QUATERNION/VECTOR DUAL SPACE ALGEBRAS APPLIED TO THE DIRAC EQUATION AND ITS EXTENSIONS

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1 Bulletin of the Transilvania University of Braşov ol 857), No Series III: Mathematis, Informatis, Physis, QUATERNION/ECTOR DUAL SPACE ALGEBRAS APPLIED TO THE DIRAC EQUATION AND ITS EXTENSIONS György DARAS 1 Comuniated to the onferene: Finsler Extensions of Relativity Theory, Brasov, Romania, August 18-23, 214 Abstrat The paper re-applies the 64-part algebra disussed by P. Rowlands in a series of FERT and other) papers in the reent years. It demonstrates that the original introdution of the γ algebra by Dira to the quantum theory of the eletron an be interpreted with the help of quaternions: both the α matries and the Pauli σ) matries in Dira s original interpretation an be rewritten in quaternion forms. This allows to onstrut the Dira γ matries in two quaternion-vetor) matrix produt representations - in aordane with the double vetor algebra introdued by P. Rowlands. The paper attempts to demonstrate that the Dira equation in its form modified by P. Rowlands, essentially oinides with the original one. The paper shows that one of these representations affets the γ 4 and γ 5 matries, but leaves the vetor of the Pauli spinors intat; and the other representation leaves the γ matries intat, while it transforms the spin vetor into a quaternion pseudovetor. So the paper onludes that the introdution of quaternion formulation in QED does not provide us with additional physial information. These transformations affet all gauge extensions of the Dira equation, presented by the author earlier, in a similar way. This is demonstrated by the introdution of an additional parameter named earlier isotopi field-harge spin) and the appliation of a so-alled tau algebra that governs its invariane transformation. The invariane of the extended Dira equation is subjet of the onvolution of the Lorentz transformation and this invariane group. It is shown that additional physial information an be expeted by extending these investigations to the Finsler-like metri proposed by Dira in 1962 for the theory of the eletron. 21 Mathematis Subjet Classifiation: 53B5, 81Q5, 81R5. Key words: quaternions, dual vetor algebra, Dira algebra, extended Dira equation, tau algebra. 1 Symmetrion, Budapest, darvasg@iif.hu

2 28 György Darvas 1 etor algebra, quaternion algebra Several statements in this paper ontinue ertain ideas set forth in reent years publiations by P. Rowlands [1, 2, 3, 4]. etors an be transformed into quaternions multiplying them by the imaginary unit, and vie versa. When one substitutes vetors with quaternions or quaternions with vetors obtained in this way in physial equations, they will not hange the physial meaning of the equations. In other words, they do not bring in extra physial information. This statement does not onern ertainly those quaternions and vetors that have not been obtained in this way.) We use the following notations: i = j = k vetors, i = j = k quaternions, i = ii. etors are subjets of the following algebra: i 2 = j 2 = k 2 = 1 ijk = i antiommutation rules: {i, j} = {j, k} = {k, i} = ommutation rules: [i, j] = 2ik, [j, k] = 2ii, [k, i] = 2ij. Quaternions are subjets of the following algebra i 2 = i) 2 i 2 = 1, j 2 = i) 2 j 2 = 1, k 2 = i) 2 k 2 = 1, ijk = i. antiommutation rules: {i, j} = {j, k} = {k, i} = ommutation rules: [i, j] = 2k, [j, k] = 2i, [k, i] = 2j. Let s apply this algebra to Pauli s and Dira s matries with the following notations the upper ase haraters refer to the notations introdued by P. Rowlings): i = σ 1 j = σ 2 k = σ 3 i = ii = iiσ 1 j = ij = ijσ 2 k = ik = iσ 3 σ i = iσ i σ i = iσ i i, j, k) ρ i = iρ i σ i = iρ i I, J, K) Using these definitions, we an interpret Dira s matries as a double vetor algebra, presented in vetor and quaternion ombinations). They an be expressed as produts of ρ and σ matries. Dira introdued [5] first matries marked α, then the matries γ, named after him. α i = ρ 1 σ i = iρ 1 σ i = iρ 1 σ i = iρ 1 σ i γ i = ρ 2 σ i = iρ 2 σ i = iρ 2 σ i = iρ 2 σ i Observe, that the α and γ matries are generated by the help of the ρ 1 and ρ 2 matries as oeffiients of the σ i matries. We extend this set of matries with matries to be alled β, generated by the help of ρ 3 : β i = ρ 3 σ i = iρ 3 σ i = iρ 3 σ i = iρ 3 σ i.

3 Quaternion/vetor dual spae algebras applied to the Dira equation 29 The latter did not appear in the system of matries introdued by Dira. Notie, that all the above produt ombinations of either vetors, quaternions or both provide vetors. The vetor and quaternion algebra of all the above listed five types of matries are summarized in the following table: We must note that they are not all independent. These five matries an be expressed by two of the others but not in all arbitrary ombinations). σ plays a distinguished role what restrits the number of allowed representations. The different pairings provide different dual representations of the Dira algebra. We see that they an be represented both in vetor and quaternion pseudovetor) forms. Note, that α, γ and β do not transform aording to the i, j, k vetor algebra, like ρ and σ do. Their ommutators are not produed from their own set, but they are a omponent of σ. Another observation is that the roles of the ρ and σ matries show similitude to the roles of U and T in the geneti algebra [6]! The roles of ρ and σ an be hanged. This allows an interesting observation and leads to a third mixed) algebra. Nevertheless, this allows another set of representations. For ρ and σ ommute, ρ i σ j = ρ j σ i, let s ompose the following ombinations: ρ 1 σ 1 = σ 1 ρ 1 = α 1 = κ 1 ρ 2 σ 1 = σ 1 ρ 2 = γ 1 = κ 2 ρ 3 σ 1 = σ 1 ρ 3 = β 1 = κ 3 ρ 1 σ 2 = σ 2 ρ 1 = α 2 = λ 1 ρ 2 σ 2 = σ 2 ρ 2 = γ 2 = λ 2 ρ 3 σ 2 = σ 2 ρ 3 = β 2 = λ 3 ρ 1 σ 3 = σ 3 ρ 1 = α 3 = µ 1 ρ 2 σ 3 = σ 3 ρ 2 = γ 3 = µ 2 ρ 3 σ 3 = σ 3 ρ 3 = β 3 = µ 3 These equalities serve as definitions of the κ, λ and µ matries. κ, λ, µ are defined by a new indexation row-olumn hange), they provide no new matries, however, allow to ompose new vetors, in the following way: They follow the algebra: κ 2 i = λ 2 i = µ 2 i = 1 σ 1 ρ i = κ i σ 2 ρ i = λ i σ 3 ρ i = µ i α 1 α 2 α 3 γ 1 = κ i γ 2 = λ i γ 3 = µ i β 1 β 2 β 3 κ i κ j = κ j κ i κ i κ j = iρ k and the same rules apply for λ and µ too). Expressing the similar equalities with quaternions: κ i = iκ i, λ i = iλ i, µ i =

4 3 György Darvas iµ i κ 2 i = λ 2 i = µ 2 i = 1 κ i κ j = κ j κ i κ i κ j = iρ k and the same rules apply for λ and µ too). The κ, λ, µ representation ensures a σ ρ mirror-symmetri algebra of the α, γ, β representation. This onfirms the geneti U T nuleotide analogy onjeture see more about it later in setion 3.1). While the generators of the α, γ, β double vetor representation were ρ, the generators of the κ, λ, µ double vetor representation are σ. Let us reluster the κ, λ, µ matries. They an define another mixed, vetorquaternion-like) Clifford algebra. They define a semi-vetor - semi-quaternion algebra. The κ, λ, µ matries - ommute; - square like quaternions; and - multiple like vetors! κ 2 i = λ2 i = µ2 i = 1 κ iλ j = λ j κ i κ i λ j = µ k κ 2 i = λ2 i = µ2 i = 1 κ iλ j = λ j κ i κ i λ j = iµ k 2 How an they applied to physis? All the above representations indiate Clifford algebras generated either by vetors or by quaternions over a spinor field. Their speifis is that the roles of the applied vetors, or quaternions, or their mixtures in the doublets) an be hanged, and may serve one as generators of the field, one as bases of the vetor, quaternion) spae. Any of them may serve as generator - at least in formal mathematial terms -, however, there is doubtful, at least at our present day knowledge of physis, whether a definite physial meaning an be assigned to all formally) so generated fields. As Wigner notied in his famous essay on the unreasonable effetiveness of mathematis in the sienes, mathematis allows more than nature an realise. This an be interpreted as underdetermination seen from one side, and overdetermination seen from the other.) Conerning the appliability of these algebras in physis, we have to answer, at least, three questions: having introdued the double vetor/quaternion matries in a physial e.g., the Dira) equation 1) an we obtain any additional physial information; 2) how an we interpret in physial terms) the results obtained by the hanged formalism; e.g., ertainly, not all matries of the presented algebra an represent spinor spaes), and

5 Quaternion/vetor dual spae algebras applied to the Dira equation 31 3) an we eliminate imaginary fators from the equations? I attempt to answer these questions applying the skethed double vetorquaternion algebra for the Dira equation. Note, QED, in its original formulation by Dira ) [5] ould not handle highly relativisti situations. It onsidered invariane of the equation under Lorentz transformation, but did not take into aount all relativisti effets e.g., mass inrease far from the rest, or effet of a veloity dependent field). As he wrote in the introdution to [5]:,,The resulting theory is therefore still only an approximation,... p. 612). Later 1951) [7], 1962) [8] he made two attempts to extend his,,theory of the eletron, but none of them ould ompletely solve the problems left open, although he applied a Finsler-like metri in the latter extension, without mentioning Finsler). 2.1 Dira equation with no EM field First, examine the onstrution of the lassial Dira equation with no eletromagneti field, in Hamiltonian representation. Originally Dira applied 4 matrix sets: α; σ and ρ; then γ. We are free to hoose any matrix representation, beause they an be transformed into eah other. For simpliity, we redue our sope to Dira s original formalism like P. Rowlands did in his ited papers. α 4 p 4 + α 4 α i p i + m)ψ = ; where α i = ρ 1 σ i, α 4 = ρ 3. Multiplying with ρ 1 : iρ 2 p 4 + ρ 3 σ i p i ρ 1 m)ψ = iρ 2 p 4 + ρ 3 σ i p i ρ 1 m)ψ =. or with quaternions: Why did we keep σ instead of onverting it also to σ? At first: does a quaternioni sigma provide us with any new information? Probably not. At seond: ould we physially interpret σ? Presumedly, not. At third: is σ less imaginary than σ? The first and third omponents of σ onsist of real elements, the seond omponent onsists of imaginaries. In the ase of σ the situation is just the opposite. Thus, both σ and σ inlude real and imaginary elements either. Now we an answer questions 1)-3): during the onstrution of the lassial Dira equation, we do not obtain any additional physial information when we replae vetor-like matries with quaternioni ones; the interpretation of the results obtained by the hanged formalism at least, in physial terms), not only do not provide extra information, even there is doubtful whether all matries of the presented algebra an represent spinor spaes; and we demonstrated that we annot eliminate imaginary fators from the equations by the appliation of quaternions. The double vetor, ρ σ representations follow Clifford algebras as developed by P. Rowlands in the reent years. The algebrai produt of the + and versions of 8 base units, e.g., 1, i, j, k,i, i, j, k, provides 64 terms. They are isomorphi to the γ algebra of the Dira equation, based on 4 4 matries.

6 32 György Darvas 2.2 Dira equation in the presene of EM field Now, let us investigate the onstrution of the lassial Dira equation in the presene of eletromagneti field. We extend the momentum omponents with the ontribution of the field omponents. We demonstrate it both in vetor and quaternion forms: [iρ 2 p 4 + e ) A 4 + ρ 3 σ, p + e ) ] A ρ 1 m ψ = [iρ 2 p 4 + e ) A 4 + ρ 3 σ, p + e ) ] A ρ 1 m ψ =. It is obvious that, similar to the no-field ase, we have not got any new physial information in the latter equation. 2.3 Dira equation in the presene of isotopi field-harges In the third step, let us investigate the onstrution of the Dira equation from the Klein-Gordon equation) in the presene of isotopi field-harges [9, 1, 11, 12, 13]). For this reason, we replae the eletri and the gravitational fieldharges with the respetive isotopi eletri harge densities ρ T, ρ ) and isotopi masses m T, m ), in both vetor and quaternion forms. refers to the potential, and T to the kineti isotopi field harges.) [ iρ 2 p 4 + ρ ) T A 4 + ρ 3 σ, p + ρ ) ] A ρ 1 m [iρ 2 p 4 + ρ ) T A 4 + ρ 3 σ, p + ρ ) ] A ρ 1 m T ψ = [ iρ 2 p 4 + ρ ) T A 4 + ρ 3 σ, p + ρ ) ] A ρ 1 m [iρ 2 p 4 + ρ ) T A 4 + ρ 3 σ, p + ρ ) ] A ρ 1 m T ψ =. Essentially we haven t got new information due to the quaternion formalism, however, we must notie that the nilpotent harater of the equations is lost. This influenes the invariane properties of the equation. We will return later to the solution of this problem in subsetion 2.5. Let us exeute the multipliation of the square brakets to obtain the modified forms) of the Dira equation: {[ p 4 + ρ ) T 2 A 4 + ρ3 σ, p + ρ ) 2 A + m ρ )) + σ, rot A ρ ) i ρ 1 σ, grad A ρ ) )] t A + +ρ 3 [ p 4 + ρ ) T A 4 + ρ 1 σ, p + ρ ) ] } A ρ 3 m m T m v ) ψ = This form of the equation is in onsistene with the usual γ 4 and γ 5 representation. See the disussion of this equation in my paper presented at the 212

7 Quaternion/vetor dual spae algebras applied to the Dira equation 33 FERT onferene held in Fryazino Russia) [1]. The same equation in quaternion form: {[ p 4 + ρ ) T 2 A 4 + ρ3 σ, p + ρ ) 2 A + m ρ )) + σ, rot A ρ ) + ρ 1 σ, grad A ρ ) )] t A + +ρ 3 [ p 4 + ρ ) T A 4 + ρ 1 σ, p + ρ ) ] } A ρ 3 m m T m v ) ψ = 2.4 Dira equation in the presene of isotopi field harges plus a kineti field D) For this reason, we extend the momentum omponents introdued in subsetion 2.2 with the ontribution of the respetive kineti field omponents and exeute again the multipliation with these additional omponents. {[ p 4 + ρ T A 4 + ρ ) 2 D 4 + ρ3 ρ )) A ρ i ρ 1 σ, grad A 4 σ, p + ρ A + ρ ) ] 2 D + m ρ ) )] A + ) + σ, rot + 1 t ρ )) ) + σ, rot D + ρ 1 σ, ρ2 2 [D jd k D k D j ] ρ ) 1 i ρ 1 σ, grad D ρ ) ) 4 t D ρ 2 ρ 1 2 σ, D 4D DD 4 ) + +ρ 3 [ p 4 + ρ T A 4 + ρ ) D 4 ρ 1 σ, p + ρ A + ρ ) ] D + ρ 3 m m T m v )} ψ = As we already mentioned in the above subsetion, the asymmetri role of the isotopi field harges in the presene of a kineti field, destroys the Lorentz invariane of the equation. This feature beame more spetaular by the presene of the kineti field omponents. However, as shown earlier, the invariane of the equation is restored by the onservation of the isotopi field harges and the assoiated symmetry, as it has been proven in [9]. A more demonstrative form of that invariane is given in [11, 12]. We will speify this invariane restoration starting in subsetion 2.5. Until that, for the sake of ontinuing the quaternion presentation, the same equation takes the following form, although, similar to the previous ases, it does not provide us with additional physial information:

8 34 György Darvas {[ p 4 + ρ T A 4 + ρ ) 2 D 4 + ρ3 ρ )) A ρ + ρ 1 σ, grad A 4 σ, p + ρ A + ρ ) + 1 ) ] 2 D + m ρ ) )] A + + σ, rot t ρ )) ) + σ, rot D + ρ 1 σ, ρ2 2 [D jd k D k D j ] + ρ ) 1 + ρ 1 σ, grad D ρ ) ) 4 t D ρ 2 iρ 1 2 σ, D 4D DD 4 ) + +ρ 3 [ i p 4 + ρ T A 4 + ρ ) D 4 + ρ 1 σ, p + ρ A + ρ ) ] D ρ 3 m m T m v )} ψ = On the basis of [11] and [12], one an get asertained that the urvature of the onnetion field is also independent of the vetor or quaternion representations, sine none of the magneti and eletri momenta ontain either vetor or quaternion matries. The full magneti moment is: and the full eletri moment is: M F ULL = ρ rota + D) + igc i jk ρ 2 D jd k, N F ULL = gradρ T A 4 + ρ A + D). t These M F ULL and N F ULL should ommute with the Hamiltonian of the interating two harges. H = ρ T A 4 ρ D 4 + iρ 1 σ, ihgrad ρ A ρ D) + iρ 3 m 2 This has been disussed in [12]. 2.5 Kineti addition to the transformation of the Lorentz fore In order to determine how one should extend the Lorentz transformation to restore the invariane of the basi equation of QED, first, we examine the roles of the extended magneti and eletri moments on a simple example, on the Lorentz fore. The presented M F ULL and N F ULL appear in the EM field tensor - in the presene of isotopi field harges and an additional kineti field: ρ F µν = M3 D M2 D iγ 5 N1 D M3 D M1 D iγ 5 N2 D M2 D M1 D iγ 5 N3 D iγ 5 N1 D iγ 5 N2 D iγ 5 N3 D

9 Quaternion/vetor dual spae algebras applied to the Dira equation 35 It inludes the iγ 5 = iρ 1 omponents, that an be replaed by their quaternion partners γ 5 or ρ 1. This tensor appears in the F µ Lorentz fore: M F µ = F µν 1 j ν = 1 3 D M2 D iγ 5 N D ẋ 1 ρ 1 T M3 D M1 D iγ 5 N2 D ẋ ρ 2 ρ M2 D M1 D iγ 5 N3 D T ẋ ρ 3 T = iγ 5 N1 D iγ 5 N2 D iγ 5 N3 D iρ M3 D = 1 ρ T ẋ2 M 2 Dρ T ẋ3 + γ 5N1 Dρ M3 D ρ T ẋ1 + M 1 Dρ T ẋ3 + γ 5N2 Dρ ρ M2 Dρ T ẋ1 M 1 Dρ T ẋ2 + γ 5N3 Dρ iγ 5 N D 1 ρ T ẋ1 + iγ 5N D 2 ρ T ẋ2 + iγ 5N D 3 ρ T ẋ3 where γ 5 = ρ 1 = iρ 1. This Lorentz fore an be further shaped in the quaternion form of ρ: M3 D F µ = 1 ρ T ẋ2 M 2 Dρ T ẋ3 iρ 1N1 Dρ M3 D ρ T ẋ1 + M 1 Dρ T ẋ3 iρ 1N2 Dρ ρ M2 Dρ T ẋ1 M 1 Dρ T ẋ2 iρ 1N3 Dρ = ρ 1 N1 Dρ T ẋ1 + ρ 1N2 Dρ T ẋ2 + ρ 1N3 Dρ T ẋ3 M3 D ẋ2 1 M 2 D ẋ3 iρ 1 N1 D M = 3 D ẋ1 + M 1 D ẋ3 iρ 1 N D [ ] 2 ρt ρ M2 D ẋ1 M 1 D ẋ2 iρ 1 N3 D = 1 H Dκl ρ ρ l. ρ ρ 1 N1 D ẋ1 + ρ 1N2 D ẋ2 + ρ 1N3 D ẋ3 [ ] In the tensor H Dκl ρt : κ = 1,..., 4; l = 1, 2 and ρ l =. This ρ l an be expressed by the help of the eigenfuntions of the state funtion ψ in the extended Dira equation presented in subsetion 2.4. Sine ρ appears in the salar potential, while ρ in the vetor potential and an serve as oeffiient of the three omponents of the vetor omponents, the latter should take the form of a dreibein. This indiates that ρ l, and aordingly the eigenfuntions, by the expressions of whom it is demonstrated, should be subjet of a speial algebra. That algebra must be represented by matries that reflet the harater of the ρ l four-vetor. 3 The algebra of the isotopi field-harge model applied to QED 3.1 The tau algebra for the extended Dira equation The ψ state funtion in the Dira equation depends on the x ν spae-time oordinates, the r and s parameters of ρ and σ, and in the extended equation on the parameter of the isotopi field-harge spin rotation matrix τ. We were more onsequent by denoting the latter parameter by t. We had two reasons not to do ρ

10 36 György Darvas so. One, t denotes time in physis. Seondly, we denoted the isotopi field-harge spin by in the previous papers.) Let s denote the eigenfuntions of σ by ε and ζ, those of ρ by ξ and η, and the eigenfuntions of τ by ϑ and χ. Then the eigenfuntions of the ρσ produt an be expressed in the following forms: 1 1 εξ = ; ζξ = ; εη = ; ζη =. and an be denoted, aording to the ± 1 2 ϕ ρ) +, ϕρ), 1 1 values of r and s, respetively, by ϕσ) +, ϕσ). Let us represent the eigenfuntions of τ in the following form. Certainly, we had some limited freedom to hoose the values in the eigenvetors. This hoie makes handling the τ transformation matries onvenient. Reall, that in the following ase, like in many other ases, the representation of the transformation group oinides with a representation of the respetive Lie algebra.) 1 1 χ = ; ϑ =. The respeting eigenfuntions belonging to the ± ϕ τ) +, ϕτ). i values of are Similar to the σ algebra in Dira s QED, we introdued [14] the τ matries with the following algebrai properties: τ 3 χ = χ τ 2 χ = ϑ τ 3 ϑ = ϑ τ 3 ϑ = χ requiring τ i τ j = iτ k i, j, k yli indies τ 1 χ = iϑ τ 1 ϑ = iχ. A set of the following matries meets the required properties: 1 i 1 τ 1 = 1 1, τ 2 = i i, τ 3 = i 1 These τ matries satisfy the following algebra: 1 τ1 2 = τ2 2 = τ3 2 = 1 1 = E L, τ i τ j + τ j τ i =, [τ i, τ j ] = 2iτ k. 1

11 Quaternion/vetor dual spae algebras applied to the Dira equation 37 The representation of the transformation group, we are looking for in order to extend the invariane group belonging to the extended form of the Dira equation, is determined by the algebra of these matries. They transform the two states of the IFCS s among eah other by a rotation in the abstrat IFCS field. The τ matries are representations of operators that ompose the three omponents of the isotopi field harge spin, 1, 2, 3 as derived in [9], setions and 4.2.3) whih follow the same non-abelian) ommutation rules like do the τ. These operators represent the harges of the isotopi field harge spin field. The algebra of the τ matries is disussed in detail in [14]. Similar to the definition of γ 4 in the Dira algebra, let us define artifiially τ 4 : 1 τ4 2 = τ 3 where τ 4 = 1 1. i τ i i = 1, 2, 3) belong to the set of dyadi shift matries [15]. τi 2 are so alled oblique projetors. The asymmetri role of the fourth omponents in the τ algebra onfirms the similitude to the asymmetri role of the nuleotides U and T of RNA and DNA, respetively, in the algebra of the moleular geneti ode. A detailed desription of suh oblique projetion operators whih are onneted with matrix representations of the geneti oding system in forms of the Rademaher and Hadamard matries is given among others in [16]. Petoukhov proposes multidimensional vetor spaes, whose subspaes are under a seletive ontrol or oding) by means of a set of matrix operators on the basis of geneti projetors. Similitude between the mathematial desription of physial fields of interations and the system of oding moleular geneti information suggests demonstrating that there are fundamental logial priniples behind these apparently different ontologial phenomena. Referring to the mathematial disussion presented in [14], we define the following set of δ matries: δ 1 = τ 2 1, δ 2 = τ 2 2, δ 3 = τ 2 3, δ 4 = τ 2 4. This hoie has two logial reasons. The γ algebra was reated by the multipliation of σ and ρ matries. The ρ matries were obtained by hanging the seond and third rows and olumns in the σ matries. In ase of the τ matries, the seond and third rows and olumns oinide, respetively. Therefore, if we onsidered the roles of the τ matries as the analogues of the σ matries in the new algebra, the analogue roles orresponding to the ρ matries must oinide with the τ matries. Sine the squares of all the three τ matries oinide, the analogue set is omplete with this hoie. This set has another important advantage that justifies our hoie. Aording to the independent freedom on σ and ρ dependene, the original Dira equation had four solutions. The introdution of the isotopi field-harges should not quadruple, only double the solutions of the Dira equation. For this reason the τ

12 38 György Darvas algebra should have been hosen in suh way that involved only one two-valued property one new parameter, instead of two). Thus the number of the solutions of the extended Dira equation will be eight, as we expeted. The ψ = ψx ν, r, s, ) state funtion separates now aording to the eigenfuntions of the spae-time dependent differential operators, so that the eigenfuntions of the σ, ρ and τ operators take disrete eigenvalues ± 1 2 eah. The differential operators at only on ψx ν ), the σ operators on the eigenfuntions of σ, the ρ operators on the eigenfuntions of ρ, and the τ operators on the eigenfuntions of. The state funtion an be dissoiated to the following eight produts aording to the eigenfuntions of the matrix operators: ψ = ψ 1 x ν )ϕ ρ) + ϕσ) + ϕτ) + + ψ 1x ν )ϕ ρ) + ϕσ) + ϕτ) + ψ 2x ν )ϕ ρ) ϕσ) + ϕτ) + + +ψ 2x ν )ϕ ρ) ϕσ) + ϕτ) + ψ 3x ν )ϕ ρ) + ϕσ) ϕτ) + + ψ 3x ν )ϕ ρ) + ϕσ) ϕτ) + +ψ 4 x ν )ϕ ρ) ϕσ) ϕτ) + + ψ 4x ν )ϕ ρ) ϕσ) ϕτ). This involves also that the solutions of the extended Dira equation s ψ state funtion an be separated into eight spae-time dependent funtions. Opposite to the four solutions of the original Dira equation, this extended one has eight solutions, in aordane with the additional bivariant opposite positions of the IFCS. 3.2 Invariane of the extended Dira equation Let us introdue the following generalised momentum notations: p 4 = i p 4 + ρ T A 4 + ρ ) D 4 ; p = p + ρ T A + ρ D. Now the above extended Dira equation an be written as: [ ip 4 γ 5 σ, p ) + γ 4 m ] [ip 4 γ 5 σ, p ) + γ 4 m T ] ψ = We onsider that γ 4 = ρ 3, γ 5 = ρ 1, and apply the tau algebra for the isotopi field-harges. The isotopi field harges - both the eletri and the gravitational - are rotated by the same transformation matries in the isotopi field-harge spin field. Let s onstrut, on the analogy of the onstrution of γ i by the help of σ 2 the upper ase tau matries T i = τ 2 τ i. So T i = τ2 2 = δ 2 = E L. Multiply the expression in the first square braket with T 2 ϑ from right, and the expression in the seond square braket with T 2 χ from right, and both expressions in the brakets with ρ 3 from the left. The T operators do not affet either the derivative operators or the spin operator, and vie versa. So, they ommute within the individual terms. Aording to a similar observation, and an be dropped behind the square brakets. [ ρ3 T 2 ϑp 4 + iρ 2 σ, T 2 ϑp ) + T 2 ϑm ] [ρ 3 T 2 ϑp 4 + iρ 2 σ, T 2 ϑp ) + T 2 χm T ] ψ =

13 Quaternion/vetor dual spae algebras applied to the Dira equation 39 and onsequently, replaing again the γ matries for ρ and σ, the equation takes the form: [ iγ4 δ 4 p 4 + iγ, δp ) + m ] [iγ 4 δ 4 p 4 + iγ, δp ) + m T ] ψ ϑχ = where we applied the following notations: T 2 p 1 = δ 1 p 1 = τ 2 1 p 1; T 2 p 2 = δ 2 p 2 = τ 2 2 p 2; T 2 p 3 = δ 3 p 3 = τ 2 3 p 3; δ 1 τ 2 1 δ 2 = τ 2 2 = δ. δ 3 τ3 2 The same extended Dira equation an be written with four-omponents: [ iγµ δp ) µ + m ] [iγ ν δp ) ν + m ] ψϑχ = Exeuting the matrix multipliation, we an write: [ iγµ p µ + m ] [iγ ν p ν + m T ] ψ = where Γ µ = γ µ δ µ. This simplified form provides the extended form of the soure of the Dira equation in the presene of isotopi field-harges with the appliation of the tau algebra. All the rest an be alulated following my previous papers [9, 11, 12, 13]. The latter form demonstrates the ovariant harater of the equation. It was a long lasting paradigm in physis that equations of the physial interations must be subjet of invariane under the Lorentz transformation, and this is not only a neessary, but also a suffiient ondition. Now, we see this is not the ase. Nevertheless, no physial priniple stated that Lorentz invariane is a suffiient ondition demanded for the equations. There are other invarianes that may appear together and ombined with Lorentz s. How does this observation present itself in this ase? As mentioned above in setion 2.4, the appearane of the opposite isotopi field-harges in the p µ generalised momentum and the mass terms destroys the Lorentz invariane of the equation. In addition, there appears a new invariane. It is represented in the equation by the δτ) operators that rotates the isotopi field-harges in an additional abstrat field. The invariane of the extended Dira equation is resulted in the onvolution of the Lorentz- and this new IFCS) invariane. In other words, the extended Dira equation is invariant under the ombined transformation [11, 12]. The ombined Lorentz IFCS) invariane of the equation allows to reinterpret our earlier imagination on the invarianes of our physial equations. This requires ertain intendment hange in the approah to the world-piture, espeially in its relativisti extension, what an be summarised in the following. The new invariane desribed algebraially in this and introdued in the ited previous) papers is interpreted in the presene of a kineti gauge field. The Lorentz invariane is a ombined symmetry in itself. The SO + 3, 1) group of the proper Lorentz transformation an be haraterised by six independent subgroups. These six independent

14 4 György Darvas subgroups an be separated to and haraterised by three [4 4] rotation matries [Rϕ)] in the spae-time, and three [4 4] veloity boosts [Λẋ)] into a given diretion in the onfiguration spae = R Λ). Sine the IFCS invariane [Λẋ)] is interpreted also in a kineti field, it seems reasonable to substitute R Λ ) for the transformation. Although this lustering is only formal, we must mention that the reason is to assoiate the veloity dependent transformations Λ and ) with eah other, and formally separate them from the spae-time rotation R. 4 Conluding remarks Does the quaternion formalism bring any new information in the extended Dira equation? The algebra of the σ, ρ, and aordingly the γ matries represented a vetor algebra, they ould be transformed in a quaternion form, and disussed parallel. The ombined ovariane of the extended Dira equation introdued the Γ matries. Although the γ and the δ matries were subjets of a definite algebra separately, their produt Γ) matries do not show up any sign of a regular vetor or quaternion algebra. This does not ause a surprise. We saw in setion 1: while σ and ρ represent a omplete algebra, the γ matries, onstruted by their ombination, deline from that f., their ommutators). One ould have expeted further delination when the upper ase Γ matries were onstruted from the lower ase γ matries multiplied by elements hosen from the τ algebra. So there is no reason to speak about a vetor- or quaternion harater onerning the properties of the Γ matries. They mix the four omponents of the vetors in the Dira equation in a speial way that expresses the ombined invariane transformation. The question about a Finsler extension - at least in the present stage of the development of this ontexture - is not atual. Nevertheless, the extension of the relativity theory in this diretion and applied to the QED was not useless. It provided us with additional information that will be useful in future studies. Let us see how! Works in this field ould be extended with investigations into the third approah of Dira to the theory of the eletron [8], where he introdued a urvilinear o-ordinate system. There, he defined an auxiliary o-ordinate system y Λ,,,whih is kept fixed during the variation proess and use the funtions y Λ x)) to desribe the x o-ordinate system in terms of the y o-ordinate system. He defined the y system so that the metri for the x system be g µν = y Λ,µ y,ν. Λ This metri, whih then appeared in the Hamiltonian of the eletromagneti interation, was,,at least, aording to me the first step to a Finsler extension of the theory. My sinere hope is that one will ombine the extension of the original [5] 1928) Dira equation presented in this and the ited) papers) with the onlusions of that third 1962) attempt of Dira [8] to reate a more preise approximation the expression borrowed from Dira) to the relativisti quantum theory of the eletron. This paper made a step towards it. That expeted next step may make the Finsler extension of the relativisti quantum theory of the eletron omplete.

15 Quaternion/vetor dual spae algebras applied to the Dira equation 41 Nevertheless, one an find a few attempts in the literature to apply Dira s 1962 approah [8] to an extended relativisti QED. However, a) none of them referred to Dira [8] and b) none of them tried to ombine the Finslerian approah with the fully relativisti alled by us [ẋ]) transformation. I d like to mention only a few reent papers from among them e.g., [17, 18]). H. Brandt published several papers on Finslerian extensions of quantum field theory e.g., [19] speified to salar quantum fields) - with the mentioned defiienes. G. Munteanu - partly together with N. Aldea - made also attempts to suh extensions e.g., [2, 21]). The latter one was presented at the 1 th FERT onferene, 214, parallel with this paper. I assume that their results ould be ombined with those presented in this paper, beause they ould extend eah other. They apply Finsler metri and tangent bundles, while this paper onsiders strong relativisti generalisation and effets of isotopi field harges. Their treatment onsiders urved spae, but I am missing in it urvature of fields although, perhaps they are onsidered, but not speified). Both their and this papers define themselves as initial attempts towards a unified field theory, but none of them are omplete without the other. A possible ombining point should be found in setion 3 of their paper [21], where they refer to Born s reiproity priniple whih is based on a symmetry between the spae-time oordinates and momentum-energy oordinates, and where they pass from the omplex spae-time oordinates... to the omplex momentumenergy oordinates without defining the transformation group of this symmetry i.e., of the transition). This transformation is speified in [22]. I hope that ombined efforts ould lead to promising results. Referenes [1] Rowlands, P., Dual spaes, partile singularities and quarti geometry, 9th FERT, 213, 1p. [2] Rowlands, P., Symmetry in physis from the foundations, Symmetry: Culture and Siene, ), 1-4, [3] Rowlands, P., A null Berwald-Moor metri in nilpotent spinor spae, Symmetry: Culture and Siene, ), 2, [4] Rowlands, P., The null Berwald-Moor metri and the nilpotent wavefuntion, 8th FERT, 212. [5] Dira, P. A. M., The quantum theory of the eletron, Proeedings of the Royal Soiety A: Mathematial, Physial and Engineering Sienes, ), [6] Petoukhov, S.., Symmetries of the geneti ode, hyperomplex numbers and geneti matries with internal omplementarities, Symmetry: Culture and Siene, 23, 212) No. 3-4,

16 42 György Darvas [7] Dira, P. A. M., A new lassial theory of eletrons, Pro. Roy. So. A, ), [8] Dira, P. A. M., An extensible model of the eletron, Pro. Roy. So. A, ), [9] Darvas, G., The isotopi field harge spin assumption, Int. J. Theor. Phys., 5 211), [1] Darvas, G., Finslerian approah to the eletromagneti interation in the presene of isotopi field-harges and a kineti field, Hyperomplex Numbers in Geometry and Physis, ), 9, [11] Darvas, G., The isotopi field-harge assumption applied to the eletromagneti interation, Int. J. Theor. Phys., a), 11, [12] Darvas, G., Eletromagneti interation in the presene of isotopi fieldharges and a kineti field, Int. J. Theor. Phys., ), 1, [13] Darvas, G., A symmetri adventure beyond the Standard Model - Isotopi field-harge spin onservation in the eletromagneti interation, Symmetry: Culture and Siene, b), No. 1-4, [14] Darvas, G. 215) Algebra of the isotopi field-harge spin transformation, submitted) 7 p.. [15] Petoukhov, S.. Dyadi groups, dyadi trees and symmetries in long nuleotide sequenes, 212). [16] Petoukhov, S.. The geneti ode, algebra of projetion operators and problems of inherited biologial ensembles, 213). [17] Brinzei, N., Siparov, S. Equations of eletromagnetism in some speial anisotropi spaes, arxiv: v1 [gr-q] 15 p, 28). [18] Siparov, S., Introdution to the anisotropi geometrodynamis, Singapore: World Sientifi, 33 p, 212). [19] Brandt, H. E., Finslerian quantum field theory, arxiv:4713v2. [2] Munteanu, G., A Yang-Mills eletrodynamis theory on the holomorphi tangent bundle, Journal of Nonlinear Mathematial Physis, 17 21), No. 2, [21] Aldea, N. and Munteanu, G., A generalized Shrodinger equation via a omplex Lagrangian of eletrodynamis, Journal of Nonlinear Mathematial Physis, 22, 3215), [22] Darvas, G., Conserved Noether urrents, Utiyama s theory of invariant variation, and veloity dependene in loal gauge invariane, Conepts of Physis, I, 1, 29), 3-16.