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

Coherence lengths for superconductivity in the two-orbital negative-U Hubbard model

We study the peculiarities of coherency in the superconductivity of two-orbital system. The superconducting phase transition is caused here by the on-site intra-orbital attractions (negative-U Hubbard model) and inter-orbital pair-transfer interaction. The dependencies of critical and noncritical correlation lengths on interaction channels and band fillings are analyzed.Comment: 5 pages, 3 figures, Acta Physica Polonica (2012) in pres

Evidence for alignment of the rotation and velocity vectors in pulsars. II. Further data and emission heights

We have conducted observations of 22 pulsars at frequencies of 0.7, 1.4 and 3.1 GHz and present their polarization profiles. The observations were carried out for two main purposes. First we compare the orientation of the spin and velocity vectors to verify the proposed alignment of these vectors by Johnston et al. (2005). We find, for the 14 pulsars for which we were able to determine both vectors, that 7 are plausibly aligned, a fraction which is lower than, but consistent with, earlier measurements. Secondly, we use profiles obtained simultaneously at widely spaced frequencies to compute the radio emission heights. We find, similar to other workers in the field, that radiation from the centre of the profile originates from lower in the magnetosphere than the radiation from the outer parts of the profile.Comment: Accepted by MNRAS. 14 page

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

Quantum-classical transition in Scale Relativity

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrodinger and Klein-Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds - ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^mu - dx^mu) and by requiring invariance under parity and time inversion. The Pauli equation is recovered as a non-relativistic-motion approximation of the Dirac equation.Comment: 28 pages, no figur