[TABLE]

Equation (37) then reduces to

[TABLE]

where $\mathbb{J}$ is described as the octonion form of generalized current associated with dyon and is already given by equation (33). Equation (37) may also be written as

[TABLE]

where $\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}$ and $\mathtt{\square}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}$-$\frac{\partial^{2}}{\partial t^{2}}=\nabla^{2}-\frac{\partial^{2}}{\partial t^{2}}$ . Using the Lorentz gauge conditions (25) and the definition of octonion valued generalized current of dyon given by equation (34), we get

[TABLE]

As such, we have obtained consistently the generalized Dirac Maxwell’s (GDM) equations from octonion wave equations on considering the non associativity of octonion variables. The advantages of present formalism are discussed in terms of compact and simpler notations of octonion valued potential, field and currents of dyons despite of non associativity of octonions. The present octonion reformulation of generalized fields of dyons represents well the invariance of field equations under Lorentz and duality transformations. It also reproduces the dynamics of electric (magnetic) charge yielding to the usual form of Maxwell’s equations in the absence of magnetic (electric charge) in compact, simpler and consistent way. Octonion element $e_{7}$ has been considered to be invariant and the theory resembles with the bi-quaternion theory of generalized fields of dyons [9]. In the forthcoming paper theory of split octonion will be discussed.