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A Study and Critique of the Mean Position Concept in Relativistic Wave Mechanics

Hugh David Gebhart

Published Jan 1, 1967• 0 citations
OpenAlex

Abstract

The basic concept to be used in studying the question of one-particle interpretations of relativistic wave equations is that of observables and operator representations that are different from the more usual classically motivated observables and representations. In particular, the concept of a mean-position observable will be used to determine to what extent the one-particle "problems" can be resolved.

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The basic concept to be used in studying the question of one-particle interpretations of relativistic wave equations is that of observables and operator representations that are different from the more usual classically motivated observables and representations. In particular, the concept of a mean-position observable will be used to determine to what extent the one-particle "problems" can be resolved.
The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n
A realistic interpretation of Schroedinger and Dirac equations for density matrices is proposed, in which the difference between the position arguments of the density matrix is considered as an objective extra space dimension. "Particle" solutions are found, which are perfectly localized both in position space and in momentum space (the position and momentum operators commute in the density matrix representation); definitions for all observable quantities are given and the values associated to the "particle" solutions are the correct ones, both for the non-relativistic and the relativistic case. Finally, a non-linear interaction (the electromagnetic one) is introduced in an attempt to single out the "particle" solutions of the Dirac equation from all other solutions; the dynamical evolution of the electromagnetic field is described by the classical (unquantized) Maxwell equations.
The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.
The relativistic phase-space representation by means of the usual position and momentum operators for a class of observables with Weyl symbols independent of charge variable (i.e. with any combination of position and momentum) is proposed. The dynamical equation coincides with its analogue in the non-local theory (generalization of the Newton-Wigner position operator approach) under conditions when particles creation is impossible. Differences reveal themselves in specific constraints on possible initial conditions.