Hyperbolic Quaternion Formulation of Electromagnetism - Springer Link

Adv. appl. Clifford alg. 20 (2010), 547–563 c 2010 Springer Basel AG, Switzerland 0188-7009/030547-17, published online March 12, 2010 DOI 10.1007/s00006-010-0209-9

Advances in Applied Clifford Algebras

Hyperbolic Quaternion Formulation of Electromagnetism S¨ uleyman Demir, Murat Tanı¸slı and Nuray Candemir Abstract. In this paper, after presenting the hyperbolic quaternion formalism, an alternative representational method is proposed for the formulation of classical and generalized electromagnetism in the case of the existence of magnetic monopoles and massive photons. Maxwell’s equations and relevant field equations are derived in compact, simpler and elegant forms. These equations are compared with their vectorial, complex quaternionic, dual quaternionic and octonionic representations. Mathematics Subject Classification (2000). 00A79, 8A25, 17D99. Keywords. Hyperbolic quaternion, electromagnetism, magnetic monopole.

1. Introduction Recent papers in literature have been demonstrated that the hypercomplex number systems with nonreal square root +1 have a wide potential to investigate the physical theories in different areas. Generally, in these papers hypernumber concepts invented by Charles Musés (1919-2000) are used. Musés also showed in his works how hypernumbers interface with the physical word, specially the quantum field theory [1]-[3]. A special case of the hypernumber arithmetic is the 16- dimensional conic sedenions which were studied by Carmody [4, 5]. K¨ oplinger has used this special hypernumber system to investigate the theories in other areas of physics. For example, the hyperbolic octonions as a subalgebra of the conic sedenions have been used to express the Dirac equation in a simple form [6]. The potential applicability of this structure has been examined in the relativity [7], gravity and electromagnetism [8, 9], as well. Similarly, Candemir et al. [10] have expressed the differential operator, the Proca-Maxwell equations and relevant field equations in terms of the hyperbolic octonions. The Musés hyperbolic octonions are isomorphic to the split octonions and Gogberashvili [11] showed that the Dirac equation can be written by using the algebra of the split octonions. As distinct

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from similar formulations [12]-[16] Gogberashvili did not use any complex numbers or bi-spinors. In general, the hyperbolic numbers are also known as paracomplex, unipodal, duplex or split-complex numbers [17, 18]. A larger list of references regarding applications of this mathematical structure has been given in Ulrych’s papers on Clifford algebra [19]-[23]. The paravectors used in these works have been introduced originally by Sobczyk in the spacetime vector analysis [24]. Similar to the Hestenes space-time formulation [25], Baylis showed that the theory of electrodynamics can be fully expressed in terms of this algebra [26]. By inserting the hyperbolic unit j (j 2 = 1) into the Baylis formulations, Ulrych has generalized ¯ 3,0 and proposed to use this structure to obtain the complex Clifford algebra C this algebra to represent the Poincaré mass operator [19, 20] and the spinors [21]. Ulrych has also proved that the complex Clifford algebra with the hyperbolic numbers is rich enough in structure to investigate the relativistic quantum physics [22] and the gravitoelectromagnetism [23]. Similarly, by introducing hyperbolic virtual unit j into Clifford algebra, Xuegang et al. have discussed the hyperbolic spherical harmonics [27], the Dirac wave equation [28], the four dimensional Lorentz transformations [29], the hyperbolic Schr¨ odinger equation [30] and the relative transformation equations of velocity and acceleration in the four dimensional hyperbolic complex space [31]. As closely related to the context of this paper, Cafaro and Ali [32] have used the hyperbolic number concept in spacetime algebra to formulate the classical electrodynamics with massive photons and magnetic monopoles. Maxwell’s equations which are the cornerstones of classical electromagnetism have been formulated in many forms since their discovery in 1873. Although, in his famous book Treatise on Electricity and Magnetism [33], Maxwell used 3dimensional vector representation to formulate electromagnetism, he also gave their quaternionic forms in a number of places. Maxwell used the V Q and SQ symbols to refer to the vector part and the scalar part of quaternion Q, respectively [34]. The classical electromagnetic theory related to quaternions can be formulated very efficiently in different forms. The complex quaternions, also named biquaternions and composed of two real quaternions, were used frequently to reformulate the classical electrodynamics and so Maxwell’s equations were reduced to a simple and compact form [35]-[45]. Similarly, the quaternion analysis of the timedependent Maxwells equations in the presence of electric and magnetic charges and the solutions for the classical problem of moving sources were obtained by Singh et al. [46] in a unique, simpler and consistent manner. The similarities in Maxwell and Dirac equations have been investigated by Rodrigues and Capelas de Oliveira [47, 48]. In their approach, how to write the Dirac and the generalized Maxwell equations (including monopoles) have also been described in the Clifford and spin-Clifford bundles over space-time. Dual quaternions that are in the similar structure to complex quaternions are used generally to investigate spatial screw motion of a rigid body. In the paper ¨ by Demir and Ozda¸ s [49], the classical electromagnetism has been reformulated by using this type quaternions. Maxwell’s equations have been rewritten in terms

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of the dual quaternions and these four equations have been combined in a single equation. Recently, a new representational model based on dual quaternionic matrices has been proposed for classical electromagnetism [50], as well. As the largest of the four normed division algebras, the classical octonions share many attractive mathematical properties with the quaternions. In spite of their nonassociativity, this mathematical structure enables us to derive alternative formulations in electromagnetism and quantum mechanics. By introducing leftright barred operators, De Leo and Abdel-Khalek have overcome the problems due to the nonassociativity of the octonionic algebra and obtained a consistent formulation of octonionic quantum mechanics [51]–[53]. In the paper by Tolan et al. [54], the field equations and Maxwell’s equations for electromagnetism have been investigated with new octonionic equations, and these equations have been compared with their vectorial representations. As a distinction in structure from classical octonions, both the split octonions and the hyperbolic octonions have hyperbolic basis instead of the complex ones. The Dirac operator and Maxwell equations in vacuum have been derived in the algebra of split octonions by Gogberashvili [55]. Similarly, Bisht and Negi have reformulated electrodynamics and the dyonic field equations in terms of the split octonion and its Zorn’s vector matrix realization along with the corresponding field equations and the equation of motion in a unique and consistent manner [56]. In their work with Dangwal, the split octonion formalism has also been extented to cover the unified theory of linear gravity and electromagnetism with the simultaneous existence of electric, magnetic, gravitational and Heavisidian charges [57]. The hyperbolic quaternions are one of the non-associative hyperbolic number systems that are very suitable for the investigation of space-time theories. Unfortunately, this system is 4-dimensional. Therefore, by using the same idea on the construction of the complex quaternions, we combined two hyperbolic quaternion to express up to 8-dimensional physical quantities. Although the obtained mathematical structure is 8-dimensional, the number of basis elements are still four. Therefore, manipulating of this new hypernumber system is simpler than similar structures such as the octonions. Furthermore, this paper also aims to contribute the potential applications of non-associative algebras in the description of physical laws. The hyperbolic quaternion formulation of electromagnetism was absent in literature. Therefore, this paper fills a gap and contains useful results. Maxwell’s equations and relevant field equations are investigated with the hyperbolic quaternions, and these equations have been given in compact, simpler and elegant forms. The derived equations are compared with their vectorial, complex quaternionic, dual quaternionic and octonionic representations, as well. The layout of the paper is as follows. In section 2, a brief introduction to the hyperbolic quaternion algebra is given. In section 3, the classical electromagnetism theory is reformulated in the absence of magnetic monopoles and massive photons. Similarly, in section 4, a compact formulation of Maxwell’s equations with photon mass in the absence of magnetic monopoles is presented. In sections 5 and 6, the

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general theory is considered in two limiting cases; i) with magnetic monopoles in the absence of photon mass and ii) massive electromagnetism with magnetic monopoles. Finally, the hyperbolic quaternions are proposed as a non-associative algebric tool for obtaining the Proca-Maxwell equations with magnetic monopoles.

2. Preliminaries Quaternions were discovered by Hamilton in 1843 during his attempt to generalize complex numbers to three dimensions as the following form [58]: q = (q0 + q1 e1 ) + (q2 + q3 e1 )e2 = q0 + q1 e1 + q2 e2 + q3 (e1 e2 ). e21

(2.1)

e22

= = −1, similar to If the basis elements e1 and e2 are chosen as imaginary an ordinary complex unit i, we get Hamilton’s real quaternions. Although these quantities are most often called Hamilton’s quaternions, actually a few years before Hamilton’s discovery, in 1840, French mathematician Olinde Rodrigues developed his parameters of rotation which are equivalent to quaternions [59]-[61]. A real quaternion with four components can be written as q = q0 e0 + q1 e1 + q2 e2 + q3 e3

(2.2)

where q0 , q1 , q2 , q3 are real numbers. e0 and e1 , e2 , e3 are quaternion basis elements that obey the following noncommutative multiplication rules e20 = 1, e21 = e22 = e23 = −1, e1 e2 = −e2 e1 = e3 , e3 e1 = −e1 e3 = e2 , e2 e3 = −e3 e2 = e1 .

(2.3a) (2.3b)

Real quaternions have the positively defined norm as ¯ q = q02 + q12 + q22 + q32 q=q Nq = q¯

(2.4)

¯ = q0 − q1 e1 − q2 e2 − q3 e3 is conjugate of q. where q In the other case in the eq.(2.1), that is e21 = e22 = +1, we obtain the splitquaternions. The basis elements of the split quaternions do not satisfy the same relations for Hamilton’s real quaternions given by eq.(2.3): e20 = e21 = e22 = 1, e23 = −1, e1 e2 = −e2 e1 = e3 , e3 e1 = −e1 e3 = −e2 , e2 e3 = −e3 e2 = −e1 .

(2.5a) (2.5b)

The norm equation, then becomes ¯ q = q02 − q12 − q22 + q32 . q=q Nq = q¯

(2.6)

Both real quaternion and split quaternion products are associative but not commutative. On the other hand, if all of the quaternion basis elements are chosen as e20 = e21 = e22 = e23 = +1,

(2.7)

we obtain the hyperbolic quaternions which was first suggested by Alexander MacFarlane in 1891 [62]. The products of basis elements of the hyperbolic quaternions satisfy the same relations for real quaternions given in eq.(2.3b).


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However, unlike the real and the split quaternions, the hyperbolic quaternion product is not associative, (pq)r = p(qr). MacFarlane’s hyperbolic quaternions are different from Musés’ hyperbolic quaternions (1, ε1 , ε2 , i) which are isomorphic to split quaternions. Similarly, the hyperbolic quaternion term which was used by G¨ odel [63] refers also to split quaternions. The norm of the hyperbolic quaternion is Nq = qq∗ = q∗ q = q02 − q12 − q22 − q32 .

(2.8)

An eight dimensional quantity, we called hyperbolic biquaternion, can be obtained as an extension of the hyperbolic quaternions. The Hyperbolic biquaternions are formed by complex combinatination of two hyperbolic quaternions; Q = q + iq

= (q0 + iq0 )e0 + (q1 + iq1 )e1 + (q2 + iq2 )e2 + (q3 + iq3 )e3 = Q0 e0 + Q1 e1 + Q2 e2 + Q3 e3 . (2.9)

Here Q0 , Q1 , Q2 and Q3 are complex numbers. A hyperbolic biquaternion can be represented in terms of its scalar and vector parts as P = P0 + P

(2.10)

where scalar and vector parts, respectively, are PS = P0 = P0 e0 , PV = P = P1 e1 + P2 e2 + P3 e3 .

(2.11) (2.12)

The product of two hyperbolic biquaternions P and Q is PQ = (P0 + P )(Q0 + Q) = P0 Q0 + P0 Q+Q0 P + P · Q + P × Q

(2.13)

where the dot and cross indicate, respectively, the usual three-dimensional scalar and vector products. As in familiar case of the complex numbers, conjugation is accomplished by changing the sign of the components of the imaginary basis elements, ¯ = Q 0 − Q = Q 0 e0 − Q 1 e1 − Q 2 e2 − Q 3 e3 . Q (2.14) Similarly, the complex conjugate of Q is also defined as Q∗ = (q0 − iq0 )e0 + (q1 − iq1 )e1 + (q2 − iq2 )e2 + (q3 − iq3 )e3 .

(2.15)

The norm of a hyperbolic biquaternion in general is a complex number and given by ¯ = QQ ¯ = Q2 − Q2 − Q2 − Q2 . (2.16) NQ = QQ 0 1 2 3 The hyperbolic biquaternions with unit norm are called as unit hyperbolic biquaternions. After this brief summary about algebra of the hyperbolic quaternions, in next sections we will utilize this useful mathematical structure in the classical and generalized electromagnetism.

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3. Hyperbolic Quaternions and Classical Electromagnetism We would like to propose this alternative formulation of electromagnetism as an example of possible applications of the hyperbolic quaternions in physics. The hyperbolic biquaternions, which are complex combination of hyperbolic quaternions, allow us to express up to 8-dimensional physical quantities in a compact and simple form. The hyperbolic biquaternion representation of the differential operator can be defined as following ∂ ∂ ∂ ∂ ∂ e1 + e2 + e3 = i + ∇ (3.1) = i e0 + ∂t ∂x ∂y ∂z ∂t which is analogous to eq.(2.9), with q0 = 0, q1 = ∂/∂x, q2 = ∂/∂y, q3 = ∂/∂z and q0 = ∂/∂t, q1 = q2 = q3 = 0. Its conjugate and complex conjugate, respectively, are ¯ = i ∂ e0 − ∂ e1 − ∂ e2 − ∂ e3 = i ∂ − ∇, (3.2) ∂t ∂x ∂y ∂z ∂t ∂ ∂ ∂ ∂ ∂ ∗ = −i e0 + e1 + e2 + e3 = −i + ∇. (3.3) ∂t ∂x ∂y ∂z ∂t In order to reformulate the classical field equations of electromagnetism in terms of the hyperbolic biquaternions, we will assume that = μ = c = 1. Then, Maxwell’s equations can be expressed as the following vectorial notation, ∂H ∇×E =− , ∇.E = ρ, (3.4a) ∂t ∂E ∇×H = J + , ∇.H = 0, (3.4b) ∂t where (3.5) E = E1 i + E2 j + E3 k, H = H1 i + H2 j + H3 k represent, respectively, the electric and magnetic fields of an electrically charged particle, while ρ is the electric charge source density and J = J1 i + J2 j + J3 k is the electric current source density. The magnetic and electric fields can be combined in the following hyperbolic biquaternion, F = H + iE

=

(H1 + iE1 )e1 + (H2 + iE2 )e2 + (H3 + iE3 )e3

=

F1 e1 + F2 e2 + F3 e3

where H=

1 ∗ (F + F) 2

(3.6) (3.7)

and

i ∗ (F − F). (3.8) 2 In this representation, H and E are regarded as the hyperbolic quaternions in the vector form: E=

H = H1 e 1 + H 2 e 2 + H 3 e 3 ,

E = E1 e1 + E2 e2 + E3 e3 .

(3.9)


The product of F with its complex conjugate F ∗ gives F F ∗ = [H + iE][H − iE] = E 2 + H 2 + 2i [E × H] . If we define new hyperbolic biquaternion as 1 S = F F ∗ = ε + iS, 2 the scalar part of S 1 2 SS = ε = E + H2 2 gives energy density while its vector part SV = S = E × H

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(3.10)

(3.11)

(3.12)

(3.13)

yields the energy-current density, known as the Poynting vector. A very useful application of the hyperbolic biquaternionic differential operator occurs in the description of the electromagnetic fields. If we operate on F , then the following vector product like equation can be written ∂ F = i + ∇ [H + iE] ∂t ∂H ∂E + ∇.H + ∇ × H . (3.14) = i ∇.E + ∇ × E + + − ∂t ∂t According to the definitions in eq.(3.4), the imaginary scalar part of (3.14) gives Gauss’ law, whereas the imaginary vector part is Ampere’s law. Similarly, the real scalar part yields the condition for the nonexistence of magnetic monopoles, and real vector part is Faraday’s law of induction. By using the definitions in eq.(3.4), then we get the hyperbolic biquaternion valued four current: F = J

(3.15)

where J = iρ + J = iρ + J1 e1 + J2 e2 + J3 e3 . (3.16) In this notation, J = J1 e1 + J2 e2 + J3 e3 is the hyperbolic quaternion in vector form which corresponds to electric current source density. Eq.(3.15) is equivalent to the old quaternion form of Maxwell’s equations [64]. Furthermore, with vector like elements, the hyperbolic quaternions should give easier and more successful generalization of the classical electromagnetism than the other non-associative algebras such as octonions. We can also describe the electromagnetic fields E and H in terms of a scalar potential φ and a vector potential A = A1 i + A2 j + A3 k as ∂A E = −∇φ − , H = ∇×A. (3.17) ∂t Then, it is possible to introduce a new hyperbolic biquaternion quantity which expresses the scalar and vector potentials together, A = iφ − A = iφ − A1 e1 − A2 e2 − A3 e3 .

(3.18)

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The dependency of E and H on the hyperbolic biquaternion potential A can be written as a single equation ∂ ¯ A = i − ∇ [iφ − A] ∂t ∂A ∂φ + ∇.A + ∇ × A + i −∇φ − = − . (3.19) ∂t ∂t According to the definitions of electric and magnetic fieds in eq.(3.17), the vector part of first parenthesis gives the magnetic field H while the second paranthesis yields the electric field E. At this point, another usefulness of hyperbolic quaternion formulation elegantly comes out. As defined in eq.(3.6), the vector part of this equation is F . Therefeore, eq.(3.19) can be represented in a more compact and elegant form as ¯ = F. (3.20) A V

4. Massive Classical Electromagnetism It is well known that Maxwells equations and Maxwell’s Lagrangian are based on the hypothesis that the photon has zero mass. If there is any deviation from zero, Maxwell’s Lagrangian need to be modified by adding mass term. This modified Lagrangian is known as the Maxwell-Proca Lagrangian. In CGS unit system, it is given by m2γ 1 1 Aμ Aμ − Jμ Aμ (4.1) L = − Fμv F μv + 16 8π c where mγ is the inverse of Compton wavelength associated with the photon’s mass, Jμ is the four-current (ρ, J ), A is the four-vector potential (φ, A) and Fμv = ∂μ Av − ∂v Aμ represents the electromagnetic field tensor [32]. The Euler-Lagrange equation is expressed as ∂L ∂L − ∂v = 0. (4.2) ∂Aμ ∂(∂v Aμ ) Then, the Proca equation is defined by 4π v J . (4.3) c After eliminating all constants except the photon mass, the Proca-Maxwell equations can be written in following vectorial formalism: ∂μ F μv + m2γ Av =

∇.E = ρ − m2γ φ, ∂H , ∂t ∇.H = 0, ∂E − m2γ A. ∇×H =J + ∂t ∇×E =−

(4.4a) (4.4b) (4.4c) (4.4d)


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Similar to eq.(3.14), we can get ∂ F = i + ∇ [H + iE] ∂t ∂H ∂E = i ∇.E + ∇ × E + + ∇.H + ∇ × H + − ∂t ∂t = i ρ − m2γ φ + J − m2γ A =

[iρ + J] − m2γ [iφ + A]

(4.5)

Using the definitions in (3.16) and (3.18), then the Proca equation can be expressed as the following hyperbolic quaternion form, ¯ = J. F + m2γ A (4.6) In standard electrodynamics, as it is seen from eq.(3.4), the magnetic current is absent. Therefore, in the next section we will investigate the generalized electromagnetism in the presence of magnetic monopole.

5. Generalized Electromagnetism The concept of symmetry plays an important role in the electromagnetism. The exchange of the electric and magnetic fields in eq.(3.4), E −→ H and H −→ −E, is known as duality transformation. Maxwell’s equations are not symmetric under the duality transformation. Although so far no experiments have revealed such particles, the existence of magnetic monopoles is postulated as a consequence of this concept. The idea of a magnetic monopole was put forward originally by Dirac in 1931 [65]. Dirac has also shown that the magnetic monopole theory requires the quantization of electric charge. By the introduction of two potentials, Cabibbo and Ferrari have first formulated Dirac monopole theory in a manifestly covariant and symmetrical way in 1962 [66]. Similarly, an algebraic derivation of Dirac’s quantization condition [67] and the first use of two potentials in a Clifford algebra approach to describe generalized electrodynamics were given by Rodrigues, Rosa, Maia and Recami [68, 69]. As pointed by K¨ uhne [70], with the concept of the magnetic monopoles, the electric and magnetic fields can be described equivalently and the quantum electrodynamics models of the monopoles are able to explain the quantization of electric charge. Therefore, modification of Maxwell’s equations is done by putting in the magnetic charge ρm and the magnetic current density J m : ∇.E = ρe ,

(5.1a)

∇ × E = −J m − ∇.H = ρm , ∇ × H = Je +

∂H , ∂t

(5.1b) (5.1c)

∂E . ∂t

(5.1d)

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Here, ρe and J e are the charge source density and the current source density due to electric charge, respectively [71]. With the assumption of existence magnetic monopoles the expressions of the electric and magnetic fields also need modification in the following manner: ∂Ae − ∇ × Am , (5.2a) ∂t ∂Am H = −∇φm − + ∇ × Ae , (5.2b) ∂t where φe and φm are scalar potentials associated, respectively, with electric and magnetic charges. Am is the extra magnetic potential obtained due to the rotation of the electric charge in the field of the magnetic monopole, while Ae is the well-known vector potential responsible for the magnetic field. The symmetrized Maxwell’s equations satisfy the following duality transformations: E = −∇φe −

E −→ H e

m

;

H −→ −E, m

(5.3a) e

J −→ J ρe −→ ρm

; ;

J −→ −J , ρm −→ −ρe ,

(5.3b) (5.3c)

φe −→ φm Ae −→ Am

; ;

φm −→ −φe , Am −→ −Ae .

(5.3d) (5.3e)

If the well-known classical scalar and vectoral product are used, four symmetric Maxwell’s equations in eq.(5.1) are reduced to the following two differential equations: (5.4a) ∇.F = ρm + iρe , ∂E ∂H ∇×F = Je + + i −J m − ∂t ∂t ∂F . (5.4b) = −i[J m + iJ e ] − i ∂t However, a more compact and simpler representation of eq.(5.4) can be obtained by using the hyperbolic biquaternion notation as the following ∂ F = i + ∇ [H + iE] ∂t ∂H ∂E = − + ∇.H + ∇ × H + i + ∇.E + ∇ × E ∂t ∂t = [ρm + iρe ] + [J e − iJ m ]. (5.5) Upon introduction of a new hyperbolic biquaternion quantity, the generalized source density, is given Jg = ρg + J g = [ρm + iρe ]e0 + (J1e − iJ1m)e1 + (J1e − iJ1m )e2 + (J1e − iJ1m )e3 (5.6) where ρg = ρm + iρe

(5.7)


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is the generalized charge density and J g = J e − iJ m = (J1e − iJ1m )e1 + (J1e − iJ1m )e2 + (J1e − iJ1m )e3

(5.8)

is the generalized current source density. Then the generalized Maxwell equations (5.1) can be represented in a single equation, F = Jg .

(5.9)

In order to reformulate the generalized electromagnetism in terms of the hyperbolic quaternion variable, the generalized potential can be defined in a following manner: Ag

= =

φg + Ag (φm + iφe )e0 +

(5.10) (−Ae1

+

iAm 1 )e1

+

(−Ae2

+

iAm 2 )e2

+

(−Ae3

+

iAm 3 )e3 ,

Here, φg = φm + iφe is the generalized scalar potential and

(5.11)

e m e m Ag = −Ae + iAm = (−Ae1 + iAm 1 )e1 + (−A2 + iA2 )e2 + (−A3 + iA3 )e3 (5.12) ¯ given by eq.(3.2) on Ag , the is the generalized vector potential. Operating following hyperbolic biquaternion equation can be derived: ∂ ¯ g = A i − ∇ [(φm − Ae ) + i(φe + Am )] ∂t ∂Am ∂φe + ∇.Ae + ∇ × Ae − ∇φm − = − ∂t ∂t ∂φm ∂Ae − ∇.Am − ∇ × Am − ∇φe − +i . (5.13) ∂t ∂t

Using the definitions in eq.(5.2), the vectoral parts in the first and the second parenthesis yield, respectively, the magnetic field H and the electric field E. Therefore, eq.(5.13) can be written in a more compact and elegant manner: ¯ g = F. (5.14) A V

This equation is also in the same form with eq.(3.20).

6. Generalized Electromagnetism with Massive Photons By adding the Proca terms to eq.(5.1), we arrive at the generalized Maxwell equations in the presence of massive photons and magnetic monopoles in the vector algebra formalism: ∇.E = ρe − m2γ φ, (6.1a) ∇ × E = −J m − ∇.H = ρm ,

∂H , ∂t

(6.1b) (6.1c)

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∂E − m2γ A. (6.1d) ∂t Here, the vector potential (φ, A) is associated with the magnetic charge of the monopole [32]. So, the following hyperbolic quaternion equation can be easily written, ∂ F = i + ∇ [H + iE] ∂t ∂H ∂E + ∇.H + ∇ × H + i + ∇.E + ∇ × E = − ∂t ∂t = ρm + J e − m2γ A + i ρe − m2γ φ − J m ∇ × H = Je +

=

[ρm + iρe ] + [J e − iJ m ] − m2γ [iφ + A] .

(6.2a)

Using the definitions (5.7) and (5.8), a more compact form can be obtained: ¯ F = ρg + J g − m2γ A.

(6.2b)

This expression is the generalized formulation of the fundamental equations of massive classical electrodynamics in the presence of the magnetic monopoles. Thus, the Proca equation can be rewritten as the following form, ¯ = Jg . F + m2γ A

(6.3)

7. Conclusions In the example of octonions, non-associative algebras have not been used frequently in physical applications as they deserve. The octonions are known as the widest normed algebra after the algebras of real numbers, complex numbers and quaternions [11]. Because of eight parameters, it is slightly harder to manipulate octonions than quaternions. The hyperbolic quaternions are 4-dimensional and non-associative mathematical structures. Therefore, we combined two hyperbolic quaternions to express up to 8-dimensional physical quantities. Although the obtained mathematical structure is 8-dimensional, the number of basis elements did not increase and remained as four. Therefore, manipulating of this new hypernumber system is simpler than the classical octonions, hyperbolic octonions and split octonions. Furthermore, this paper contributes potential applications of non-associative hypernumber systems in the description of physical laws. Maxwell’s equations and the relevant field equations are investigated with new hyperbolic biquaternions and these equations have been given in compact, simpler and elegant forms. By using hyperbolic number formalism, we have combined Maxwell’s classical four vectorial equations in a single equation (3.15). Furthermore, we also derived the generalized electromagnetism equations in the existence of the magnetic monopoles and massive photons. For this aim, we have defined 8-dimensional generalized source density Jg which composed of the generalized charge density ρg = ρm + iρe and the generalized current


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source density J g = J e − iJ m . This structure provided us to express the generalized Maxwell equations in a compact and simple form. After defining another 8-dimensional quantity, the generalized potential Ag , we have obtained similar equations for the generalized and the classical electromagnetism, as well. In this work, the hyperbolic quaternions are proposed as a useful tool to formulate the Proca-Maxwell equations. The obtained equations (4.6) and (6.3) are the same as usual the Proca-Maxwell equations. The expressions of hyperbolic biquaternion equations related with the classical electromagnetism are similar to the complex quaternionic [35]-[46], dual quaternionic [49, 50], octonionic [54, 55] and Clifford algebra formulations [25, 26, 32, 71]. But, because of their associative algebraic properties, the complex quaternion and Clifford algebra formalism give more successful generalization than the nonassociative hyperbolic quaternions. However, the hyperbolic quaternionic formulation can be preferable to the octonionic representations. The advantages of this formalism can be summarized as following. Firstly, as mentioned before, if the octonion formalism is used, in this case eight basis elements must be used instead of four. Therefore, manipulating of this system is easier than the octonions. Secondly, the hyperbolic biquaternion formalism of the generalized electromagnetism enhances the dimension of magnetic monopole equations from 4-D to 8-D. These equations can be represented either in complex 4-dimensional vector space or 8dimensional real space. In other words, the foregoing formalism serves two different ways to formulate generalized electromagnetism. Thirdly, in contrast to the octonionic formalism, because of having useful symbol i, the hyperbolic biquaternions provide us to separate quantities in different physical nature explicitly. Thus, it can be seen easily how to relate independent results after a mathematical operation. For example, after operation on F , real parts of eqs.(3.15) and (5.9) are associated with components of magnetic field H while complex parts with symbol i are related to components of electric field E. Finally, the hyperbolic biquaternion algebra facilitates to connect the derived equations with vectors. The products of hyperbolic biquaternions as in eq.(2.13) can be written in terms of scalar and vectorial products. Therefore, correlation between the derived hyperbolic biquaternion equations and vectors is also very easy. Furthermore, this paper also contributes the potential applications of hypernumbers in the description of physical laws. Hyperbolic biquaternion formulation of electromagnetism was absent in literature. Consequently, this paper fills a gap and the hyperbolic biquaternion version of the electromagnetism appears to be a fruitful choice.

References [1] C. Musés, Applied hypernumbers: Computational concepts. Appl. Math. Comput. 3 (1976), 211–216. [2] C. Musés, Hypernumbers II-Further concepts and computational applications. Appl. Math. Comput. 4 (1978), 45–66.

560


AACA

[3] C. Musés, Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries. Appl. Math. Comput. 6 (1980), 63–94. [4] K. Carmody, Circular and hyperbolic quaterternions, octonions, and sedenions. Appl. Math. Comput. 28 (1988), 47–72. [5] K. Carmody, Circular and hyperbolic quaterternions, octonions, and sedenions-furher results. Appl. Math. Comput. 84 (1997), 27–48. [6] J. K¨ oplinger, Dirac equation on hyperbolic octonions. Appl. Math. Comput. 182 (2006), 443–446. [7] J. K¨ oplinger, Hypernumbers and relativity. Appl. Math. Comput. 188 (2007), 954– 969. [8] J. K¨ oplinger, Signature of gravity in conic sedenions. Appl. Math. Comput. 188 (2007), 942–947. [9] J. K¨ oplinger, Gravity and electromagnetism on conic sedenions. Appl. Math. Comput. 188 (2007), 948–953. ¨ [10] N. Candemir, K. Ozda¸ s, M. Tanı¸slı, S. Demir, Hyperbolic octonionic Proca-Maxwell equations. Z. Naturforsch. A 63 (2008), 15–18. [11] M. Gogberashvili, Octonionic version of Dirac equations. Int. J. Mod. Phys. A 21 (2006), 3513–3524. [12] S. De Leo, W. A. Rodrigues Jr., Quaternionic electron theory: Dirac’s equation. Int. J. Theor. Phys. 37 (1998), 1511–1530. [13] S. De Leo, W. A. Rodrigues Jr., Quaternionic electron theory: geometry, algebra and Dirac’s spinors. Int. J. Theor. Phys. 37 (1998), 1707–1720. [14] S. De Leo, W. A. Rodrigues Jr., J. Vaz Jr., Complex geometry and Dirac equation. Int. J. Theor. Phys. 37 (1998), 2415–2431. [15] S. De Leo, G. Ducati, Quaternionic groups in physics: a panoramic review. Int. J. Theor. Phys. 38 (1999), 2197–2220. [16] S. De Leo, Z. Oziewicz, J. Vaz Jr., W. A. Rodrigues Jr., The Dirac–Hestenes lagrangian. Int. J. Theor. Phys. 38 (1999), 2349–2369. [17] D. Hestenes, P. Reany, G. Sobczyk, Unipodal algebra and roots of polynomials: report. Adv. appl. Clifford alg. 1 (1991), 51–64. [18] J. Keller, Quaternionic, complex, duplex and real Clifford algebras. Adv. appl. Clifford alg. 4 (1) (1994), 1–12. [19] S. Ulrych, The Poincaré mass operator in terms of hyperbolic algebra. Phys. Lett. B 612 (2005), 89–91. [20] S. Ulrych, Symmetries in the hyperbolic Hilbert space. Phys. Lett. B 618 (2005), 233–236. [21] S. Ulrych, Spinors in the hyperbolic algebra. Phys. Lett. B 632 (2006), 417–421. [22] S. Ulrych, Relativistic quantum physics with hyperbolic numbers. Phys. Lett. B 625 (2005), 313–323. [23] S. Ulrych, Gravitoelectromagnetism in a complex Clifford algebra. Phys. Lett. B 633 (2006), 631–635. [24] G. Sobczyk, Spacetime vector analysis. Phys. Lett. A 84 (1981), 45–48. [25] D. Hestenes, Space–time algebra. Gordon and Breach, 1966.


561

[26] W. E. Baylis, Electrodynamics: a modern geometric approach. Birkh¨ auser, 1999. [27] Y. Xuegang, L. Wuming, G. Junli, The four dimensional hyperbolic spherical harmonics. Adv. appl. Clifford alg. 10 (2) (2000), 163–171. [28] Y. Xuegang, Z. Shuna, H. Qiunan, Clifford algebraic spinors and the Dirac wave equations. Adv. appl. Clifford alg. 11 (1) (2001), 27–38. [29] Y. Xueqian, H. Qiunan, Y. Xuegang, Clifford algebra and the four dimensional Lorentz transformations. Adv. appl. Clifford alg. 12 (1) (2002), 13–19. [30] Z. Zheng, Y. Xueqang, Hyperbolic Schr¨ odinger equation. Adv. appl. Clifford alg. 14 (2) (2004), 207–213. [31] Z. Wei, Y. Xueqang, Some physics question in hyperbolic complex space. Adv. appl. Clifford alg. 17 (1) (2007), 137–144. [32] C. Cafaro, S. A. Ali, The spacetime algebra approach to massive classical electrodynamics with magnetic monopoles. Adv. appl. Clifford alg. 17 (1) (2007), 23–36. [33] J. C. Maxwell, A treatise on electricity and magnetism. Clarendon Press, 1873. [34] I. Abonyi, J. F. Bito, J. K. Tar, A quaternion representation of the Lorentz group for classical physical applications. J. Phys. A: Math. Gen. 24 (1991), 3245–3254. [35] K. Imaeda, A new formulation of classical electrodynamics. Nuovo Cimento B. 32 (1976), 138–162. [36] F. Colombo, P. Loustaunau, I. Sabadini, D. C. Struppa, Regular functions of biquaternionic variables and Maxwell’s equations. J. Geom. Phys. 26 (1998), 183–201. [37] J. Lambek, If Hamilton had prevailed: quaternions in physics. Math. Intelligencer 17 (1995), 7–15. [38] J. P. Ward, Quaternions and Cayley numbers. Kluwer, 1996. [39] V. V. Kassandrov, Biquaternionic electrodynamics and Weyl-Cartan geometry of space-time. Grav. & Cosm. 3 (1995), 216–222. [40] A. Gsponer, J. P. Hurni, Comments on formulating and generalizing Dirac’s, Proca’s, and Maxwell’s equations with biquaternions or Clifford numbers. Found. Phys. Lett. 14 (2001), 77–85. [41] V. V. Kravchenko, On a quaternionic reformulation of Maxwell’s equations for inhomogeneous media and new solutions. Z. f. Anal.u. ihre Anwend. 21 (2002), 21–26. [42] K. J. Van Vlaenderen, A. Waser, On a quaternionic reformulation of Maxwell’s equations for inhomogeneous media and new solutions. Hadronic J. 27 (2004), 673– 691. [43] S. M. Grudsky, K. V. Khmelnytskaya, V. V. Kravchenko, On a quaternionic Maxwell equation for the timedependent electromagnetic field in a chiral medium. J. Phys. A: Math. Gen. 37 (2004), 4641–4647. [44] M. Tanı¸slı, Gauge transformation and electromagnetism with biquaternions. Europhys. Lett. 74 (2006), 569–573. [45] O. P. S. Negi, S. Bisht, P. S. Bisht, Revisiting quaternion formulation and electromagnetism. Nuovo Cimento B. 113 (1998), 1449–1467. [46] J. Singh, P. S. Bisht, O. P. S. Negi, Quaternion analysis for generalized electromagnetic fields of dyons in an isotropic medium. J. Phys. A: Math. Theor. 40 (2007), 9137–9147.

562


AACA

[47] W. A. Rodrigues Jr., E. Capelas de Oliveira, Dirac and Maxwell equations in Clifford and spin-Clifford bundles. Int. J. Theor. Phys. 29 (1990), 397–412. [48] W. A. Rodrigues Jr., E. Capelas de Oliveira, The many faces of Maxwell, Dirac and Einstein equations. A Clifford bundle approach. Lecture notes in physics. 722, Springer, 2007. ¨ [49] S. Demir, K. Ozda¸ s, Dual quaternionic reformulation of classical electromagnetism. Acta Phys. Slovaca 5 (2003), 429–436. [50] S. Demir, Matrix realization of dual quaternionic electromagnetism. Cent. Eur. J. Phys. 5 (2003), 487–506. [51] S. De Leo, K. Abdel-Khalek, Octonionic quantum mechanics and complex geometry. Prog. Theor. Phys. 96 (1996), 823–832. [52] S. De Leo, K. Abdel-Khalek, Octonionic Dirac equation. Prog. Theor. Phys. 96 (1996), 833–846. [53] S. De Leo, K. Abdel-Khalek, Towards an octonionic world. Int. J. Theor. Phys. 37 (1998), 1945–1985. ¨ [54] T. Tolan, K. Ozda¸ s, M. Tanı¸slı, Reformulation of electromagnetism with octonions. Nuovo Cimento B. 121 (2006), 43–55. [55] M. Gogberashvili, Octonionic electrodynamics. J. Phys. A: Math. Gen. 39 (2006), 7099–7104. [56] P. S. Bisht, O. P. S. Negi, Split octonion electrodynamics. Indian J. Pure and Appl. Phys. 31 (1993), 292–296. [57] P. S. Bisht, S. Dangwal, O. P. S. Negi, Unified split octonion formulation of dyons. Int. J. Mod. Phys. A, online first, DOI: 10.1007/s10773-008-9662-9 [58] W. R. Hamilton, Elements of quaternions. Chelsea, 1899. [59] S. L. Altmann, Rotations, quaternions, and double groups. Clarendon Press, 1986. [60] S. L. Altmann, Hamilton, Rodrigues, and the quaternion scandal. Mathematics Magazine 62 (1989), 291–308. [61] J. S. Dai, An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist. Mech. Mach. Theor. 41 (2006), 41–52. [62] A. MacFarlane, Hyperbolic quaternions. Proc. Roy. Soc. Edinburgh 23 (1900), 169. [63] K. G¨ odel, An example of a new type of cosmological solutions of Einstein’s field equations of Gravitation. Rev. Mod. Phys. 21 (1949), 447–450. [64] L. Silberstein, Quaternionic form of relativity. Philos. Mag. 23 (1912), 790–809. [65] P. A. M. Dirac, Quantized singularities in the electromagnetic field. Proc. Roy. Soc. London A. 133 (1931), 60–72. [66] N. Cabibbo, E. Ferrari, Quantum electrodynamics with Dirac monopoles. Nuovo Cimento 23 (1962), 1147–1154. [67] W. A. Rodrigues Jr., E. Recami, A. Maia Jr., M. A. F. Rosa, The classical theory of the charge and pole motion. A satisfactory formalism by Clifford algebras. Phys. Lett. B 220 (1989), 195–199. [68] M. A. F. Rosa, E. Recami, W. A. Rodrigues Jr., A satisfactory formalism for magnetic monopoles by Clifford algebras. Phys. Lett. B 173 (1986), 233–236.


563

[69] A. Maia Jr., E. Recami, W. A. Rodrigues Jr., M. A. F. Rosa, Magnetic monopoles without string in the K¨ ahler-Clifford algebra bundle: a geometrical interpretation. J. Math. Phys. 31 (1990), 502–505. [70] R. W. K¨ uhne, A model of magnetic monopoles. Mod. Phys. Lett. A 12 (1997), 3153– 3159. [71] C. Doran, A. Lasemby, Geometric algebra for physicists. Cambridge Univ. Press, 2003.

Acknowledgment We kindly thank to the anonymous referee for his/her valuable comments. S¨ uleyman Demir, Murat Tanı¸slı and Nuray Candemir Anadolu University Physics Department 26470 Eski¸sehir Turkey e-mail: [email protected] [email protected] [email protected] Received: June 19, 2008. Accepted: August 10, 2008.