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

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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) Formulae for Energy and Momentum of Relativistic Particle Regular at Zero-Mass State 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-mo- mentum 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 Copyright © 2011 SciRes. JMP 850 R. M. YAMALEEV 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 quan- tity 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) ddmc0 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- ddmm 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) dddmc 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.
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