New representation for energy-momentum and its applications to relativistic dynamics
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ELEMENTARY PARTICLES AND FIELDS Theory
New Representation for Energy–Momentum and its Applications to Relativistic Dynamics* R. M. Yamaleev** Universidad Nacional de Mexico, Facultad de Estudios Superiores, Mexico; Joint Institute for Nuclear Research, Dubna, Russia Received July 19, 2010
Abstract—In this paper we introduce the concept counterpart of rapidity and define energy and momentum of the relativistic particle as functions of the counterpart of rapidity. Formulae of the relativistic mechanics defined in such a way are regular near the zero-mass and speed of light state. This representation admits to attain a correct limit of the formulae of the relativistic mechanics, including the Dirac equation, at zero-mass point and explains violation of the parity at this state. On the other hand, the representation for energy–momentum can be realized as a mapping from the massless state onto the massive one which looks like a “q deformation”. Hypothesis on quantization of the energy–momentum and the velocity near the light speed is suggested. The group of transformations using the counterpart of rapidity as a parameter of transformation is constructed. DOI: 10.1134/S1063778811070180
1. INTRODUCTION In the present paper we elaborate new expressions for the energy, momentum, and velocity of a relativistic particle regular at the point m = 0, v = c. The formulae for the energy–momentum are presented as functions of some hyperbolic angle χ dual to the rapidity ψ, which forms a counterpart of the rapidity. In one-dimensional case the hyperbolic angles χ and ψ are reciprocal quantities. However, in general, the rapidity and its counterpart physically and geometrically are quite different each from an other. The rapidity, ψ, is equal to zero at the rest state, v = 0, and goes to infinity when the velocity tends to v = c, whereas its counterpart, the hyperbolic angle χ, is equal to zero at the point v = c and becomes infinity at the rest state, v = 0. Another important property of the counterpart of rapidity is its dependence of the proper mass: χ = mc/π0 . The quantity cπ0 is interpreted as an energy of the massless state. Thus, the formulae for energy–momentum can be realized as a mapping from an energy of the massless state onto the energy of a particle with mass. These formulae look like as formulae of “q deformation”. An analysis of this observation prompts to introduce a hypothesis on quantization of the velocity near the speed of light. In a similar manner as translations of the rapidity form a part of the Lorentz group of transformations, ∗ **
The text was submitted by the author in English. E-mail: [email protected]
translations of the counter-rapidity form some group of transformations. Usefulness of the present theory we demonstrate exploring the Dirac equation at the limit m = 0. The representation for the energy–momentum via counter-rapidity allows to reach a correct limit at m = 0 explicitly displaying violation of the parity in the Dirac equation at this limit. 2. REPRESENTATIONS OF ENERGY–MOMENTUM AS FUN
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