Formulae for Energy and Momentum of Relativistic Particle Regular at Zero-Mass State

Journal of Modern Physics, 2011, 2, 849-856 doi:10.4236/jmp.2011.28101 Published Online August 2011 (http://www.SciRP.org/journal/jmp)

Robert M. Yamaleev1,2 1Joint Institute for Nuclear Research, LIT, Dubna, Russia 2Departamento de Fsica, Facultad de Estudios Superiores, Universidad Nacional Autonoma de Mexico Cuautitlán Izcalli, Campo 1, México E-mail: [email protected] Received October 22, 2010; revised March 2, 2011; accepted April 13, 2011

Abstract

In this paper we substantiate a necessity of introduction of a concept the counterpart of rapidity into the framework of relativistic physics. It is shown, formulae for energy and momentum defined via counterpart of rapidity are regular near the zero-mass and speed of light states. The representation for the energy-momentum is realized as a mapping from the massless-state onto the massive one which looks like as a “q”-deformation. Quantization of the energy, momentum and the velocity near the light-speed is presaged. An analogue between the relativistic dynamics and the statistical thermodynamics of a micro-canonical ensemble is brought to light.

Keywords: Relativistic Dynamics, Complex Algebra, Rapidity, Energy-Momentum, Background Energy, Statistical Thermodynamics

1. Introduction of the relativistic massive particle are singular near the velocity equal to speed of light, whereas they are well de- The developments of the basic theories in the fields of fined for the rest state. In general, it is supposed that near solid state and elementary particles exhibit a crucial im- the speed of light the dynamics of a massive elementary portance of the behavior of the physical systems near the particle more does not obey the classical mechanics of a critical points. The experimental results and their theo- single particle and one must work in the scope of the retical treatments have discovered the new physical phe- quantum field theory. In fact, acceleration of the charged nomena named as spontaneously broken symmetries near particle induces radiation of electromagnetic fields and the ground state. That a ground state of a quantal system leads to cumulative process of creation of a cascade of need not possess the Hamiltonian’s symmetries, and elementary particles. Noteworthy a crucial gap between therefore degenerate, was first appreciated and realized in massive and massless states: the particle with extremely non-relativistic many body systems and in many con- small value of mass can stay in the rest state, meanwhile densed matter situations. Also that had been realized, the its neighbour, the particle with m =0, has to move with state of superconductivity possesses with lower entropy the speed of light. (hence, with higher order) then a normal state. Following The irregular behavior of the conventional representa- this understanding Heisenberg [1] and Nambu [2] made tions for energy-momentum near the zero-mass point is a the seminal suggestion that this may also be true for the well-known problem of the relativistic mechanics. In fact, vacuum state of a relativistic quantum field theory. the celebrated formulae for the energy-momentum pp0 , For the relativistic mechanics the state with m =0, defined via velocity v , vc= , i.e., the state with proper mass equal to zero and mv mc2 pcp=, = (1.1) the velocity equal to speed of light, is a singular point of 0 vv22 the theory. The massive particles, according to laws of 11 cc22 the relativistic mechanics, cannot attain the velocity equal to speed of light. The formulae of the Lorentz could not be used to obtain any reasonable limit at the transformations and formulae for the energy-momentum points vc= , m =0, because the indeterminacy of type

0 An important feature of the counterpart of rapidity is , meanwhile, the energy and momentum of the particle 0 that the hyperbolic angle is proportional to the proper with vcm=, =0 are given by a certain finite value. In mass of a particle. The part of the hyperbolic angle in- fact, in order to prove this assertion we may use the dependent of the mass can be interpreted via the concept expression of the energy via momentum of background energy. The relativistic energy-momentum within the framework of the new representation formally Ecpmc= 222 (1.2) coincides with formulae of statistical thermodynamics of an single oscillator which allows one to give an inter- For small values of the mass, mc p , we can use pretation of the background energy by introducing the the following approximation quantities of the state like temperature and entropy. mc22 mc 23 The paper besides of the Introduction and Conclusions Epc=1 = pc  (1.3) p2 2 p is presented by the following sections. Section 2 presents elements of the relativistic dynamics

Hence, when m  0 , pp0  the velocity tends to of charged particle. In Section 3, an evolution generated speed of light, vc/= p / p0  1. The energy and mo- by mass-shell equation is explored. Transmission between mentum at this limit numerically are equal to each other translation and hyperbolic rotation is established. In Sec- and equal some finite value. tion 4, the representation for the momenta is modified by The following question gives arise: what kind of the introducing a fundamental constant of mass. The modified law of relativistic dynamics can help us to solve this formula is interpreted as a mapping from massless state indeterminacy? Or, in the other words, how we must mo- onto the state with nontrivial mass. A hypothesis on quan- dify, or generalize, the conventional frames of the theory tization of the velocity near light velocity is suggested. In in order to elaborate some pathway from the state with Section 5, an analogue between formulae of relativistic non-vanishing proper mass to the state with zero-mass? mechanics and formulae of statistical thermodynamics is In order to answer these questions we have to deepen demonstrated. the concept of rapidity. The rapidity is well-known quantity of the relativistic kinematics, this is an hyperbolic 2. Elements of Relativistic Dynamics of angle used as a parameter of the Lorentz-boost in the Charged Particle Lorentz group of transformations. Most physicists main- tained that the rapidity is a merely formal quantity be- In this section we will remind only selected elements of cause the rapidity did not receive any physical interpre- the relativistic dynamics of charged particle necessary in tation in the framework of the relativistic kinematics [3]. subsequent sections. Nevertheless, it had been noted that the rapidity plays an Consider a motion of the relativistic particle with important role in establishing some link between the charge e in the external electromagnetic fields E and hyperbolic geometry and relativistic kinematics. This link B . The relativistic equations of motion with respect to firstly had been discovered by V. Varicak [4], H. Hergoltz the proper time  are given by the Lorentz-force [5] and A. A. Rob [6]. Contemporary model of relativistic equations [8]: kinematics based on non-associative algebras and on the dp eedp e concept of gruppoids had been constructed by A. Ungar =EpBEp p  ), 0 = (2.1) ddmc0 m mc [7]. According to our point of view, the rapidity is one of ddrpct p =, =0 (2.2) principal notions of the relativistic dynamics. An ess- ddmm ential role plays a quantity dual to the rapidity, we de- These equations imply the first integral of motion nominate as the counterpart of rapidity. Introduction of 22 22 the counterpart of rapidity is related with discovery of a ppMc0  = (2.3) new features of dynamical variables of the relativistic The constant of motion has obtained its interpretation mechanics. The counterpart of rapidity is a rapidity re- as a square of proper mass of the relativistic particle, so lated with the systems of references coming from the that, M 22= m . In the case of stationary potential field, light-speed state toward to the rest state. Therefore the ex- i.e. when eVrE =(), the equations imply the other pressions for energy, momentum and velocity expressed constant of motion, the energy of the relativistic particle via the counterpart of rapidity are regular at the state with mvc= 0, = . The new representation for energy EcpVr=()0  (2.4) and momentum can be considered as a mapping from the In order to give a main idea we shall restrict ourselves massless state onto the state with mass. by considering only lengths of the momenta. For that

Copyright © 2011 SciRes. JMP R. M. YAMALEEV 851 purpose let us consider the projection of Equation (2.1) 22222 Yppmc==.0  onto direction of motion. In this way we come to the following set of equations Observation 2.1 The mass-shell equation remains invariant under ddpedp0  additive changes (simultaneous translations) of the pair = pp0 , = , = E (2.5) dddmc of quantities qq12, , where qq11=(0),  qq 2=(0). 2 (2.13) p These translations result an additive change of the E =,nE n= (2.6) p kinetic part of the energy pp=(0) (2.14) Equation (2.5) can be integrated with respect to para- 00 meter  . We find In the same spirit as a solution of the quadratic equation x2 =1 , the imaginary i , generates algebra pA0 =cosh B sinh,  (2.7) of complex numbers, the solution of quadratic Equation pA= sinh B cosh  (2.12) generates an algebra of general complex numbers [12,13]. In references [14,15], some geometrical and Let  = 0 for the rest state where p =0. Then (2.7) algebraical properties of the evolution governed by by is redefined as follows quadratic Equation (2.12) had been explored. Since qq12,

pmc00= cosh  , pmc = sinh  0 (2.8) are roots of the quadratic Equation (2.10), the following Euler formulae hold true Velocity with respect to coordinate time is defined by exp x = g ;, pp xg ;,, pp  k=1,2. vp kk00 10 ==tanh  (2.9) (2.15) cp 0 0 Form the following ratio Comparing this formula with the formulae used in the qD relativistic kinematics we come to conclusion that the exp 2mc = 2 (2.16) qD parameter coincides with hyperbolic angle denominated 1 in Lorentz-kinematics as the rapidity. In Equation (2.5) where the rapidity  is presented as a complementary dy- g ;,pp namical variable. D = 00 (2.17) g ;,pp The evolution governed by Lorentz-force Equation 10

(2.1) changes the quantities pp0 , in a such way that re- Translations qqkk=,=1,2  k remain invariant main invariant the mass-shell Equation (2.3). The rapidity m , hence during of this evolution undergoes to translations. qD Notice, however this is not unique form of variation of exp 2mc = 2 (2.18) qD1  pp0 , remaining invariant the mass-shell Equation (2.3). Now let us explore another form of evolutions of the Let,  = D . Then, energy-momentum which also remain invariant the mass- q2 shell equation. exp 2mc0 = (2.19) q Following references [9,10], let us introduce two 1 quantities qq21>>0 by The formula (2.19) is a Key-formula of the hyperbolic calculus. This formula establishes some interrelation be- 112 pqqmcqqpqq021=, =, 2112 = (2.10) tween simultaneous translations of denominator and nu- 22 merator of the fraction and the hyperbolic rotation. Inversely,

q10=, p mc q 20=. p mc (2.11) 3. Counterpart of Rapidity and Representations for Energy-Momentum The quantities qq, form the set of eigenvalues of 12 Regular Near Zero-Mass State the quadratic polynomial 22222 XcpXcppp200 =0, 0 (2.12) From Key-formula (2.19) by taking into account (2.11) we get with respect to which relationships (2.10) play the role of

Vieta’s formulae. Under translation YXp=  0 the pmcmc00 = coth  0 (3.16) quadratic polynomial takes the form of mass-shell equa- 2 22 tion: Any set of momenta {,pp0 } with pp0 > real

Copyright © 2011 SciRes. JMP 852 R. M. YAMALEEV positive finite numbers may serve as a generator of the or, evolution where the initial point is defined by p2 2 Ecpmc()nonrel == 0 (3.25) q2 pmc0  2m exp 2mc =2 = (3.17) q1 pmc0  where E()nonrel is expression for kinetic energy of New- and tonian mechanics. The high-speed representation allows to obtain analo- mc pmcmcp= coth, = (3.18) gous expression for energy near the light-speed state. For 0 sinh mc small values of   1 we come to the following expansion From these equations the following useful relationship is derived 11 1 pmc0 =coth= mc , pmc= . 2 pmc  expmc = 0 (3.19) p Removing from these equations  , we get mc22 With respect to  the evolution of energy- mo- pp=. (3.26) mentum is given by following equations: 0 2 p

dd2 From comparison of formulae (3.25) and (3.26) we pp=, ppp = (3.20) dd00 conclude that the mass and momentum at these limits are mutually replaced. Formulae (3.23) near the light-speed The evolution of kinetic energy is described by non- state are given by linear equation 11mc22 d 222 pmcmc=coth= , p=. ppmc =0 (3.21) 0  2  d 00 It is seen, at the limit m =0 we have Thus, we possess now with two different repre- sen- 11 tations for the energy and momentum via hyperbolic pm=0 = p m =0 = . (3.27) trigonometry. 00 c One is a function of the rapidity  given by Here we introduced the new quantity 0 which is (Ip ) 0 = mc cosh( ), pmc = sinh( ). (3.22) equal to energy-momentum of the relativistic system at And the other one describes an evolution with respect the state mvc=0, = . In the rest state the energy is to counterpart of rapidity  = mc : equal to the proper inertial mass (in energy units) and, in the same manner, in the state of the light-speed the mc energy is equal to  . Thus, the relativistic dynamics of ()II p = mc coth mc , p = (3.23) 0 0 sinh mc the relativistic particle beside the inertial mass m con- tains a parameter dual to the proper mass (we suggest to

The former is regular at the rest state, these formulae denominate this value as light-mass). The parameter 0 we shall denominated as the low-speed representation. determines the value of the kinetic energy of the motion. The latter is regular at the state of speed of light, these This quantity corresponds to the energy of the particle in formulae, correspondingly, we shall named as highspeed its massless state. representation. Now let us underline some reciprocity between two From the low-speed representation we come to the hyperbolic angles  and  . For that purpose intro- formula for energy near the rest state for slow motion, duce complementary to v velocity v obeying the fo- this is, so-called, non-relativistic limit. llowing equation In order to obtain the non-relativistic limit we use vv222 = c (3.28) expansion of (3.22) for small values of   1 : Notice that v is related with hyperbolic parameter 1 2 (  ) in a manner quite similar as v is expressed via pmc0 =1  , pmc= . 2 rapidity () : 2 2222p 2 Removing from these equations  , we get vcvc==1= ctanh 2  (3.29) p2 p2 0 pmc=  (3.24) Interrelation between  and  is expressed by the 0 2mc

Copyright © 2011 SciRes. JMP R. M. YAMALEEV 853 following formulae of reciprocity Noteworthy, the constant  now is not an arbitrary constant, but it has to be understood as a fundamental   exp = coth , exp = coth  (3.30) constant of the theory. Let us remember the formula of 22  q-deformation of a quantity N : From these formulae it follows that  =0 and  =  qqN  N when v =0, and  =  and  =0 when vc = . In ():=N q 1 (4.7) some sense the pairs (,v  ) and (,v  ) are reciprocal qq to each other. From this point of view the last expression in (4.5) is q-deformation of  with parameter of deformation 4. q-Deformation and Quantization of the qm=exp( / ). In notations of (4.7), Equation (4.5) can Energy-Momentum be written as follows

Let us introduce some parameter in unit of mass and label cm  =( )q , q =exp (4.8) this parameter by  . Define a dimensionless variable p   by Notice,  =1 =1 for any q . There fore  = c (4.1) q pm  0, = 1 = c . Re-write formulae for the energy-momentum (3.18) in these variables On the other hand, if m =0 and q =1 then from mc m (4.8) it follows ppmc=, 0 = coth (4.2) m   c sinh  pm=0 =0 = (4.9)  c  Here we should notice that the formula for the mo- Hence at the point  =1 momentum of the particle mentum admits the following integral representation with mass m is equal to the momentum of the massless mc2 particle sinh  2  0 1/2 mc  0 == expdx x (4.3) 0 2 1/2  pm 0, = 1 = pm = 0, = 1 = = c (4.10) cp mc 0 c  0 Notice, however, the velocity of the particle at this In [16] it has been shown that  1 geometrically can point is not equal to the light velocity. In fact, these for- be interpreted as a curvature of a hyperbolic space. In mulae imply existence of a point on the axis of momenta this space the length of the circle with radius m is de- where the momenta of the massive and massless particles fined by formulae [17]: are equal. At this point the energy and the velocity are given by m L=1 :=2 sinh (4.4a) mc  2  cp0 = 1 = c cosh , v = (4.11)  m Correspondingly, the length of the circle with radius cosh   m is equal  m Now remember on integral representation (4.3). Now L =2 sinh (4.4b) this fraction was replaced by  Taking into account this correspondence let us per- m sinh  form the following modifications in the formulae for  c  0  =  (4.12) energy-momentum cp p m sinh  m  2 sinh  mc mc c  =sinh = (4.5) It is interesting to observe that for the new fraction pp0 m (4.12) we shall obtain a sum instead of an integral if we sinh   assume that  is an integral number. Let J be a half-

2 integer number with J = 0,1/ 2,1,3 / 2,2, , and pp00mc m  m  = coth = sinh coth   (4.6)  = 2J  1 = 1,2,3, . Then the following equation mc0 c   holds true.

m where sinh 2J  1 cm  J  1 ==exp. n  =, p m nJ=  T sinh   with absolute temperature T given in energy unit. Con- This formula prompts us to introduce a hypothesis on tinuing the observation let us notice that the expression quantization of  . Experimentally the quantization can for the energy cp0 of the form be observed near the light velocity where the velocity of the massive particle brings nearer the light velocity spas- 2  11 cp0 =2 mc    (5.4) modically according to law  2exp2mc  1 c v =. is quite similar the expression for the mean energy of the m cosh 2J  1 single oscillator   11 U =< >, < >=    (5.5) 5. Analogue with Statistical  2exp  1 Thermodynamics These observations lead us to the following corres- In this section we are going to observe an interesting pondence between dynamic variables of the relativistic analogy between the formulae obtained in the previous mechanics and the statistical thermo-dynamics: 111 sections for the energy-momentum and the well-known 2mc2   ,  ,  (5.6) formulae of the statistical thermodynamics. This ana- cT0 logue exhibits a correspondence between the present re- In reference [15] we have justified the following representation of energy-momentum and the average energy lationship between energy and momentum of the single quantum mechanical oscillator. In this way we will come to the interpretation of  as an absolute d  cp=ln. p (5.7) temperature of a heat reservoir. 0 d Let us start from formula for the momentum of the This equation quite analogous to the well-known equa- form tion of thermodynamics connecting internal energy U p 1 with the partition function Z : = (5.1) 22sinhmc mc d UZ=ln (5.8) Notice that this expression can be represented as a sum d of geometrical series. In fact, The evolution with respect to  now obtains its interpretation via statistical thermodynamics. In (5.4), 1 expmc  ==exp Ecn  2 2sinhmc 1 exp 2 mc n=0 (5.5) the terms , as well as quantity mc , are 2 (5.2) exactly the contribution of the zero-point energy to the here the expression total energy. Furthermore, now we are able to introduce the notions of the free energy and the entropy. The free 2 1 Emcnn =2  energy is 2 can be considered as a spectrum of the quantum 2mc Fmc=ln2sinh00 =ln  (5.9)  P oscillator where the role of zero point energy is 2 The entropy is (in unit k =1 , k -constant of taken over by the energy at the rest mc2 of the particle. Boltzmann) Furthermore, it is puzzled that the formula for the length of momentum per unit of mass is quite similar to the Smcmc=cothln2sin    mc  (5.10) formula for single-particle partition function Z . The Consider N distinguishable oscillators [19]. It is partition function for the single oscillator is [18]: very instructive to calculate from the partition function  exp mc N 2  e  ZTV,,1= (5.3) ZN()= 2mc  (5.11) 1exp 1e 

Copyright © 2011 SciRes. JMP R. M. YAMALEEV 855 the corresponding density of states of the N -oscillator It is interesting to compare the limiting cases for sta- system. With the aid of the binomial expansion tistical thermodynamics and for the relativistic me- 1  Nl1! chanics. In the case of high temperatures N =  x 1 x l =0 lN!( 1)!   T , 0, we write kT the classical limit is recovered, because the characteristic  Nl1)!  1  Z NmcNl=exp2   parameter   measures the ratio of the energy levels lN!1! 2 l =0  of the oscillator compared to the mean thermal energy Comparing this with the general formula for Z with kT which is available. In the mechanics the limit discrete enumerable energies  =0 corresponds to the state of massless particle, or to the state of light. At the limit of low temperature Z NdEgEEc=exp    , one has the largest deviations from the classical we find case, where N 2 NNl1)! U =. Emclll=2 , g = (5.12) 2!1!lN 2 In the mechanics the limit   corresponds to the The energies El are just the zero-point energies of 2 2 the N oscillators plus l quanta of energy 2mc . rest state p =0 with the rest energy cp0 = mc .

There are exactly gl ways to distribute these indistinguishable energy quanta among the N distinguishable 6. Conclusions and Comments oscillators. The distribution of indistinguishable quanta instead of enumerated particles is the starting point of In the present paper we gave a ground for new repre- quantum statistics. Consider a collection of N identical sentation for energy, momentum and velocity via hy- quantum oscillators, supposed their distinguishable. The perbolic trigonometry the (hyperbolic) angle of which is total number of distinguishable states  , corresponding proportional to the proper mass. The hyperbolic angle to the energy E , and all the probabilities turn out to be dual to the rapidity was denominated as counterpart of rapidity. We have worked primarily in the scopes of the equal, as expected. It is simply  = gl . Thus we can check whether the entropy S =ln coincides with dynamics, not kinematics, and restricted ourselves only expression (5.10). Using lN,1 and Stirling's formula with one dimensional case. It is well known that in the we get covariant formulation the rapidity is presented by anti- symmetric tensor in four-dimensional Minkowski space. Ml1! S =ln , = , In that context the question gives arise: what kind tensorial Nl1!! object will present the counterpart of rapidity within the SlNlNllNN=lnlnln. framework of covariant formulation? This question has a certain answer: in the covariant formulation the counter- To obtain SEN(, ) the lEmcN=/22  /2 must be part of rapidity is presented by the four-vector [20]. inserted: 7. References EN EN S =ln22  22mc22 mc [1] W. Heisenberg, “Zur Theorie de Elementarteilchen,” Zeit (5.13) EN EN Naturforsch, Vol. 14a, 1959, pp. 441-451. 22lnNN ln [2] Y. Nambu, “Dynamical Model of Elementary Particles 22mc22 mc Based on an Analogy with Superconductivity,” Physical If we want to compare this with Equation (5.10), we Review, Vol. 117, 1960, pp. 648-659. must express the energy in terms of the “temperature”, doi:10.1103/PhysRev.117.648 [3] J.-M. Levy-Leblond, “A Gedankenexperiment in Science SENmc1 2 History,” American Journal of Physics, Vol. 48, 1980, pp.  =|=ln , E N 2mc22 E Nmc 345-354. 2 (5.15) [4] V. Varicak, “Application of Lobachevskian Geometry in E exp 2mc   1 or = the Theory of Relativity,” Physikalische Zeitschrift, Vol. Nmc2 exp 2mc2  1 11, 1910, pp. 93-96. [5] H. Hergoltz, “Geometrical Aspects of Relativity Theory,” which is identical to Equation (5.4). Ann.d.Phys, Vol.10, 1910, pp. 31-45.

[6] A. A. Robb, “Optical Geometry of Motion, a New View doi:10.1007/BF03042040 of the Theory of Relativity,” Cambridge, 1911. [14] R. M. Yamaleev, “Multicomplex Algebras on Polynomi- [7] A. A. Ungar, “Hyperbolic Trigonometry and Its Applica- als and Generalized Hamilton Dynamics,” Journal of Ma- tions in Poincaré Ball Model of Hyperbolic Geometry,” thematical Analysis and Applications, Vol. 322, No. 2, Computers & Mathematics with Applications, Vol. 41, 2006, pp. 815-824. doi:10.1016/j.jmaa.2005.09.073 No. 1-2, 2001, pp. 135-147. [15] R. M. Yamaleev, “Geometrical and Physical Interpreta- doi:10.1016/S0898-1221(01)85012-4 tion of Evolution Governed by General Complex Alge- [8] A. O. Barut, “Electrodynamics and Classical Theory of bra,” Journal of Mathematical Analysis and Application, Fields and Particles,” Dover Publications, Inc., New York, Vol. 340, No. 2, 2008, pp. 1046-1057. 1980. doi:10.1016/j.jmaa.2007.09.018 [9] R. M. Yamaleev, “Extended Relativistic Dynamics of [16] R. M. Yamaleev, “Complex Algebras on N-order Poly- Charged Spinning Particle in Quaternionic Formulation,” nomials and Generalizations of Trigonometry, Oscillator Advances in Applied Clifford Algebras, Vol. 13, No. 2, Model and Hamilton Dynamics,” Advances in Applied 2003, pp. 183-218. doi:10.1007/s00006-003-0015-8 Clifford Algebras, Vol. 15, No. 1, 2005, pp. 123-125. [10] R. M. Yamaleev, “Relativistic Equations of Motion within [17] V. F. Kagan, “Gostexizdat,” Foundations of Geometry, Nambu’s Formalism of Dynamics,” Annals of Physics, Part I,” Moscow, 1949, pp. 331-336. Vol. 285, No. 2, 2000, pp. 141-160. [18] W. Greiner, L. Neise and H. Stöcker, “Thermodynamics doi:10.1006/aphy.2000.6075 and Statistical Mechanics,” Springer-Verlag, New York, [11] A. R. Rodriguez-Dominguez, “Ecuaciones de fuerza de 1995, pp. 208-214, ISBN 0-387-94299-8. Lorentz Como Ecuaciones de Heisenberg,” Revista Mexi- [19] H. S. Robertson, “Statistical Thermodynamics,” PTR Pren- cana de Fisica, Vol. 53, No. 4, 2007, pp. 270-280. tice Hall, Englewood Cliffs, 1993, pp. 110-112, ISBN [12] I. M. Yaglom, “Complex Numbers in Geometry,” Aca- 0-13-845603-8. demic Press, New York, 1968. [20] R. M. Yamaleev, “New Representation for Energy- Mo- [13] P. Fjelstad and S. Gal, “Two Dimensional Geometries, mentum and Its Applications to Relativistic Dynamics,” Topologies, Trigonometries and Physics Generated by Nuclear Physics, Vol. 74, No. 7, 2011, pp. 1-8, (in Rus- Complex-Type Numbers,” Advances in Applied Clifford sian). Algebras, Vol. 11, No. 1, 2001, pp. 81-90.