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arXiv:1108.0125

Quantum mechanics for relativistic bosons

Soon-Tae Hong

Published Jul 31, 2011• 0 citations
arXivHTMLPDF

Abstract

We construct a relativistic quantum mechanics for a boson. To do this we exploit two component wave functions in Dirac type equations of motion. In our formalism we fix the pathological aspect of particle probability density which appears in Klein-Gordon theory. Our solutions possess a negative solution as well as a positive one. We also formulate a diagonal Hamiltonian of the relativistic quantum mechanics for the boson.

DOI
10.48550/arxiv.1108.0125

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We construct a relativistic quantum mechanics for a boson. To do this we exploit two component wave functions in Dirac type equations of motion. In our formalism we fix the pathological aspect of particle probability density which appears in Klein-Gordon theory. Our solutions possess a negative solution as well as a positive one. We also formulate a diagonal Hamiltonian of the relativistic quantum mechanics for the boson.
The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.
Department ofScience Educationand Research Institute forBasicSciences, EwhaWomansUniversity, Seoul 120-750 Korea(Dated: August 15, 2011)We construct a relativistic quantum mechanics for a boson which sustains the positive definiteparticle probability for the relativistic boson. In this theory, we settle down the well-known 85 yearsold unsolved problem on the particle probability of the Klein-Gordon quantum mechanics. Thesolutions for a boson obtained from our theory possess a negative solution as well as a positive one,similar to those of the Dirac quantum mechanics for a fermion. We next study Lorentz transforma-tion, and a vacuum state with all fermion and boson negative energy levels filled and all fermion andboson positive energy levels empty, so that we can have unification of the bosonic sea and the Diracfermionic one. In this unified sea we investigate the bosonic and fermionic antiparticles associatedwith the unified vacuum on an equal footing, and we obtain a charge conjugation operator to studydecay modes of the particles and antiparticles.I. INTRODUTION
We show that, in the relativistic regime, scalar bosons satisfy a quantum wave equation which is quite analogous to the Dirac equation. In contrast with the Klein-Gordon equation it is first order with respect to time derivation. It leads in a regular way to the standard Schrödinger equation in the non-relativistic limit. There are two components for the wave function in this representation for the scalar boson, in a way completely analogous to the four components for the spin 1/21/21/2 fermion in the Dirac equation.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.