aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug Electrodynamics in flat spacetime of six dimensions
Yurij Yaremko ∗ Institute for Condensed Matter PhysicsSvientsitskii Str. 1, 79011 Lviv, Ukraine (Dated: August 11, 2020)We consider the dynamics of a classical charge in flat spacetime of six dimensions. The mass shellrelation of a free charge admits nonlinear oscillations. Having analyzed the problem of on eigenvaluesand eigenvectors of Faraday tensor, we establish the algebraic structure of electromagnetic fieldin 6D. We elaborate the classification scheme based on three field’s invariants. Using the basicalgebraic properties of the electromagnetic field tensor we analyze the motion of a charge in constantelectromagnetic field. Its world line is a combination of hyperbolic and circular orbits which lie inthree mutually orthogonal sheets of two dimensions. Within the braneworld scenario, we projectthe theory on the de Sitter space of four dimensions. Actually, as it turns out, spins of elementaryparticles themselves are manifestations of extra dimensions.
I. INTRODUCTION
In the string theory, extra dimensions are required toensure the mathematical consistency of the theory [1].They are typically thought to be very small, close tothe Planck length. Dimensions higher than four arise inbraneworld scenarios where the observable (3 + 1) Uni-verse is embedded in a higher dimensional bulk (see [2]and references therein). The extra dimensions can bevery large or even infinitely extended. Search of manifes-tations of extra dimensions (e.g., tiny black holes [3, 4])continues at the Large Hadron Collider [5]. It is likelythat a higher energy is needed for the experiments to besuccessful.Thus. the question arises what are possible manifes-tations of extra dimensions when we put an electromag-netic interaction into a spacetime of higher dimensionappeared in braneworld scenarios. In the modern lan-guage of differential geometry the Maxwell equations inMinkowski space of D dimensions look as followsd ˆ F = 0 , d ⋆ ˆ F = Ω D − ∗ ˆ j, (1)where Ω D − is the area of unit ( D − M D , a givencurrent density ˆ j generates the electromagnetic fieldwhich is determined by the Faraday 2-form ˆ F and Hodgedual ( D − ⋆ ˆ F . As the the Faraday 2-form is ex-act ˆ F = d ˆ A , the equations can be easily transformed intowave equation (cid:3) A µ = − Ω D − j µ on the components ofthe 1-form potential ˆ A . Symbol (cid:3) denotes d’Alembertdifferential operator. In curved spacetime the problem isanalyzed in Ref. [6].For a point-like source the wave equation has beensolved with the help of either Green’s function method[7, 8] or elegant iterative procedure [9]. The solutionsextremely grow at the immediate vicinity of the pointwhere the charge is placed. To obtain sensible equation ∗ Electronic mail: [email protected] of motion of a point-like charge under the influence ofan external electromagnetic field as well as its own field,the regularization procedure is necessary. In Ref. [11]the derivation of analogue of the Lorentz-Abraham-Diracequation in six dimensions is patterned after Dirac]s sem-inal paper [10]. Kosyakov produced a proper relativis-tic equation of motion of radiating charge via analysisof energy-momentum conservation in 6 D . Alternatively,the method of removing inevitable infinities from Green’sfunction which uses functional analysis tools is developedin Ref. [12] for electrodynamics in an arbitrary dimen-sions.Consistent elimination procedure can be based on theconserved quantities arising from the symmetry of theproblem. Minkowski spacetime isometries constitute thegroup of symmetry of the electrodynamics in D di-mensions. The Poincar´e group denoted as P (1 , D − R ,D − and generalized orthogonalgroup, or Lorentz group O (1 , D − D ( D + 1) / P (1 , D − D . Theirelimination is not a trivial matter. Conservation laws arean immovable fulcrum about which tips the balance oftruth regarding renormalization, radiation reaction, andparticle’s individual characteristics. It turned out that insix dimensions, for a given δ -like current density ˆ j , tworenormalization parameters are necessary to absorb in-evitable divergencies coming from particle’s self-action.Apart from usual “bare mass” m , the particle actionterm involves an additional renormalization constant, say µ , which absorbs one extra divergent term. The parti-cle’s dynamics is governed by the action S part = Z d λγ − ( ˙ z ) (cid:18) − m − µ K ( ˙ z, ¨ z ) (cid:19) , (2)where γ = 1 / √− ˙ z is the Lorentz factor. The Lagrangiandepends on the squared curvature of the particle’s worldline K ( ˙ z, ¨ z ) = γ (cid:0) ¨ z + γ ( ˙ z · ¨ z ) (cid:1) . (3)Six functions z α ( λ ) where index α runs from 0 to 5 pa-rameterize the points z ( λ ) of particle’s world line ζ : R → M . The derivatives ˙ z α ( λ ) = d z α ( λ ) / d λ are the compo-nents of a tangent vector for ζ at the point z ( λ ) ∈ ζ .Two points over z α ( λ ) denotes the second derivatives¨ z α ( λ ) = d z α ( λ ) / d λ .It is worth noting that the action (2) governs the dy-namics of the rigid particle [15]. It is one-dimensionalapproximation [16, 17] of Polyakov-Kleinert string whoseaction integral contains, apart from Nambu-Goto term,an additional term depending on the curvature of theworld sheet of the string [15, 18, 19]. A classical pointcharge in renormalizable 6 D electrodynamics is not struc-tureless.Our concern in this paper is with the electrodynam-ics in six dimensions. We consider the behaviour of a test charge , i.e. a point charged particle which itself doesnot influence the field. In Section II we study inertialproperties of a free charge in six dimensions. They arehighly non-trivial because the Lagrangian for the point-like analogue of the Polyakov-Kleinert string depends onacceleration. In Section III we find particular solutionsof the Lorentz force equation in 6 D . In Section IV thealgebraic structure of electromagnetic field in six dimen-sions is investigated. We propose classification schemebased on three field’s invariants. In Section V we reducetwo extra dimensions by means of specific surjective dif-feomorphism that projects flat spacetime M onto four-dimensional se Sitter space. In Section VI, we summarizethe main ideas and results. II. FREE RIGID PARTICLE
The aim of this Section is to explain the mechanicalproperties of an elementary charge in six dimensions.The obvious difference from elementary charged parti-cles in four dimensions is that they have at their disposaltwo extra dimensions. It is important, but not princi-pled. The crucial difference between the two objects isthat the action (2) contains higher derivatives. For thisreason the relation between energy and momentum of afree rigid particle has highly non-trivial form. Before go-ing to the mass shell relation we consider the equationsof motion.
A. Hamiltonian equations of motion
For reader’s convenience we recall briefly the dynam-ics governed by the particle action (2). We follow theanalysis presented in Ref. [20] with the only differencebeing that the particle moves in flat spacetime of six di-mensions (see also [21, § z µ of particle’s six-velocityas new coordinates q µ we pass to the Lagrangian whichdepends on the first-order derivatives:˜ L = − p − q (cid:18) m + 12 µK ( q, ˙ q ) (cid:19) − Θ µ ( ˙ z µ − q µ ) , (4)where Θ µ are Lagrange multipliers. We assume that bothalready renormalized constants, m and µ , are finite andobservable parameters. Let us define the Hamiltonianfunction.Defining the canonical momenta P µ = ∂ ˜ L/∂ ˙ z µ and p Θ µ = ∂ ˜ L/∂ ˙Θ µ , we obtain two primary constraints P µ +Θ µ ≈ p Θ µ ≈
0. They have a non-zero Poissonbracket. Passing to the Dirac’s brackets, we exclude thepair (Θ µ , p Θ µ ) of canonically conjugated variables.For future convenience we introduce six-vector X µ ⊥ = X µ + γ ( q )( X · q ) q µ . As the scalar product ( q · q ) = − γ − ( q ), the vector is orthogonal to six-velocity, namely( X ⊥ · q ) = 0. Differentiating the Lagrangian (4) withrespect to ˙ q µ we obtain the six-momentum canonicallyconjugated to q µ : π µ = − µγ ( q ) ˙ q ⊥ ,µ . (5)There exists the constraint φ = ( π · q ) ≈ . (6)Having performed the Legendre transform we derivethe Hamiltonian function from the Lagrangian (4). Usingthe expression π = µ γ ( q ) K ( q, ˙ q ) we construct canon-ical Hamiltonian H = ˙ z µ P µ + ˙ q µ π µ + ˙ λ µ p λµ − ˜ L = P · q + γ − (cid:18) m + q π µ (cid:19) . (7)There is the secondary constraint φ = H ≈ φ ≈
0. As the Poisson bracket { φ , φ } = − φ , other constraints could not be found.The total Hamiltonian H ′ = H + υ φ produces thefollowing equations of motion˙ z µ = q µ , (8)˙ P µ = 0 , (9)˙ q µ = − γ − ( q ) π µ µ + υ q µ , (10)˙ π µ = − P µ + mγ ( q ) q µ − π µ γ − ( q ) q µ − υ π µ . (11)Comparing the equation (10) with the momentum de-fined by eq. (5) we obtain the multiplier υ = − γ ( q )( ˙ q · q ) . (12)Inserting this into expression (11) and substituting thetime derivative of the right hand side of eq. (5) for ˙ π µ wederive the total particle’s six-momentum P as functionof q , ˙ q , and ¨ q : P µ = γ (cid:18) m − µK ( q, ˙ q ) (cid:19) q µ + 3 µγ ( q · ˙ q ) ˙ q ⊥ ,µ + µγ ¨ q ⊥ ,µ . (13)We restrict ourselves to the sector of time-like worldlines ˙ z <
0, ¨ z ⊥ ≥
0. Since K = γ ¨ z ⊥ , the squaredcurvature (3) is positively defined here. Passing to theparametrization by the proper time τ , we express theconserved momentum in terms of normalized six-velocity u ν ≡ d z ν / d τ = γ ˙ z ν , six-acceleration a ν ≡ d u ν / d τ = γ ¨ z ν ⊥ , and its proper time-derivative ˙ a ν ≡ d a ν / d τ = γ (cid:2) ... z ν ⊥ + 3 γ (¨ z · ˙ z ) ¨ z ν ⊥ + γ ¨ z ⊥ ˙ z ν (cid:3) : P ν = mu ν + µ (cid:18) ˙ a ν −
32 ( a · a ) u ν (cid:19) . (14)By the normalized six-velocity u ( τ ) we mean the time-like tangent vector with norm 1: ( u · u ) ≡ η αβ u α u β = − K = ( a · a ). B. Energy, momentum and mass shell
Adapting [22, eq.(12)], we express the mass shell rela-tion in the form of the second order differential equationon the squared curvature − ( P · P ) = (cid:18) m − µK (cid:19) (cid:18) m − µK (cid:19) − µ ¨ K. (15)The other important relation is( P · u ) = − m + 12 µK. (16)The particular solution K = 0 yields the standard massshell of structureless particle: − ( P · P ) = m where six-momentum P and six-velocity u are collinear: P µ = mu µ . Further we are interesting in the mass shell of rigidcharged particle with positively defined squared curva-ture. In this case the momentum of charge and its six-velocity are not collinear. It depends on the particle’sacceleration and its time derivative (see eq. (14)).The equation (15) defines the non-linear oscillationsparameterized by Jacobi elliptic functions [23]. We over-multiply it on ˙ K and then integrate both the left sideand the right side of this equation. We express the re-sult as the sum of kinetic energy µ ˙ K / W ( K ) = − µ K / mµK − m + P ) K . Since thevibration of free rigid charge has no friction nor forcing, the energy (constant of integration) is conserved quan-tity: µ ˙ K / W ( K ) = E .To simplify further consideration we introduce the di-mensionless variables τ ′ = r mµ τ, k = µ m K, (17)and ruling parameters: δ = − ( P · P ) m , ε = µE m . (18)We obtain the first order differential equation (cid:18) d k d τ ′ (cid:19) = k − k + (1 − δ ) k + ε, (19)which simply expresses the fact that the total energy isthe sum of kinetic and potential energies. The roots ofthe cubic polynomial in the right hand side define thesolution to this equation. The oscillations take place ifand only if all the roots are distinct and real. They canbe expressed in terms of trigonometric functions k j ( δ, ε ) = 23 (cid:20) √ δ cos (cid:18) ψ ( δ, ε ) + 2 π j (cid:19)(cid:21) , (20)where index j runs from 0 to 2 and the phase is ψ ( δ, ε ) = 13 arccos − δ − ε/ δ ) / ! . (21)Energy parameter changes from minimum ε min ( δ ) = − h − δ + (1 + 3 δ ) / i , (22)where phase ψ = 0, to maximum ε max ( δ ) = 227 h − δ + (1 + 3 δ ) / i , (23)where ψ = π/
3. They are extrema points of potentialfunction V ( k ) which is pictured in Fig. 1.Factoring the cubic polynomial in the right hand sideof eq. (19) we rewrite it as follows (cid:18) d k d τ ′ (cid:19) = ( k − k ) ( k − k ) ( k − k ) . (24)According to the Handbook [24, 17.4.61], the solution is k = k + ( k − k ) sn ( w \ α ) , (25)where the argument of the elliptic sine is w = 12 p k − k τ ′ + φ , (26)and the modular anglesin α = k − k k − k . (27) V ( k ) ε k k k ε min ε max k FIG. 1. Graph of scaled potential V ( k ) = − k + 2 k − (1 − δ ) k with parameter 0 < δ <
1. Energy changes from ε min (minimum) to ε max (maximum). If ε = ε min the squaredcurvature is constant: k = k ( δ, ε min ) = k ( δ, ε min ). For agiven value of energy ε min < ε ≤ k oscillates between roots k and k . Together with zeroth root, k , they define the period of oscillation and its frequency. The phase φ is given by initial conditions. The solutionis presented in Ref. [23, eq. (17)].According to eq. (25), for a given parameter ε the vari-able k “moves” forward and backward from k to k (seeFig. 1). If ε is greater than 0, the orbit contains thesegment where k <
0. Such a trajectory is non-physical.The form of a curvature orbit heavily depends on thevalues of renormalization constant and parameter δ . If µ > δ >
1, the scaled potential V ( k ) does notadmit a bounded orbit without a “negative” segment. Inthis case positively defined solution [23, eq. (16)] containsthe term which is inversely proportional to the squaredelliptic sine; its argument and modular angle are given byeqs. (26) and (27), respectively. The solution describesnon-linear oscillations when the curvature goes to infinityover a period. C. The rest frame of a free rigid particle
In this Paragraph we solve the system of non-lineardifferential equations (14) where the squared curvature( a · a ) is given by eq. (25). To simplify the equations asmuch as possible we pass to the coordinate system wherethe translational momentum is P = 0. In this privilegedreference frame the expression (16) depends on the zerothcomponent of the normalized six-velocity only. Solvingit we obtain u ( τ ′ ) = 1 − k ( τ ′ ) √ δ , = 1 √ δ (cid:2) − k − ( k − k ) sn ( w \ α ) (cid:3) (28) where δ = ( P /m ) . Direct calculations show, that thefunction satisfies the equation of motiond u d τ ′ + 1 − k ( τ ′ )2 u = √ δ , (29)which is the zeroth part of eq. (14) in terms of dimen-sionless variables.To find out the zeroth coordinate function we integratethe zeroth component over the proper time variable. Us-ing the definition [24, 16.25.1] we obtain z ( τ ′ ) = z + 1 − k √ δ τ ′ − k − k ) p δ ( k − k ) Sn( w \ α ) . (30)Using the relation [24, 16.26.1] one can substitute the el-liptic integral of the second kind E( w \ α ) for the functionSn( w \ α ). z ( τ ′ ) τ ′ FIG. 2. Depiction of the evolution of time coordinate of freerigid particle. Space coordinates perform periodic non-linearoscillations parameterized by Jacobi elliptic functions.
At the minimum of the cubic potential pictured inFig. 1 the roots k = k = k min and, therefore, thesquared curvature given by eq. (25) does not change withtime. Inserting k min = (2 − √ δ ) / u = 1 + √ δ √ δ . (31)Rigid particle moves uniformly in time coordinate.In terms of dimensionless variables the space part ofeq. (14) looks as follows:d u i d τ ′ + 1 − k ( τ ′ )2 u i = 0 , (32)If ε = ε min , k ( τ ′ ) = k min and we deal the second or-der differential equation on the space components of six-velocity. The solution is u min = A cos ( ωτ ) + B sin ( ωτ ) , (33)where the squared frequency ω = 12 (cid:16) √ δ − (cid:17) . (34)By symbols A and B we denote the orthogonal five-vectors of equal magnitudes | A | = | B | = | u min | . Themagnitude of spatial five-vector of six-velocity is simply | u | = p ( u − u + 1). Inserting eq. (31) we obtain | u min | = s (cid:0) − √ δ (cid:1) (cid:0) √ δ − (cid:1) . (35)Recall that the parameter 0 < δ <
1. At the minimum ofcurvature the rigid particle moves along the helical line[22, 23].Near the minimum of potential pictured in Fig. 1 where ε > ε min but k − k ≪
1, the elliptic sine in the expres-sion for squared curvature (25) can be approximated bytrigonometric sine. Putting this in equation of motion(32) we obtain the Mathieu equation [24, S. 20]. We donot bother with it. Instead, we present the exact solutionof eq. (32) for zeroth energy level ε = 0. In this specificcase the roots k = 0, k = 1 − √ δ , and k = 1 + √ δ .The zeroth component of six-velocity takes the form u ( τ ′ ) = 1 √ δ h − (cid:16) − √ δ (cid:17) sn ( w \ α ) i , (36)where the argument w = p √ δ τ ′ + φ and the mod-ular angle sin α = (1 − √ δ ) / (1 + √ δ ). The magnitudeof five-vector u depends on elliptic functions: | u ( τ ′ ) | = r − δδ cn( w \ α )dn( w \ α ) . (37)We assume u ( τ ′ ) = C cn( w \ α )dn( w \ α ) where C isan arbitrary constant five-vector with length | C | = p (1 − δ ) /δ . Having integrated this function over timewe obtain coordinate five-vector z ( τ ′ ) = z + 2 C p √ δ sn( w \ α ) . (38)The solution describes the periodic orbit, repeating it-self in a sinusoidal fashion with constant amplitude andconstant frequency. III. RIGID CHARGE IN ELECTROMAGNETICFIELD
Following [11, 12], we introduce minimal coupling be-tween rigid charge e and an external electromagneticfield. We add the interaction term S int = e Z d λA µ ˙ z µ , (39)where A µ ( z ) are the components of the electromagneticone-form potential ˆ A = A µ d x µ evaluated at point z ( λ ) on the world line ζ where the charge e is placed. Varia-tion of the total action S = S part + S int yields the equa-tions of motion d P µ / d λ = eF µν q ν where the momen-tum P µ is given by eq. (13). Passing to the proper timeparametrization we obtain the analogue of the Lorentz-force equation of motion in six dimensions:dd τ (cid:20) mu µ + µ (cid:18) − ¨ u µ + 32 ( a · a ) u µ (cid:19)(cid:21) = eF µν u ν . (40)Here e is the magnitude of electric charge and F µν = ∂ ν A µ − ∂ µ A ν are the components of the electromag-netic field 2-form ˆ F = d ˆ A . If the Minkowski rectangu-lar coordinates are adapted, the state of electromagneticfield at point x ∈ M is specified by five components( E , E , E , E , E ) of electric field five-vector and tenelements F ik , i, k = 1 , i < k of the skew symmetricmatrix:( F αβ ) = − E − E − E − E − E E F − F F − F E − F F − F F E F − F F − F E − F F − F F E F − F F − F . (41)We rise index µ in equation (40) by means of the metrictensor η = diag( − , , , , , τ (cid:20) mu µ + µ (cid:18) − ¨ u µ + 32 ( a · a ) u µ (cid:19)(cid:21) = eF µν u ν . (42)Our task is to solve the equations of motion in the specificcase of static and spatially uniform electromagnetic field.First, we suppose µ ≪ a µ = em F µν u ν . (43)We replace the kinematic variables un the non-linearterms by differential consequences of this expression:˙ a µ = em F µν a ν = (cid:16) em (cid:17) F µν F ν α u α ; (44)¨ a µ = (cid:16) em (cid:17) F µν F ν α F αβ u β . For the squared curvature and its time derivative we ob-tain ( a · a ) = (cid:16) em (cid:17) η αβ F αµ F βν u µ u ν ; (45)d( a · a )d τ = (cid:16) em (cid:17) η αβ F ακ F κδ F βν u δ u ν . We take into account that tensor ˆ F is supposed to beconstant.The expressions look terribly horrible, but the matrix( F µν ) = E E E E E E F − F F − F E − F F − F F E F − F F − F E − F F − F F E F − F F − F (46)describes not only the electromagnetic field. It bears theimprint of the inertial frame which is used to determinethe components of electromagnetic field tensor. To solvethe non-linear differential equation (42) we apply thetechnique of projection operators [25, 26] based on theeigenvectors and eigenvalues of the electromagnetic fieldtensor (46). Eigenvectors constitute the basis definingthe linear transformation which makes this tensor diag-onal. The transformation can be easily transformed intoLorentz transformation which simplifies the equations ofmotion substantially.Looking ahead, there are opportunities to simplifysome specific field tensors to the form( F αβ ) ′ = b b , (47)or ( F αβ ) ′ = a − a , (48)etc. The most complicated tensor has the form( F αβ ) ′ = b b a − a c − c . (49)Field strengths a , b , and c are associated with eigenvaluesof the field tensor (46) (see eq. (85) in Section IV D).As the field strengths which constitute the tensor (46)are static and spatially uniform, its eigenvectors do notchange with points of Minkowski space. The subspacesspanned by these eigenvectors constitute foliation of M by two-dimensional planes. Traveling from point topoint, we construct the coordinate grid from these planeswhich covers all the flat spacetime. The particle’s worldline may be decomposed into three orbits in these mutu-ally orthogonal two-dimensional sheets. We insert matrix (49) into into equations of mo-tion (40) where kinematic variables in non-linearterms are replaced by the right-hand sides ofeqs. (44) and (45). We obtain three pairs of equa-tions defining the orbits in three mutually orthog-onal planes: M ( b ) = (cid:8) x ∈ M | ( x , x , x , x ) = 0 (cid:9) , M ( a ) = (cid:8) x ∈ M | ( x , x , x , x ) = 0 (cid:9) , and M ( c ) = (cid:8) x ∈ M | ( x , x , x , x ) = 0 (cid:9) :d u d τ = λ b (cid:16) − µ m υ + µm λ b (cid:17) u d u d τ = λ b (cid:16) − µ m υ + µm λ b (cid:17) u ; (50)d u d τ = ω a (cid:16) − µ m υ − µm ω a (cid:17) u d u d τ = − ω a (cid:16) − µ m υ − µm ω a (cid:17) u ; (51)d u d τ = ω c (cid:16) − µ m υ − µm ω c (cid:17) u d u d τ = − ω c (cid:16) − µ m υ − µm ω c (cid:17) u . (52)Here λ b = ( e/m ) b , ω a = ( e/m ) a , ω c = ( e/m ) c , and υ = λ b (cid:2) ( u ) − ( u ) (cid:3) + ω a (cid:2) ( u ) + ( u ) (cid:3) + ω c (cid:2) ( u ) + ( u ) (cid:3) . (53)The first pair produces the relation a u − a u = 0. Thisimmediately yields ( u ) − ( u ) = const . The solutionto eqs. (50) defines the hyperbolic orbit in plane ( x , x ): u ( τ ) = B cosh (Λ b τ + χ ) , u ( τ ) = B sinh (Λ b τ + χ ) . (54)The frequencyΛ b = λ b (cid:20) − µ m (cid:0) λ b B + ω a A + ω c C (cid:1) + µm λ b (cid:21) . (55)Constant parameter B and phase χ are defined by initialconditions.The others, eqs. (51) and (52), define circular orbits in( x , x )-plane and in ( x , x )-plane: u ( τ ) = A sin (Ω a τ + ϕ ) , u ( τ ) = A cos (Ω a τ + ϕ ) ; u ( τ ) = C sin (Ω c τ + φ ) , u ( τ ) = C cos (Ω c τ + φ ) . (56)The frequencies are as followsΩ a = ω a (cid:20) − µ m (cid:0) λ b B + ω a A + ω c C (cid:1) − µm ω a (cid:21) Ω c = ω c (cid:20) − µ m (cid:0) λ b B + ω a A + ω c C (cid:1) − µm ω c (cid:21) , (57)where constants A and C , as well as phases ϕ and φ ,are defined by initial conditions. Having integrated thesolutions with respect to proper time variable we obtainthe coordinate functions which define the charge’s worldline.Putting either ( a = 0 , c = 0) or ( b = 0 , c = 0) weobtain the solutions corresponding to field tensors (47)or (48), respectively.It is worth noting that the solutions (54) and (56) sat-isfy general equation of motion (40). The frequencies Λ b ,Ω a , and Ω c should satisfy the following system of spectralequations:Λ b h µ m (cid:0) Λ b B + Ω a A + Ω c C (cid:1) − µm Λ b i = λ b Ω a h µ m (cid:0) Λ b B + Ω a A + Ω c C (cid:1) + µm Ω a i = ω a Ω c h µ m (cid:0) Λ b B + Ω a A + Ω c C (cid:1) + µm Ω c i = ω c . (58)Crack these equations and you have got the frequencies.(Recall that eqs. (55) and (57) give approximated values.)It is not a trivial matter because the algebraic equationsare related to each other. The normalization condition( u · u ) = − B − A − C = 1. Λ b ( λ b ) λ b µ = 0 µ > FIG. 3. Illustration of inertial properties of a rigid chargeacted upon a constant electromagnetic field. Solid curvedepicts the graph of hyperbolic frequency function (60) for µ > Let us consider the motion of a rigid charge inthe field of electric type (47). The obvious particu-lar solution is u ν ( τ ) = ( B cosh( λ b τ + χ ) , B sinh( λ b τ + χ ) , A sin ϕ , A cos ϕ , C sin φ , C cos φ ). The frequencyΛ b is the root of the depressed cubicΛ b + µm (cid:18) B − (cid:19) Λ b = λ b . (59)As the parameter µ >
0, the only real root can beexpressed in terms of hyperbolic functions, i.e. Λ b = A b sinh ψ b where A b = s m µ (3 B − , ψ b = 13 arcsinh (cid:18) mλ b µ (3 B − A b (cid:19) . (60) If the electromagnetic field is switched off λ b = 0, thenΛ b = 0. If the parameter µ = 0, then Λ b = λ b . Therefore,the solution is of true physical sense.For the field of magnetic type (48) the particular so-lution of the Lorentz-force equation (40) is u ν ( τ ) =( B cosh( χ ) , B sinh( χ ) , A sin(Ω a τ + ϕ ) , A cos(Ω a τ + ϕ ) , C sin φ , C cos φ ). The frequency Ω a is the root ofthe depressed cubicΩ a + µm (cid:18) A + 1 (cid:19) Ω a = ω a . (61)The solution is Ω a = A a sinh( ψ a ) where amplitude A a and phase ψ a are given by eqs. (60) where ω a and 3 A +2are substituted for constants λ b and 3 B −
2, respectively.
IV. ALGEBRAIC STRUCTURE OF GENERALELECTROMAGNETIC FIELDS IN 6D
An elegant algebraic description of the electromagneticfield is based on its invariants, i.e. on scalar functions ofindependent components of Faraday tensor (41) whichserve as the parameters. The invariants are intimatelyconnected with eigenvalues and eigenvectors of this ten-sor. For a given square matrix (46) an eigenvector u isa non-zero column vector that only change by a scalar λ due to linear transformation described by this matrix:ˆ F u = λ u . (62)Here λ is known as the eigenvalue or characteristic rootassociated with the eigenvector u . The scalar is one ofroots of characteristic polynomial p ( λ ) = det | ˆ F − λ ˆ I | : λ + S λ − Q λ − P = 0 . (63)The coefficients S , Q , and P are the invariants of theelectromagnetic field in six dimensions. The coefficientsdo not depend on the choice of basis in M . Due tothe antisymmetry of tensor (41), the vector u is a null-vector. For a stationary and spatially homogeneous fieldthe eigenvectors are the “bricks” from which the Lorentztransformation to privileged inertial frame is designedwhere field strengths are functions of the invariants only. A. Invariants of electromagnetic field in 6D
It is straightforward to see that the invariants are the0-forms ⋆ ( ˆ F ∧ ⋆ ˆ F ), ⋆ ( ˆ F ∧ ˆ F ∧ ⋆ ( ˆ F ∧ ˆ F )), and ⋆ ( ˆ F ∧ ˆ F ∧ ˆ F ). Symbols ∧ and ⋆ denote the wedge product and theHodge star operator, respectively. We are manipulatingdifferential forms with the help of rules presented in theHandbook [27].We define 2-form ˆ G = ⋆ ( ˆ F ∧ ˆ F ) as the Hodge dual ofthe wedge product of the Faraday 2-form ˆ F = F αβ d x α ∧ d x β on itself. In terms of the Minkowski rectangularcoordinates the form ˆ G = G αβ d x α ∧ d x β is specified bythe skew symmetric matrix( G αβ ) = B B B B B − B G − G G − G − B − G G − G G − B G − G G − G − B − G G − G G − B G − G G − G . (64)The elements of this matrix involve the magneticfield five-vector which is specified by components( B , B , B , B , B ) where B i = G i . For example, B = F F − F F + F F . More generally, G αβ = 1(2!) ε αβγδµν F γδ F µν , (65)where the components F µν = η µα η νβ F αβ define 2-vector F = F αβ e α ∧ e β where ( e α , α = 0 ,
5) is the basis offor Minkowski space M and (d x α , α = 0 ,
5) is the dualbasis of 1-forms. The Levi-Civita symbol ǫ αβγδµν is atensor of rank six and is defined by 0 if any two labelsare the same, +1 if α, β, γ, δ, µ, ν is an even permutationof 0 , , , , ,
5, and − G contain components of electric field,for example G = E F + E F + E F .Direct calculations of det | ˆ F − λ ˆ I | result the followingcoefficients of characteristic polynomial (63): S = − ⋆ ( ˆ F ∧ ⋆ ˆ F )= 14! ε αβγδµν F αβ ε κσγδµν F κσ = − X i =1 ( E i ) + X i We substitute w for λ in the characteristic polynomial(63): w + S w − Q w − P = 0 . (71)Vieta’s formulae relate the coefficients of this cubic poly-nomial and its roots w , w , and w : w + w + w = −S , (72) w w + w w + w w = −Q , (73) w w w = P . (74)The coefficients are the functions (66)-(68) of the com-ponents of the Faraday tensor (41) which serve as theparameters. In this Paragraph we establish their domain.Cardano’s discriminant∆ = 18 SQP + 4 S P + S Q + 4 Q − P (75)determine the type of roots of the algebraic equation (71).The values of invariants of electromagnetic field S , Q ,and P do not depend on the choice of inertial frame ofreference where the electromagnetic field strengths aremeasured. When considering the general case P 6 = 0, itis convenient to use the Lorentz frame where the electricfield five-vector E and the magnetic field five-vector B arecollinear. In this specific reference frame the invariants(70) take the form S = − ( E ) + B ′ , Q = − ( B ) + ( E ) B ′ , P = E B , (76)where B ′ = ( F ) + ( F ) + ( F ) + ( F ) + ( F ) +( F ) . Inserting these into eq. (75), canceling like terms,and factoring we express the Cardano’s discriminant inthe form∆ = h ( F − F ) + ( F − F ) + ( F + F ) i × h ( F + F ) + ( F + F ) + ( F − F ) i × h(cid:0) E (cid:1) B ′ + (cid:0) E (cid:1) + (cid:0) B (cid:1) i . (77)Since ∆ > 0, the equation (71) has three real distinctroots.The invariants S and Q can not be negative simul-taneously. If S < E ) > B ′ and, therefore, Q = − ( B ) + ( E ) B ′ > − ( B ) + ( B ′ ) . Factoring thisexpression we obtain − ( B ) + ( B ′ ) = (cid:0) f − + f +53 + f − (cid:1) (cid:0) f +23 + f − + f +34 (cid:1) , where all the terms f ± = F ± F F + F , f ± = F ± F F + F , and f ± = F ± F F + F arenon-negative. Consequently, the condition S < Q > 0. If we choose S > 0, then Q can be either negativeor positive. Taking into account the Vieta’s formulae(72)-(74) we conclude that the characteristic polynomial(71) possesses one positive root, say w , and two negativeones, w and w .Our next task is to establish the domain of invariantsof electromagnetic field in six dimensions. To visualizethe tree-dimensional domain of function (75) we reduceit to a flat map. We pass to the dimensionless variables ω = w |S| , x = QS , y = P S . (78)The polynomial (71) simplifies ω + sqn( S ) ω − xω − sqn( S ) y = 0 . (79)The transformed Cardano’s discriminant ∆ ′ = ∆ / S be-comes ∆ ′ = 18 xy + 4 y + x + 4 x − y = 27 (cid:2) y + ( x ) − y (cid:3) (cid:2) y − y − ( x ) (cid:3) , (80)where y ± ( x ) = 227 (cid:20) x ± (1 + 3 x ) / (cid:21) . (81)The domain is the area in the ( xy )-plane bounded by thecurves y − ( x ) and y + ( x ) (see Fig. 4).The limiting curves y + ( x ) and y − ( x ) are projections ofthe two-dimensional surface P = 227 (cid:20) S + 92 QS + (cid:0) S + 3 Q (cid:1) / (cid:21) . (82)on the regions S + = (cid:8) ( Q , S ) ∈ R |Q > −S / , S > (cid:9) and S − = (cid:8) ( Q , S ) ∈ R |Q > , S < (cid:9) , respectively. Onthe limiting surface (82) the Cardano’s discriminant (75)vanishes. The cubic polynomial (71) has a simple rootand a double root in this case. The regions S + and S − are depicted in Fig. 5. C. Classification of fields We base our classification of electromagnetic fields insix dimensions on the analysis of the characteristic poly-nomial (63). Fields are divided into classes according towhether P 6 = 0 or P = 0 as well as whether S < S > 0. The other invariant is Q > −S / y ✻ x ✲ − 14 427 FIG. 4. Domain of Cardano’s discriminant (80). The shadedarea pictures the region of possible values of field’s invari-ants. It is bounded by the curve y + ( x ), x ∈ [ − / , + ∞ [, theinterval [ − / , 0] on the x -axis, and the curve y − ( x ) where x ∈ [0 , + ∞ [. (A) P 6 = 0 . There are three distinct real roots of thereduced characteristic polynomial, one positive andtwo negative. The set of eigenvalues consists ofthree pairs, one pair of real numbers, and two pairsof pure imaginary numbers.(B) P = 0 , Q 6 = 0 . There is one zero root of the reducedcharacteristic polynomial.Since the sign of Q is invariant, there are two pos-sibilities:(a) If Q > 0, there are two non-trivial roots ofthe reduced characteristic polynomial, posi-tive and negative. The set of eigenvalues con-sists of double degenerate non-defective zeroand two pairs, one real and one pure imagi-nary. Such a field has four-dimensional ana-logue which belongs to the set of crossed fields.(b) If −S / < Q < 0, there are two negativenon-trivial roots of the reduced characteristicpolynomial. The set of eigenvalues consists ofdouble degenerate non-defective zero and twopairs of pure imaginary numbers. The fieldhave not analogue in conventional electrody-namics.(C) P = 0 , Q = 0 , S 6 = 0 . There are double degeneratezero and one non-trivial real root of the reducedcharacteristic polynomial. Characteristic polyno-mial has fourth degenerate zero eigenvalue whichcan be either defective or non-defective.If zero is non-defective eigenvalue, there are twopossibilities:(a) If S < 0, the non-trivial root of the reducedcharacteristic polynomial is positive. The set0of eigenvalues consists of fourth degeneratezero and pair of real numbers, negative andpositive. Such a field has four-dimensionalanalogue which is said to be of electric type .(b) If S > 0, the non-trivial root of the reducedcharacteristic polynomial is negative. Theset of eigenvalues consists of fourth degen-erate zero and pair of pure imaginary num-bers. Such a field has four-dimensional ana-logue which is called magnetic type field.If zero is defective eigenvalue, the field matrix innon-diagonalizable. The Jordan normal form con-sists of two Jordan blocks: one-dimensional zeroblock and 3 × P = 0 , Q = 0 , S = 0 . If all the “six-dimensional” invariants are equal tozero, we deal with the null field. D. Eigenvalues Let us consider general case (A) P 6 = 0. The sub-set of domain od discriminant (75) consists of all thepoints on the map projection Fig. 4, excepting the ray x ∈ [ − / , + ∞ [ on the abscissa axis. The roots of cubicpolynomial (71) has three distinct real roots which canbe expressed in terms of trigonometric functions w k = − S + 23 p S + 3 Q cos (cid:18) ψ + 2 π k (cid:19) . (83)The angle ψ is given by ψ ( S , Q , P ) = 13 arccos P − S − QS ( S + 3 Q ) / ! . (84)The roots are ordered as w < w < w . The largest root w = b is positive, the others are negative: w = − a and w = − c .The set of eigenvalues consists of three pairs, one realand the others pure imaginary { λ } = { + b, − b, +i a, − i a, +i c, − i c } . (85)Any characteristic root from this set is associated withcorresponding eigenvector being the solution of matrixequation (62). In Appendix A we construct a squarematrix U whose columns are the six linearly independenteigenvectors ranged according to the range of eigenvaluesin the list (85). As the matrix invertible, we diagonalizethe field tensor F αβ = η αµ F µβ as follows L = U − ˆ F U. The matrix L is composed from the eigenvalues (85) onthe diagonal. However, the diagonal matrix does not describe an electromagnetic field. To build the electro-magnetic field tensor we modify the transformation toeigenbasis by additional linear transformation defined bythe matrix( J αβ ) = 1 √ − − i 1 0 00 0 0 0 i 10 0 0 0 − i 1 . (86)We obtain the electromagnetic field tensor ˆ F ′ = J − LJ which depends on the field’s invariants only (see eq.(49)).Looking at the chain of transformationsˆ F ′ = J − LJ = J − U − ˆ F U J = Λ − ˆ F Λ (87)we see that the tensor (49) defines the electromagneticfield in the privileged reference frame which is related tothe initial inertial frame by the Lorentz matrix Λ = U J which is presented in Appendix A. P = 0 , Q 6 = 0 : ( Q , S ) plane If the electric field and the magnetic field are mutuallyorthogonal in a given inertial frame, they are orthogonalin any other frame of reference. If P = 0. In the pro-jection map Fig. 4 the points lie on the ray x ≥ − / y = 0. It is worth noting that the coordinate origin isthe punctured point. The ray corresponds to the shadedregion in the ( Q , S ) plane which is depicted in Fig. 5. S ′ Q FIG. 5. Domain of Cardano’s discriminant (75) in ( QS )-plane. The shaded area pictures the region of possible valuesof field’s invariants. It is bounded by the curve Q = −S / S ∈ [0 , + ∞ [, and the negative ordinate half-axis. If P = 0, the cubic polynomial (71) simplifies: w (cid:0) w + S w − Q (cid:1) = 0 . (88)1The roots are 0 and w ± = (cid:0) −S + √S + 4 Q (cid:1) / 2. So faras arranging is concerning, there are two possibilities. a. Q > . The roots are arranged as follows: w ( S , Q , P ) | P =0 = w + ( S , Q ) ,w ( S , Q , P ) | P =0 = 0 ,w ( S , Q , P ) | P =0 = w − ( S , Q ) . The spectrum consists of two pairs, one real and the otherpure imaginary, and double degenerate zero { λ } = { + b, − b, +i a, − i a, , } , (89)where b = p w + ( S , Q ) and a = p − w − ( S , Q ). This elec-tromagnetic field has analogue in conventional electrody-namics. In the limiting case (69) we obtain the so-called“crossed field”. In the privileged inertial frame the elec-tromagnetic field tensor simplifies( F αβ ) ′ = b b a − a . (90) b. −S / < Q < , S > . Both the non-trivialroots are negative: w ( S , Q , P ) | P =0 = 0 ,w ( S , Q , P ) | P =0 = w + ( S , Q ) ,w ( S , Q , P ) | P =0 = w − ( S , Q ) . The spectrum consists of double degenerate zero and twopairs, both pure imaginary { λ } = { , , +i a, − i a, +i c, − i c } , (91)where a = p − w + ( S , Q ) and c = p − w − ( S , Q ). Thiselectromagnetic field has no analogue in conventionalelectrodynamics. In the privileged inertial frame the elec-tromagnetic field tensor takes the form( F αβ ) ′′ = a − a c − c . (92)The Lorentz matrices transforming to fields (90) and (92)are presented in Appendix A. P = 0 , Q = 0 : S -line If both the coefficients Q and P in the characteristicpolynomial (63) vanish, the spectrum contains fourth de-generate zero: λ (cid:0) λ + S (cid:1) = 0 . (93) It contains also the pair of distinct eigenvalues, eitherreal ±√−S if S < ± i √S if S > rank ( F ) | Q =0 , P =0 = 4 . The geometric multiplicity is less than the algebraic one.It is easy to show that rank ( F ) (cid:12)(cid:12) Q =0 , P =0 = 3 , rank ( F ) (cid:12)(cid:12) Q =0 , P =0 = 2 , and ranks of the matrix F in higher powers no longerdecrease. Therefore, zero is defective eigenvalue. TheJordan normal form consists of two Jordan blocks: one-dimensional zero block and 3 × S ,the field matrix in the eigenbasis takes the form, either L (cid:12)(cid:12) S < = + b − b , (94)or L (cid:12)(cid:12) S > = c 00 0 0 0 0 − i c . (95)Here b = √−S and c = √S . Such an electromagneticfield has no limit in four dimensions. We are interestedin the diagonalizable matrices because diagonal matriceshave equivalents in four dimensions. In this case zeroeigenvalue has four-dimensional eigenspace and the Jor-dan normal form is 4 × P = 0 and Q = 0, additional condi-tions on Faraday tensor (46) are necessary which provide rank ( F ) = 2.To establish them we pay attention to the matrix G µβ = η µα G αβ where G αβ is given by eq. (64). Wesubtract Λ from the diagonal to find G − Λ I and calcu-late its determinant. The characteristic polynomial mustbe zero: Λ + Q Λ − SP Λ − P = 0 . (96)Putting Λ = W we transform it in the cubic polyno-mial. Its Cardano’s discriminant is proportional to thediscriminant (75) of the characteristic polynomial (63): D = P ∆. If the invariant P 6 = 0, there are three real dis-tinct roots, one positive and the others negative. Trivialanalysis of corresponding Vieta’s relations yields W = w w , W = w w , W = w w , P = 0, the matrix F becomes degen-erate: rank ( F ) = 4. The conjugated matrix (64) is de-generate too: rank ( G ) = rank ( F ) − 2. To illustrate thesituation we consider the field tensor of rank 4:( F µν ) = E E E E F − F E − F F E F − F . (97)Its Hodge dual counterpart( G µν ) = G − G . (98)is of rank 2.If both the invariants Q and P vanish, the spectrumequation (63) admits fourth degenerate zero while thespectrum equation (96) becomes simple Λ = 0. Impos-ing the condition G ≡ E F + E F + E F = 0on the electromagnetic field (97) we obtain the matrixwhich has rank 2. All the elements of the conjugatedmatrix (98) are identically equal to zero in this case.To construct the field matrix (46) of rank 2 we shouldprovide resetting to zero of elements of the matrix (64). Itis sufficient to equate to zero the elements G ij , G ki , G jk and B i , B j , B k , and solve the algebraic equations withrespect to the elements F ij , F ki , F jk and E i , E j , E k ofmatrix (46). There are ten triples ( ijk ) if the indices runfrom 1 to 5, e.g. E = − E F + E F F , F = F F − F F F E = E F + E F F , F = F F − F F F ,E = − E F + E F F , F = F F − F F F . Diagonalization of this matrix yields the Jordan normalform being 4 × S < 0, the primed electromagnetic field tensor hasthe form (47). Such a field is said to be of electric type . If S > 0, the field tensor is given by eq. (48). Its analoguein the conventional electrodynamics is called magnetictype field. With the help of additional linear transfor-mations they are rearranged into skew symmetric matri-ces describing electromagnetic fields in specific inertialframes (see Appendix A). V. COMPACTIFICATION OF TWO EXTRADIMENSIONS Our task in this Section is to reduce the dimensional-ity of six-dimensional Minkowski space and pass to theordinary electrodynamics in four dimensions. We shallperform the procedure in two steps. At first we map theflat spacetime M onto the five-dimensional hyperboloid H η AB χ A χ B = 1 H , (99)and then repeat the surjection via passing to the four-dimensional hyperboloid in H . The hyperboloid rep-resents the de Sitter spacetime which is the solution ofthe Einstein field equation with cosmological constant Λ.The constant Λ = 3 H is related to the inverse lengthparameter involved in eq. (99). The de Sitter space withconstant curvature R = 4Λ has ten Killing vectors, i.e.is maximally symmetric. The parameter H is involvedin the de Sitter group governing the kinematics in the deSitter space. In the cosmological parameter limit Λ → M . In-deed, Λ = 0 means the absence of gravitation: the flatspacetime is a solution of the sourceless Einstein equa-tion.To parameterize the points of hyperboloid (99) we usethe five-dimensional generalization of stereographic coor-dinates [28] χ a = Ω( X ) X a ,χ = − H Ω( X ) (cid:18) − H (cid:19) , (100)whereΩ( X ) = (cid:18) H (cid:19) − , Σ = η ab X a X b . (101)Small Latin indices run from 0 to 4. Further we mapthe 5-dimensional hyperboloid H onto the 4-dimensionalhyperboloid H defined by the equation η ab X a X b = 1 h . (102)It is parameterized by the four-dimensional stereographiccoordinates [28]: X α = ω ( x ) x α ,X = − h ω ( x ) (cid:18) − h σ (cid:19) , (103)where ω ( x ) = (cid:18) h σ (cid:19) − , σ = η αβ x α x β . (104)3Small Greek indices run from 0 to 3. Composition ofthese surjections is written as χ α = Ω( κ ) ω ( x ) x α ,χ = − h Ω( κ ) ω ( x ) (cid:18) − h σ (cid:19) ,χ = − H − κ κ , (105)where Ω( κ ) = (cid:0) κ (cid:1) − depends on the dimensionlessconstant κ = H/ (2 h ). The de Sitter metric on H isinduced by the metric in the flat spacetime M :d s = η AB d χ A d χ B (cid:12)(cid:12) H = Ω ( κ ) ω ( x ) η αβ d x α d x β . (106)Let us define the cosmological constant limit when boththe constants H and h converge to 0 simultaneously,while their fraction κ remains a finite constant. In thisassumption the function ω ( x ) in eq. (106) is equal to 1.To obtain the standard flat space metric we scale theMinkowski coordinates Ω( κ ) x α → x α .Now restricting the electromagnetic field two-form (41)to the de Sitter space H we obtain the field 2-form infour dimensionsˆ f = 12 Ω ( κ ) ω ( x ) (cid:20) F αβ + hω ( x ) ( x α F β + F α x β ) − h ω ( x ) ( x α F νβ x ν + F αν x ν x β ) (cid:21) d x α ∧ d x β . (107)In the cosmological constant limit it takes the form F αβ d x α ∧ d x β where we use the scaled Minkowski co-ordinates.In six dimensions the particle dynamics is governedby the action (2). In order to restrict the dynamics onthe hyperboloid one should take into account two holo-nomic constraints: z = const and η ab z a z b = const . Re-call that small Latin indices run from 0 to 4. We donot bother with Lagrange multipliers. For treating con-straints we use the transformation (105) and pass to thecoordinates that are perfectly adapted to the above con-straints. In the cosmological constant limit the trans-formed Lagrangian takes the form L part = γ − ( ˙ z ) (cid:18) − m − µK D ( ˙ z, ¨ z ) (cid:19) , (108)where γ ( ˙ z ) is the Lorentz factor and K D ( ˙ z, ¨ z ) is the four-dimensional counterpart of the curvature (3). In order toobtain the standard Lagrangian of a structureless pointcharge in four dimensions we should put µ = 0. How-ever, we can interpret this renormalization constant asthe desired manifestation of extra dimensions. Not onlythe constant, but also the term coupled with it. Thelatter yields the additional terms to the standard orbitalmomentum: M αβ = z α P β − z β P α + q α π β − q β π α . After quantization, the squared curvature produces thespin one-half states [15]. According to this paper, “...the algebra of the Dirac brackets between the dynamicalvariables associated with velocity and acceleration con-tains the spin tensor”, namely s αβ = q α π β − q β π α = µ (cid:0) a α u β − a β u α (cid:1) . (109)Let us evaluate the magnitude of constant µ for electron.The square of the spin tensor is s αβ s αβ = − µ ( a · a ) := − µ K . According to Section II C, the squared curva-ture K = (2 m/µ ) k where dimensionless variable k < µ e ∼ ~ m e ≃ . · − Js , where ~ is the Planck constant and m e is electron’s massin SI units. (We take into account that the squared ac-celeration contains the factor c .) VI. CONCLUSIONS Within the braneworld scenario, we consider dynamicsof a classical charge in flat spacetime of six dimensions.Charge’s electromagnetic field satisfies the Maxwell equa-tions. A consistent regularization procedure which ex-ploits the Poincar´e symmetry of the theory results theparticle action functional which contains, apart fromusual “bare” mass, an additional renormalization con-stant coupled with the squared curvature of particle’sworld line [11, 13]. The mass shell of a free charge de-pends on the squared six-acceleration. This circumstanceyields non-trivial inertial properties of rigid charge. Themass shell admits the time-like periodic orbit parameter-ized by the Jacobi elliptic functions.We study the algebraic properties of electromagneticfield in 6D. We find three invariants of this field, estab-lish their domain, and elaborate the classification scheme.We study the evolution of a charge acted upon a statichomogeneous electromagnetic field. Privileged referenceframes are defined for various types of field where it issimplified substantially. In this frame the world line isthe combination of hyperbolic and circular orbits in threemutually orthogonal sheets of two dimensions.Extra dimensions have been compactified by means ofprojection onto the four-dimensional de Sitter space em-bedded in the flat spacetime of six dimensions. We ob-tain the action integral point particle with rigidity/ Themodel is quantized in Ref. [15]. It is shown that thesquared curvature leads to the spin states. Therefore,spins of elementary particles indicate presence of extradimensions.4 ACKNOWLEDGMENTS This research has been supported by Grant No.0117U002093 of the National Academy of Science ofUkraine. Appendix A: Privileged reference frames In this Appendix we present Lorentz matrices definingthe inertial frames where the field strengths are functionsof invariants (66)-(68) only. We start with general case P 6 = 0 (see item (A) in the Classification list IV C). P 6 = 0 . Solving the matrix equation ˆ F u = λ u for a giveneigenvalue λ from the set (85) we derive the eigenvec-tor u λ = ( u λ , u iλ ) where zeroth component is completelyarbitrary and the space components are u iλ = u λ λ (cid:2) λ + (cid:0) S + E (cid:1) λ + B (cid:3) ×× (cid:0) λ E i + λ A i + λ C i + λC i + P B i (cid:1) . (A1) Here E i and B i are the components of the electric field E and magnetic field B five-vectors, respectively. The othercapital letters denote the components of five-vectors A , C , and C : A i = X j = i F ij E j ,C i = ( − i +1 X j 0, the spectrum consists of double degeneratezero and two pairs, one real and the other pure imaginary(see eq. (89)). We design the auxiliary matrix as follows( J αβ ) ′ = 1 √ − − i 1 0 00 0 0 0 √ √ . (A15)Putting c = 0 in eqs. (A4), (A5), (A8), and (A9) we ob-tain four columns of matrix Λ = U J ′ defining the trans-formation to privileged inertial frame. Keeping in mindeq. (A14) we assume the fifth column of matrix U as u = 0 , u i = B i √ B . (A16) The sixth column is u = √ B ab , u i = C i ab √ B . (A17)In the privileged inertial frame the electromagnetic fieldtensor is given by eq. (90).If −S / < Q < 0, both the squared eigenvalues (A12)are negative. The spectrum consists of the double degen-erate zero and two pairs of pure imaginary numbers (seeeq. (91)). We define the auxiliary matrix( J αβ ) ′′ = 1 √ √ √ − i 1 0 00 0 0 0 i 10 0 0 0 − i 1 , (A18)The first column of the Lorentz matrix defining transfor-mation to the privileged reference frame is u = √ B ac , u i = C i ac √ B , (A19)while the second column is given by eq. (A16). The otherscan be derived from the expressions (A5), (A6), (A9), and(A10) where b = 0. In the privileged inertial frame theelectromagnetic field tensor is given by eq. (92). P = 0 , Q = 0 , S 6 = 0 . In this Paragraph we consider the item (C) of theClassification list IV C. We restrict ourselves to the non-defective zero eigenvalue when Jordan canonical formconsists of four Jordan blocks composed from zeroes.Direct calculations produce the following components ofzero eigenvectors: u = (cid:18) u i , E u + F k u k F , − E u + F k u k F (cid:19) T , (A20)where four components u i , i = 0 , , , 3, are completelyarbitrary and index k runs from 1 to 3.As all the elements of Hodge dual matrix (64) van-ish, five-vectors C and C are identically equal to zero.Putting these and Q = 0 in eq. (A13) we obtain the space6components of eigenvectors associated with the non-zeroeigenvalues: u iλ = u λ E (cid:0) λE i + A i (cid:1) . (A21)Zeroth components u λ are completely arbitrary. The for-mula gives two columns of matrix U which diagonalizesthe electromagnetic field tensor. The scalar products are( EA ) = 0 and A = E (cid:0) S + E (cid:1) .According to the Classification list IV C, there are twopossibilities.If S < 0, the spectrum consists of pair + b and − b where b = √−S and forth degenerate zero. We definethe auxiliary matrix in the form( J αβ ) = 1 √ − √ √ √ √ . (A22)Simple calculations yield the first column of the Lorentz matrix Λ = U J Λ = | E | b , Λ i = A i b | E | , (A23)while the second column isΛ = 0 , Λ i = E i | E | . (A24)The others should be composed from eigenvectors (A20)by means of Gram-Schmidt process. In the privilegedreference frame the electromagnetic field tensor is givenby eq. (47).If the invariant S > 0, the Lorentz matrix is com-posed from four columns obtained with the help of Gram-Schmidt process, with column (A24), and with modifiedcolumn (A23) where b should be replaced by a = √S . 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