The Dirac Equation in Classical Statistical Mechanics

The Dirac equation, usually obtained by `quantizing' a classical stochastic model is here obtained directly within classical statistical mechanics. The special underlying space-time geometry of the random walk replaces the missing analytic continuation, making the model `self-quantizing'. This provides a new context for the Dirac equation, distinct from its usual context in relativistic quantum mechanics.Comment: Condensed version of a talk given at the MRST conference, 05/02, Waterloo, Ont. 7 page

Entwined Paths, Difference Equations and the Dirac Equation

Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are `self-quantizing'. We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process in the same sense that the Diffusion and Telegraph equations are phenomenological descriptions of stochastic processes.Comment: 15 pages, 5 figures Replacement 11/02 contains minor editorial change

The Feynman chessboard model in 3 + 1 dimensions

The chessboard model was Feynman’s adaptation of his path integral method to a two-dimensional relativistic domain. It is shown that chessboard paths encode information about the contiguous pairs of paths in a spacetime plane, as required by discrete worldlines in Minkowski space. The application of coding by pairs in a four-dimensional spacetime is then restricted by the requirements of the Lorentz transformation, and the implementation of these restrictions provides an extension of the model to 4D, illuminating the relationship between relativity and quantum propagation

Scale relativity and fractal space-time: theory and applications

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to system biology.Comment: 63 pages, 14 figures. In : First International Conference on the Evolution and Development of the Universe,8th - 9th October 2008, Paris, Franc

Modelling annual and orbital variations in the scintillation of the relativistic binary PSR J1141−-6545

We have observed the relativistic binary pulsar PSR J1141$-$6545 over a period of $\sim$6 years using the Parkes 64 m radio telescope, with a focus on modelling the diffractive intensity scintillations to improve the accuracy of the astrometric timing model. The long-term scintillation, which shows orbital and annual variations, allows us to measure parameters that are difficult to measure with pulsar timing alone. These include: the orbital inclination $i$; the longitude of the ascending node $\Omega$; and the pulsar system transverse velocity. We use the annual variations to resolve the previous ambiguity in the sense of the inclination angle. Using the correct sense, and a prior probability distribution given by a constraint from pulsar timing ($i=73\pm3^\circ$), we find $\Omega=24.8\pm1.8^\circ$ and we estimate the pulsar distance to be $D=10^{+4}_{-3}$ kpc. This then gives us an estimate of this pulsar's proper motion of $\mu_{\alpha}\cos{\delta}=2.9\pm1.0$ mas yr$^{-1}$ in right ascension and $\mu_{\delta}=1.8\pm0.6$ mas yr$^{-1}$ in declination. Finally, we obtain measurements of the spatial structure of the interstellar electron density fluctuations, including: the spatial scale and anisotropy of the diffraction pattern; the distribution of scattering material along the line of sight; and spatial variation in the strength of turbulence from epoch to epoch. We find that the scattering is dominated by a thin screen at a distance of $(0.724\pm0.008)D$, with an anisotropy axial ratio $A_{\rm r} = 2.14\pm0.11$.Comment: 17 pages, 8 figures, 2 tables. Accepted for publication in MNRA