Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2014, Article ID 450262, 6 pages http://dx.doi.org/10.1155/2014/450262

Research Article Field Equations in the Complex Quaternion Spaces

Zi-Hua Weng

School of Physics and Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China

Correspondence should be addressed to Zi-Hua Weng; [email protected]

Received 7 May 2014; Accepted 19 July 2014; Published 6 August 2014

Academic Editor: Shao-Ming Fei

Copyright © 2014 Zi-Hua Weng. 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 paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents of two fields, which were considered to be independent of each other in the past. J. C. Maxwell applied simultaneously the vector terminology and the quaternion analysis to depict the electromagnetic theory. This method edified the paper to introduce the quaternion and octonion spaces into the field theory, in order to describe the physical feature of electromagnetic and gravitational fields, while their coordinates are able to be the complex number. The octonion space can be separated into two subspaces, the quaternion space and the 𝑆-quaternion space. In the quaternion space, it is able to infer the field potential, field strength, field source, field equations, andsoforth,inthegravitationalfield.Inthe𝑆-quaternion space, it is able to deduce the field potential, field strength, field source, and so forth, in the electromagnetic field. The results reveal that the quaternion space is appropriate to describe the gravitational features; meanwhile, the 𝑆-quaternion space is proper to depict the electromagnetic features.

1. Introduction Since the vector terminology can describe the electro- magnetic and gravitational theories, the quaternion should J. C. Maxwell described the physical feature of electromag- be able to depict these two theories also. In the paper, the netic field with the vector as well as the quaternion. In 1843 quaternionspaceissuitabletodescribethegravitational W. R. Hamilton invented the quaternion, and J. T. Graves features, and the 𝑆-quaternion space is proper to depict the invented the octonion. Two years later, A. Cayley reinvented electromagnetic features. And it may be one approach to fig- the octonion independently in 1845. During two decades ure out some puzzles in the electromagnetic and gravitational from that time on, the scientists and engineers separated the theories described with the vector. quaternion into the scalar part and vector part, to facilitate its application in the engineering. In 1873 Maxwell mingled In recent years applying the quaternion to study the naturally the quaternion analysis and vector terminology to electromagnetic feature has been becoming one significant depict the electromagnetic feature in his works. Recently research orientation, and it continues the development trend some scholars begin to study the physics feature of gravita- of gradual deepening and expanding. Increasingly, the focus tional field with the algebra of quaternions. is being placed on the depiction discrepancy of electromag- The ordered couple of quaternions composes the octo- neticfeaturesbetweenthequaternionandvector.Therelated nion (Table 1). On the contrary, the octonion can be separated research into the discrepancy is coming out all the time. into two parts. This case is similar to the complex number, Some scholars have been applying the quaternion analysis which can be separated into the real number and the to study the electromagnetic and gravitational theories, try- imaginary number. The octonion is able to be divided into ing to promote the further progress of these two field theories. two parts as well, the quaternion and the 𝑆-quaternion (short Majernik [1] described the electromagnetic theory with the for the second quaternion), and their coordinates are able complex quaternion. Honig [2]andSingh[3]applied,respec- to be complex numbers. For the convenience of description, tively,thecomplexquaterniontodeducedirectlyMaxwell’s the quaternion described in the following context sometimes equations in the classical electromagnetic theory. Demir et al. includes not only the quaternion but also the 𝑆-quaternion. [4] studied the electromagnetic theory with the hyperbolic 2 Advances in Mathematical Physics

Table 1: The multiplication table of octonion. quaternion, it is able to deduce directly Maxwell’s equations 1 in the classical electromagnetic theory, without the help of i1 i2 i3 I0 I1 I2 I3 the current continuity equation. Similarly some scholars have 11 i1 i2 i3 I0 I1 I2 I3 been applying the complex quaternion to research the gravi- −1 − − − i1 i1 i3 i2 I1 I0 I3 I2 tational theory. A comparison of the two description methods i2 i2 −i3 −1 i1 I2 I3 −I0 −I1 indicates that there are quite a number of sameness between i3 i3 i2 −i1 −1 I3 −I2 I1 −I0 the field theory described by the complex quaternion and

I0 I0 −I1 −I2 −I3 −1 i1 i2 i3 the classical field theory described by the vector terminology,

I1 I1 I0 −I3 I2 −i1 −1 −i3 i2 except for a little discrepancy. In the existing studies that dealt with the quaternion I2 I2 I3 I0 −I1 −i2 i3 −1 −i1 field theory up to now, most of researches introduce the I3 I3 −I2 I1 I0 −i3 −i2 i1 −1 quaternion to rephrase simplistically the physics concept and deduction in the classical field theory. They considered the quaternion as one ordinary substitution for the complex quaternion. Morita [5] researched the quaternion field theory. numberorthevectorinthetheoreticalapplications.Thisisa Grudsky et al. [6] utilized the quaternion to investigate the far cry from the expectation that an entirely new method can time-dependent electromagnetic field. Anastassiu et al.7 [ ] bring some new conclusions. In the past, the application of applied the quaternion to describe the electromagnetic fea- one new mathematic description method each time usually ture. Winans [8] described the physics quantities with the enlarged the range of definition of some physics concepts, quaternion. Meanwhile the complex quaternion has certainly bringing in new perspectives and inferences. Obviously the piqued scholars’ interest in researching the gravitational existing quaternion studies in the field theory so far have not theory. Edmonds [9] utilized the quaternion to depict the achieved the expected outcome. wave equation and gravitational theory in the curved space Making use of the comparison and analysis, a few primal time. Doria [10] adopted the quaternion to research the problems from the preceding studies are found. (1) The gravitational theory. Rawat and Negi [11]discussedthe preceding studies were not able to describe simultaneously gravitational field equation with the quaternion treatment. the electromagnetic and gravitational fields. These existing Moreover a few scholars applied the octonion analysis to researches divide the electromagnetic and gravitational fields study the electromagnetic and gravitational theories. Gog- into two isolated parts and then describe, respectively, the two berashvili [12] discussed the electromagnetic theory with fields.Inthepaper,theelectromagneticfieldandgravitational the octonion. V. L. Mironov and S. V. Mironov [13]applied field can combine together to become one unitary field in the octonion analysis to represent Maxwell’s equations and the theoretical description, depicting the physics features of relevant physics features. Tanisli et al. [14] applied the algebra two fields simultaneously. (2) The preceding studies were of octonions to study the gravitational field equations. In difficult to unify similar physics quantities of two fields into the paper, we will focus on the application of the complex the single definition. In the existing researches, the field quaternion on the electromagnetic and gravitational theory potential (or field strength, field source) of gravitational field field. is different from that of electromagnetic field, and it is not At present there are mainly three description methods for able to unify them into the single definition. But the paper the electromagnetic theory. (1) Vector Analysis.Theelectro- is able to unify the field potential (or field strength, field magnetic theory described with the three-dimensional vector source) of electromagnetic and gravitational fields into the is ripe enough. However the theory has a questionable single definition. logicality, because the deduction of Maxwell’s equations In the paper, the author explores one new description dealt with the participation of the current continuity equa- method, introducing the complex quaternion space into the tion. Logically, the electromagnetic strength determines the field theory, to describe the physical feature of electromag- electromagnetic source via Maxwell’s equations, and then netic and gravitational fields. This method’s inferences cover the electromagnetic source concludes the current continu- the most of conclusions of electromagnetic and gravitational ity equation. On the contrary, the electromagnetic source fields described with the vector. decides the displacement current in Maxwell’s equations via the current continuity equation. This description method results in the problem of logic circulation. The puzzle urges 2. Field Equations other scholars to look for the new description method. (2) Real Quaternion Analysis. In the electromagnetic theory The octonion space O canbeseparatedintotwoorthogonal described with the real quaternion, it is able to deduce directly subspaces, the quaternion space H𝑔 and the 𝑆-quaternion the electromagnetic field equations. But its displacement space H𝑒.AndthequaternionspaceH𝑔 is independent of the current’s direction and the gauge condition of field potential 𝑆-quaternion space H𝑒.ThequaternionspaceH𝑔 is suitable are, respectively, different from that in the classical electro- to describe the feature of gravitational field, while the 𝑆- magnetic theory. The limited progress arouses the scholar’s quaternion space H𝑒 is proper to depict the property of enthusiasm to reapply the complex quaternion to depict the electromagnetic field. electromagnetic theory. (3) Complex Quaternion Analysis. In the quaternion space H𝑔 for the gravitational field, In the electromagnetic theory described with the complex the basis vector is H𝑔 =(i0, i1, i2, i3),theradiusvectoris Advances in Mathematical Physics 3

R𝑔 =𝑖𝑟0i0 +Σ𝑟𝑘i𝑘,andthevelocityisV𝑔 =𝑖V0i0 +ΣV𝑘i𝑘.The Table 2: The multiplication of the operator and octonion physics gravitational potential is A𝑔 =𝑖𝑎0i0 +Σ𝑎𝑘i𝑘,thegravitational quantity. strength is F𝑔 =𝑓0i0 +Σ𝑓𝑘i𝑘, and the gravitational source is S𝑔 =𝑖𝑠0i0 +Σ𝑠𝑘i𝑘.Inthe𝑆-quaternion space H𝑒 for the Definition Expression meaning H =( , , , ) electromagnetic field, the basis vector is 𝑒 I0 I1 I2 I3 , ∇⋅a −(𝜕1𝑎1 +𝜕2𝑎2 +𝜕3𝑎3) the radius vector is R𝑒 =𝑖𝑅0I0+Σ𝑅𝑘I𝑘,andthevelocityisV𝑒 = ∇×ai1(𝜕2𝑎3 −𝜕3𝑎2)+i2(𝜕3𝑎1 −𝜕1𝑎3)+i3(𝜕1𝑎2 −𝜕2𝑎1) 𝑖𝑉0I0 +Σ𝑉𝑘I𝑘.TheelectromagneticpotentialisA𝑒 =𝑖𝐴0I0 + ∇𝑎0 i1𝜕1𝑎0 + i2𝜕2𝑎0 + i3𝜕3𝑎0 Σ𝐴𝑘I𝑘, the electromagnetic strength is F𝑒 =𝐹0I0 +Σ𝐹𝑘I𝑘,and the electromagnetic source is S𝑒 =𝑖𝑆0I0 +Σ𝑆𝑘I𝑘. Herein H𝑒 = 𝜕0ai1𝜕0𝑎1 + i2𝜕0𝑎2 + i3𝜕0𝑎3 H𝑔 ∘ I0 ∘ .Thesymbol denotes the octonion multiplication. ∇⋅A −(𝜕1𝐴1 +𝜕2𝐴2 +𝜕3𝐴3)I0 𝑟𝑗, V𝑗,𝑎𝑗,𝑠𝑗,𝑅𝑗,𝑉𝑗,𝐴𝑗,𝑆𝑗,𝑓0,and𝐹0 are all real. 𝑓𝑘 and 𝐹𝑘 are ∇×A −I1(𝜕2𝐴3−𝜕3𝐴2)−I2(𝜕3𝐴1−𝜕1𝐴3)−I3(𝜕1𝐴2−𝜕2𝐴1) the complex numbers. 𝑖 is the imaginary unit. i0 =1. 𝑗 = 0, 1, ∇∘ 𝜕 𝐴 + 𝜕 𝐴 + 𝜕 𝐴 2, 3. 𝑘 = 1, 2, 3. A0 I1 1 0 I2 2 0 I3 3 0 H H These two orthogonal quaternion spaces, 𝑔 and 𝑒, 𝜕0AI1𝜕0𝐴1 + I2𝜕0𝐴2 + I3𝜕0𝐴3 compose one octonion space, O = H𝑔 + H𝑒.Intheoctonion space O for the electromagnetic and gravitational fields, R = R +𝑘 R the octonion radius vector is 𝑔 𝑒𝑔 𝑒 and the 2.1. Gravitational Field Equations. In the quaternion space octonion velocity is V = V𝑔 +𝑘𝑒𝑔V𝑒,with𝑘𝑒𝑔 being one H𝑔 =(1,i1, i2, i3),thegravitationalpotentialisA𝑔 =𝑖𝑎0 + a, coefficient. Meanwhile the octonion field potential is A = the operator is ◻=𝑖𝜕0 +∇, and the gravitational strength is A𝑔 +𝑘𝑒𝑔A𝑒;theoctonionfieldstrengthisF = F𝑔 +𝑘𝑒𝑔F𝑒.From F𝑔 =◻∘A𝑔. The definition of gravitational strength can be here on out, some physics quantities are extended from the expressed as quaternion functions to the octonion functions, according to the characteristics of octonion. Apparently V, A,andtheir F𝑔 =(−𝜕0𝑎0 +∇⋅a)+𝑖(𝜕0a +∇𝑎0)+∇×a, (6) differential coefficients are all octonion functions of R. The octonion definition of field strength is where a =Σ𝑎𝑘i𝑘; 𝑎0 =𝜑/V0,with𝜑 being the scalar potential F =◻∘A, (1) of gravitational field. For the sake of convenience the paper adopts the gauge where F𝑔 =◻∘A𝑔 and F𝑒 =◻∘A𝑒.Thegaugeconditionsof condition (Table 2), 𝑓0 =−𝜕0𝑎0 +∇⋅a =0. Further the above field potential are chosen as 𝑓0 =0and 𝐹0 =0.Theoperatoris can be written as f =𝑖g/V0 + b,withf =Σ𝑓𝑘i𝑘.One ◻=𝑖i0𝜕0 +Σi𝑘𝜕𝑘. ∇=Σi𝑘𝜕𝑘; 𝜕𝑗 =𝜕/𝜕𝑟𝑗. V0 is the speed of component of gravitational strength is the gravitational light. acceleration, g/V0 =𝜕0a +∇𝑎0.Theotherisb =∇×a,which The octonion field source S of the electromagnetic and is similar to the magnetic flux density. gravitationalfieldscanbedefinedas In the definition of gravitational source equation (3), 𝑖F ∗ S =𝑖𝑠+ 𝜇S =−( +◻) ∘ F the gravitational source is 𝑔 0 s.Substitutingthe S F V0 gravitational source 𝑔 and gravitational strength 𝑔 into the (2) definition of gravitational source can express the definition as 𝑖F ∗ =𝜇 S +𝑘 𝜇 S −( ) ∘ F, 𝑔 𝑔 𝑒𝑔 𝑒 𝑒 V 𝑖 0 −𝜇 (𝑖𝑠 + ) = (𝑖𝜕 +∇)∗ ∘ ( g + ) , 𝑔 0 s 0 V b (7) where 𝜇, 𝜇𝑔,and𝜇𝑒 are coefficients. 𝜇𝑔 <0,and𝜇𝑒 >0. ∗ 0 denotes the conjugation of octonion. In general, the contri- ∗ 2 bution of the tiny term, (𝑖F ∘ F/V0),intheabovecouldbe where the gravitational coefficient is 𝜇𝑔 =1/(𝜀𝑔V0),and neglected. 𝜀𝑔 = −1/(4𝜋𝐺),with𝐺 being the gravitational constant. The =Σ𝑠 According to the coefficient 𝑘𝑒𝑔 and the basis vectors, H𝑔 density of linear momentum is s 𝑘i𝑘. and H𝑒, the octonion definition of the field source can be The above can be rewritten as separated into two parts, ∗ ∗ ∗ ∇ × g ∇ ⋅ g 𝜇𝑔S𝑔 =−◻ ∘ F𝑔, (3) −𝜇𝑔 (𝑖𝑠0 + s)=𝑖(𝜕0b + )+𝑖( ) V0 V0 ∗ (8) 𝜇𝑒S𝑒 =−◻ ∘ F𝑒, (4) ∗ 𝜕0g ∗ +(∇ × b − )+∇ ⋅ b. where (3) is the definition of gravitational source, while (4)is V0 the definition of electromagnetic source. The octonion velocity V is defined as Comparing both sides of the equal sign in the above will yield V = V0𝜕0R, (5) ∗ whereinthecaseforsingleoneparticle,acomparisonwith ∇ ⋅ b =0, (9) the classical electromagnetic (and gravitational) theory S =𝑚V S =𝑞V 𝑚 ∇∗ × reveals that 𝑔 𝑔 and 𝑒 𝑒. is the mass density, 𝜕 + g =0, 𝑞 0b (10) while is the density of electric charge. V0 4 Advances in Mathematical Physics

∇∗ ⋅ g =−𝜇𝑠 , The above can be rewritten as V 𝑔 0 (11) 0 ∇∗ × ∇∗ ⋅ −𝜇 (𝑖 + )=𝑖(𝜕 + E)+𝑖( E) 𝜕 𝑒 S0 S 0B V V ∇∗ × − 0g =−𝜇 , 0 0 b V 𝑔s (12) (15) 0 𝜕 +(∇∗ × − 0E)+∇∗ ⋅ . 𝑠 =𝑚V =Σ𝑔 =Σ𝑏 B B where 0 0. g 𝑘i𝑘; b 𝑘i𝑘. V0 Equations (9)–(12) are the gravitational field equations. Because the gravitational constant 𝐺 is weak and the velocity Comparingthevariablesonbothsidesoftheequalsign ratio k/V0 is tiny, the gravity produced by the linear momen- in the above will reason out Maxwell’s equations as follows: tum s can be ignored in general. When b =0and a =0,(11) ∗ ∇ ⋅ B =0, can be degenerated into Newton’s law of universal gravitation ∗ in the classical gravitational theory. ∇ × E By all appearances the quaternion operator ◻,gravita- 𝜕0B + =0, V0 tional potential A𝑔,gravitationalsourceS𝑔, radius vector R𝑔, V ∗ velocity 𝑔, and so forth are restricted by the component ∇ ⋅ E (16) selection in the definitions. That is, the coordinates with =−𝜇𝑒S0, V0 the basis vector i0 are the imaginary numbers, while the coordinates with the basis vector i𝑘 are all real. In the 𝜕 ∇∗ × − 0E =−𝜇 , paper they can be written as the preceding description form B V 𝑒S in the context, in order to deduce the gravitational field 0 equations. Those restriction conditions are the indispensable where S0 =𝑆0I0 and 𝑆0 =𝑞𝑉0. E =Σ𝐸𝑘I𝑘 and B =Σ𝐵𝑘I𝑘. components of the classical gravitational theory. The density of electric current is S =Σ𝑆𝑘I𝑘.Forthecharged The deduction approach of the gravitational field equa- particles, there may be V𝑒 = V𝑔 ∘ I(I𝑗).AndtheunitI(I𝑗) is ∗ tions can be used as a reference to be extended to that of one function of I𝑗,withI(I𝑗) ∘ I(I𝑗)=1. electromagnetic field equations. In a similar way the electromagnetic potential A𝑒,elec- tromagnetic source S𝑒, radius vector R𝑒,velocityV𝑒,andso 2.2. Electromagnetic Field Equations. In the electromagnetic forth are restricted by the component selection of the defi- theory described with the complex quaternion, it is able nitions. That is, the coordinates with the basis vector I0 are to deduce directly Maxwell’s equations in the classical elec- the imaginary numbers, while the coordinates with the basis 𝑆 tromagnetic theory. In this approach, substituting the - vector I𝑘 are all real. In the paper they can be written as the quaternion for the quaternion, one can obtain the same preceding description form in the context, in order to deduce conclusions still. Maxwell’s equations. Those restriction conditions are the In the 𝑆-quaternion space H𝑒 =(I0, I1, I2, I3),theelec- indispensable components of the classical electromagnetic tromagnetic potential is A𝑒 =𝑖A0 + A; the electromagnetic theory. strength is F𝑒 =◻∘A𝑒. The definition of electromagnetic On the analogy of the coordinate definition of complex strength can be expressed as coordinate system, one can define the coordinate of octonion, 𝑆 F = (−𝜕 +∇⋅ ) +𝑖(𝜕 +∇∘ ) +∇× , which involves the quaternion and -quaternion simulta- 𝑒 0A0 A 0A A0 A (13) neously. In the octonion coordinate system, the octonion physics quantity can be defined as {(𝑖𝑐0 +𝑖𝑑0I0)∘i0 + Σ(𝑐𝑘 + where the vector potential of electromagnetic field is A = ∗ 𝑑𝑘I0 )∘i𝑘}. It means that there are the quaternion coordinate Σ𝐴𝑘I𝑘. A0 =𝐴0I0; 𝐴0 =𝜙/V0,with𝜙 being the scalar poten- ∗ 𝑐𝑘 and the 𝑆-quaternion coordinate 𝑑𝑘I for the basis vector tial of electromagnetic field. 0 i𝑘, while being the quaternion coordinate 𝑐0 and the 𝑆- For the sake of convenience the paper adopts the gauge quaternion coordinate 𝑑0I0 for the basis vector i0. Herein 𝑐𝑗 condition, F0 =−𝜕0A0 +∇⋅A =0. Therefore the above can be and 𝑑𝑗 are all real. written as F =𝑖E/V0 + B. Herein the electric field intensity is E/V0 =𝜕0A +∇∘A0, meanwhile the magnetic flux density is B =∇×A. F0 =𝐹0I0. F =Σ𝐹𝑘I𝑘. 3. Equivalent Transformation In the definition of electromagnetic source equation (4), 3.1. Gravitational Field Equations. Making use of the trans- the electromagnetic source is S𝑒 =𝑖S0 + S.Substitutingthe forming of the basis vectors of ∇ in the quaternion operator ◻, electromagnetic source S𝑒 and electromagnetic strength F𝑒 it is able to translate further the gravitational field equations, into the definition of electromagnetic source can separate the (9)–(12), from the quaternion space H𝑔 into that in the three- definition as dimensional vector space. In the three-dimensional vector 𝑖 ( , , ) 󳶃=Σ 𝜕 2 =1 −𝜇 (𝑖 + ) = (𝑖𝜕 +∇)∗ ∘( E + ), space j1 j2 j3 ,theoperatoris j𝑘 𝑘,withj𝑘 . 𝑒 S0 S 0 B (14) ∇ V0 In the gravitational field equations, the operator should be substituted by the operator 󳶃.Meanwhileb, g,ands are 2 󸀠 󸀠 󸀠 where the electromagnetic coefficient is 𝜇𝑒 =1/(𝜀𝑒V0),with𝜀𝑒 substituted by b =Σ𝑏𝑘j𝑘, g =Σ𝑔𝑘j𝑘,ands =Σ𝑠𝑘j𝑘,respect- being the permittivity. ively. The cross and dot products of quaternion are substituted Advances in Mathematical Physics 5 by that of vector, respectively. And the gravitational field Expressing the above into the scalar equations will reveal equations, (9)–(12),canbetransformedinto that (16)and(22) are equivalent to each other. And the defi- 󸀠 󸀠 󸀠󸀠 󸀠 󸀠 nition of (13)requiressubstitutingE and B by E =−E and −󳶃 ⋅ b =0, 󸀠󸀠 󸀠 B =−B , respectively, in order to approximate to Maxwell’s 󸀠 󸀠 󳶃×g equations in the classical electromagnetic theory as near as 𝜕0b − =0, V0 possible. Therefore the electromagnetic field equations, (22), 󸀠 are able to be transformed into that in the three-dimensional 󳶃⋅g (17) =−𝜇𝑔𝑠0, vector space, V0 󳶃⋅ 󸀠󸀠 =0, 𝜕 󸀠 B 󳶃× 󸀠 + 0g =𝜇 󸀠, b V 𝑔s 󸀠󸀠 0 󸀠󸀠 󳶃×E 𝜕0B + =0, 󸀠 󸀠 󸀠 󸀠 󸀠 V where b =󳶃×a and g /V0 =𝜕0a +󳶃𝑎0. a =Σ𝑎𝑘j𝑘. 0 Expressing the above into the scalar equations will reveal 󸀠󸀠 󳶃⋅E (23) that (9)–(12)and(17) are equivalent to each other. And the =𝜇𝑆 , 󸀠 󸀠󸀠 󸀠 V 𝑒 0 definition of (6)requiressubstitutingg by g =−g , in order 0 to approximate to Newton’s law of universal gravitation in the 󸀠󸀠 󸀠󸀠 𝜕0E 󸀠 classical gravitational theory as near as possible. Therefore the 󳶃× − =𝜇 . B V 𝑒S gravitational field equations, (17), are able to be transformed 0 into that in the three-dimensional vector space, Comparing (23) with Maxwell’s equations in the classical 󸀠 󳶃⋅b =0, (18) electromagnetic theory states that those two field equations are equivalent to each other. It means that in the elec- 󸀠󸀠 󸀠 󳶃×g tromagnetic theory described with the quaternion, without 𝜕0b + =0, (19) V0 the participation of the current continuity equation, it is still able to reason out Maxwell’s equations in the classical 󸀠󸀠 󳶃⋅g electromagnetic theory. =𝜇𝑔𝑠0, (20) V0 󸀠󸀠 4. Conclusions and Discussions 󸀠 𝜕0g 󸀠 󳶃×b − =𝜇𝑔s . (21) V0 Applying the complex quaternion space, the paper is able to describe simultaneously the electromagnetic field equations Comparing (20) with Poisson equation for Newton’s law andthegravitationalfieldequations.Thespacesofgravita- of universal gravitation reveals that those two equations are tional and electromagnetic fields both can be chosen as the the same formally. In case the gravitational strength compo- quaternion spaces. Furthermore some coordinates of those nent b is weak and the velocity ratio k/V0 is quite tiny, (20) two quaternion spaces and relevant physics quantities may be willbereducedintoNewton’slawofuniversalgravitationin the imaginary numbers. the classical gravitational theory. In the definitions of the quaternion operator, field poten- tial,fieldsource,radiusvector,velocity,andsoforth,the 3.2. Electromagnetic Field Equations. By means of the trans- I0 i0 ∇ coordinates with the basis vector or are all imaginary formation of basis vectors of in the quaternion operator, it is numbers. However this simple case is still able to result in able to translate further the electromagnetic field equations, many components of other physics quantities to become (16), from the 𝑆-quaternion space H𝑒 into that in the three- ( , , ) the imaginary numbers, even the complex numbers. In the dimensional vector space j1 j2 j3 . In the electromagnetic definitions, only by means of the confinement of the com- field equations, the operator ∇ is substituted by 󳶃.AndB, E, 󸀠 󸀠 󸀠 ponent selection can we achieve the quaternion field theory and S are substituted by B , E ,andS , respectively. Therefore approximatingtotheclassicalfieldtheory,enablingittocover the electromagnetic field equations,16 ( ), can be transformed the classical field theory. Therefore it can be said that the into component selection is the essential ingredients for the field 󸀠 󳶃⋅B =0, theory too. 󸀠 Inthequaternionspace,itisabletodeducethegravita- 󸀠 󳶃×E 𝜕0B + =0, tional field equations as well as the gauge condition of V0 gravitational potential and so on. Newton’s law of universal 󳶃⋅ 󸀠 (22) gravitation in the classical gravitational theory can be derived E =−𝜇𝑆 , V 𝑒 0 from the gravitational field equations described with the 0 quaternion. Two components of the gravitational strength 𝜕 󸀠 are the counterpart of the linear acceleration and of the 󳶃× 󸀠 − 0E =−𝜇 󸀠, B V 𝑒S precession angular velocity. In general, the component of 0 the gravitational strength, which is corresponding to the 󸀠 󸀠 󸀠 󸀠 󸀠 where B =−󳶃×A and E /V0 =𝜕0A +󳶃𝐴0. A =Σ𝐴𝑘j𝑘 precession angular velocity of the gyroscopic torque which 󸀠 and S =Σ𝑆𝑘j𝑘. dealt with the angular momentum, is comparatively tiny. 6 Advances in Mathematical Physics

In the 𝑆-quaternionspace,itisabletoreasonoutdirectly electromagnetics: a lower order alternative to the Helmholtz the electromagnetic field equations as well as the gauge equation,” IEEE Transactions on Antennas and Propagation,vol. condition of electromagnetic potential and so forth. The elec- 51, no. 8, pp. 2130–2136, 2003. tromagnetic field equations described with the 𝑆-quaternion [8] J. G. Winans, “Quaternion physical quantities,” Foundations of can be equivalently translated into Maxwell’s equations of Physics, vol. 7, no. 5-6, pp. 341–349, 1977. the classical electromagnetic theory in the three-dimensional [9] J. Edmonds, “Quaternion wave equations in curved space-time,” vector space. Moreover the definition of electromagnetic International Journal of Theoretical Physics,vol.10,no.2,pp.115– strength and the gauge condition of electromagnetic potential 122, 1974. both can be equivalently translated, respectively, into that in [10] F. A. Doria, “A Weyl-like equation for the gravitational field,” the classical electromagnetic theory. Lettere al Nuovo Cimento,vol.14,no.13,pp.480–482,1975. Itshouldbenotedthatthepaperdiscussedonlytheelec- [11] A. S. Rawat and O. P.S. Negi, “Quaternion gravi-electromagnet- tromagnetic field equations and the gravitational field equa- ism,” International Journal of Theoretical Physics,vol.51,no.3, tions and so forth, described with the complex quaternion, pp. 738–745, 2012. and the equivalent transformation of those field equations [12] M. Gogberashvili, “Octonionic electrodynamics,” Journal of into that in the three-dimensional vector space. However it Physics A,vol.39,no.22,pp.7099–7104,2006. clearly states that the introducing of the complex quaternion spaceisabletoavailablydescribethephysicsfeaturesof [13]V.L.MironovandS.V.Mironov,“Octonicrepresentationof electromagnetic and gravitational fields. This will afford the electromagnetic field equations,” Journal of Mathematical Phys- ics,vol.50,no.1,ArticleID012901,10pages,2009. theoreticalbasisforthefurtherrelevanttheoreticalanalysis and is helpful to apply the complex quaternion analysis [14] M. Tanisli, M. E. Kansu, and S. Demir, “Reformulation of elec- to study the angular momentum, electric/magnetic dipole tromagnetic and gravito-electromagnetic equations for Lorentz moment, torque, energy, force, and so forth in the electro- system with octonion algebra,” General Relativity and Gravita- tion,vol.46,article1739,2014. magnetic and gravitational fields.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The author is indebted to the anonymous referees for their valuable and constructive comments on the previous paper. This project was supported partially by the National Natural Science Foundation of China under Grant no. 60677039.

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