Garnet N. Ord

Department of Math, Physics and Computer Science

 
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The Dirac Equation

This section contains abstracts of representative papers:

A Classical Analog of Quantum Phase

A modified version of the Feynman relativistic chessboard model (FCM) is investigated in which the paths involved are spirals in space-time. Portions of the paths in which the particle's proper time is reversed are interpreted in terms of antiparticles. With this interpretation the particle-antiparticle field produced by such trajectories provides a classical analog of the phase associated with particle paths in the unmodified FCM. It is shown that in the nonrelativistic limit the resulting kernel is the correct Dirac propagator and that particle-antiparticle symmetry is in this case responsible for quantum interference. (Int. J. Theo. Phys. v31 1992) 
 

Time Reversal in Stochastic Processes and the Dirac Equation

We consider the motion of a classical particle in 1+1 dimensional space-time. Four probability distributions govern the trajectory of the particle; these give the probability of moving to the left or right in space while moving backwards or forwards in time. If these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forward s in time in a prescribed manner, then the master equations for these probabilities give rise to the Dirac equation without recourse to direct analytic continuation. In contrast, when a particle always moves forward in time, an analytic continuation is required to recover the Dirac eqation. (D.G.C. McKeon and G.N. Ord, Phys. Rev. Lett. v69 1992) 
 

The Schroedinger and Dirac Free Particle Equations without Quantum MechanicsEinstein's theory of Brownian Movement has provided a well accepted microscopic model of diffusion for many years. Until recently the relationship between this model and quantum mechanics has been completely formal. Brownian motion provides a microscopic model for diffusion, but quantum mechanics and diffusion are related by a formal analytic continuation, so the relationship between Brownian motion and quantum mechanics has been correspondingly vague. Some recent work has changed this picture somewhat and here we show that a random walk model of Brownian motion produces the diffusion equation or the telegraph equations as a descriptions of particle densities, while at the same time the correlations in the space-time geometry of these same Brownian particles obey the Schroedinger and Dirac equations respectively. This is of interest because the equations of quantum mechanics appear here naturally in a classical context without the problems of interpretation they have in the usual context. (Annals of Physics 250, August 1996) 
 

Fractal Space-Time and the Statistical Mechanics of Random Walks

Some of the ideas involved in fractal-space-time are illustrated using familiar deterministic fractals. Starting with the objective of reproducing the Heisenberg uncertainty principle for point particles, we use the Peano-Moore curve to help visualize the qualitative behaviour of particles moving on fractal trajectories in space and time. With this qualitative picture in mind we then explore exactly solvable models to verify that our qualitative models are mathematically consistent. We find that the Schroedinger equation describes ensembles of classical particles moving on fractal random walk trajectories. This shows that the Schroedinger equation has a straight forward microscopic model which is not however appropriate for quantum mechanics. The free particle Dirac equation is also derivable in terms of ensembles of classical particles and this unites the two equations conceptually in a very direct way. In both cases what we discover is a many-particle simulation of quantum mechanics and this confirms in a graphic way that the mysteries surrounding quantum mechanics lie not in the equations, but in interpretation and the theory of measurement. 

Finally we discuss an exactly solvable model which incorporates fractal time. The calculation produces the Dirac equation in 1+1 dimensions and because of intrinsic space-time loops, constitutes a model with the potential to exhibit the wave-particle duality found in nature. (Chaos, Solitons and Fractals, v7, 5, 1996)