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Covariant Relativistic Quantum Mechanics Analysis of a Linearly Accelerated Scalar Particle

Karol McDonald

Published Jan 1, 2009• 0 citations
Publisher

Abstract

A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.

DOI
10.21427/d7qw2f

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A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.
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.
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A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function Q(x,p)Q(x,p)Q(x,p) is derived and expanded in power series of the charge of the particle. Then, solutions of the zeroth order in the charge of the iterative equations for Q(x,p)Q(x,p)Q(x,p) are considered. For them, it follows that the space-time probability density ρ(x)ρ(x)ρ(x) and the function S(x)S(x)S(x) which gradient defines the mean value of the momentum at the space time point xxx, define a complex function ψ(x)ψ(x)ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. It is argued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. It follows that the total number of particles conserves when ψ(x)ψ(x)ψ(x) composed alternatively of pure positive or negative energy modes. In addition, solutions for the joint distribution function in lowest order, satisfying the positive condition are also found. They suggests a possible link with the Couder's experiments on droplet movements over oscillating liquid surfaces.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
A derivation is presented of the quantummechanical wave equations based upon the Equity Principle of Einstein's General Relativity Theory. This is believed to be more generic than the common derivations based upon Einstein's energy relationship for moving particles. It is shown that Schrodinger's Equation, if properly formulated, is relativisticly covariant. This makes the critisized Klein-Gordon Equation for spinless massparticles obsolete. Therefore Dirac's Equation is presented from a different viewpoint and it is shown that the relativistic covariance of Schrodinger's Equation gives a natural explanation for the dual energy outcome of Dirac's derivation and for the nature of antiparticles. The propagation of wave functions in an energy field is studied in terms of propagation along geodesic lines in curved space-time, resulting in an equivalent formulation as with Feynman's path integral. It is shown that Maxwell's wave equation fits in the developed framework as the massless limit of moving particles. Finally the physical appearance of electrons is discussed including a quantitative calculation of the jitter phenomenon of a free moving electron.