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arXiv:hep-th/0110128

The particle invariance in particle physics

H. Y. Cui

Published Oct 15, 2001• 0 citations
arXivHTMLPDF

Abstract

Since the particles such as molecules, atoms and nuclei are composite particles, it is important to recognize that physics must be invariant for the composite particles and their constituent particles, this requirement is called particle invariance in this paper. But difficulties arise immediately because for fermion we use Dirac equation, for meson we use Klein-Gordon equation and for classical particle we use Newtonian mechanics, while the connections between these equations are quite indirect. Thus if the particle invariance is held in physics, i.e., only one physical formalism exists for any particle, we can expect to find out the differences between these equations by employing the particle invariance. As the results, several new relationships between them are found, the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found, it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed. An improved model is proposed to gain a better understanding on elementary particle interactions. This approach offers enormous advantages, not only for giving the first physically reasonable interpretation of quantum mechanics, but also for improving quark model.

DOI
10.48550/arxiv.hep-th/0110128

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Since the particles such as molecules, atoms and nuclei are composite particles, it is important to recognize that physics must be invariant for the composite particles and their constituent particles, this requirement is called particle invariance in this paper. But difficulties arise immediately because for fermion we use Dirac equation, for meson we use Klein-Gordon equation and for classical particle we use Newtonian mechanics, while the connections between these equations are quite indirect. Thus if the particle invariance is held in physics, i.e., only one physical formalism exists for any particle, we can expect to find out the differences between these equations by employing the particle invariance. As the results, several new relationships between them are found, the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found, it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed. An improved model is proposed to gain a better understanding on elementary particle interactions. This approach offers enormous advantages, not only for giving the first physically reasonable interpretation of quantum mechanics, but also for improving quark model.
Since particle such as molecule, atom and nucleus are composite particle, it is important to recognize that physics must be invariant for both the composite particle and its constituent particles, this requirement is called particle invariance. But difficulties arise immediately because for fermion we use the Dirac equation, for boson we use the Klein-Gordon equation. Therefore, the particle invariance demands there is a general wave equation for describing particle motion regardless particle class. In this paper, three advances in this subject are reported: (1) momentum-wavefunction relation is a general relation shared by both fermion and boson, meets the requirement of the particle invaiance. As a test, the momentum-wavefunction relation was directly applied to hydrogen atom, and get the correct fine structure and spin effect for the electron. (2) the Dirac equation and Klein-Gordon equation can be derived out from the momentum-wavefunction relation when we abandon some higher order terms. (3) according to the momentum-wavefunction relation a path integral method was developed, differing from Feynman's path integral, it simplfies quantum computation.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.
The principle of invariance(also called phase harmonic law) in classical wave dynamics as a basic physical principle is discussed and an underlying intuitive reason which lies behind it——the order invariance of physical quantities is proposed.Such invariance reflects the principle of relativity.From phase harmonic law we deduced the transformation laws for wave vector,frequency,and group velocity of wave packets while space-time coordinates satisfying Galileo and Lorentz transformation,respectively.Furthermore we discussed the problem of transformation for Schrdinger equation and Klein-Gordon equation.For Schrdinger equation,we believe that the complex nature of wave function releases the wave function from invariance which is inevitable for classical waves(where only the real part of wave function have physical meaning),thus expanded the physical region.It becomes the basic quantum equation of nonrelativistic particles.Klein-Gordon equation,however,satisfies phase harmonic law,we believe that there should be some kind of order invariance for physical quantities behind it.We discussed one possibility from professor Guangjiong Ni's two-component(particle and antiparticle) notion to identify such an order.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.
We derive the equations of relativistic quantum mechanics from a modified version of classical electrodynamics, where probability is replaced by potential. As a result, a particle is not a localized entity, in the classical sense, but has a localized energy extremum. The particle/wave aspect of matter is inherent in the particle/wave equation describing elementary particles. Furthermore, the Heisenberg uncertainty and Planck-Einstein-de Broglie relations, and the Klein-Gordon, Dirac and Proca equations follow naturally from the particle/wave equation. In addition, we incorporate a new and more physical interpretation of spin angular momentum.
Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a generalized Hamiltonian dynamics and to a covariant Schrödinger type equation is shown. Examples for relativistic potential problems are solved. Mathematically and physically acceptable probability densities for the Klein-Gordon equation and for the Dirac equation are derived, and a new continuity equation for each case is given. The quantum distribution for mass is discussed, and unambiguous representations of four-velocity and four-acceleration operators are given.