Considers a 'thought experiment' in which particles are confined to move on fractal trajectories in both space and time, treating the case of a Peano-Moore trajectory in detail. Generalising these results, the author uses the classical principle of relativity and a correspondence principle to show that fractal trajectories in space with Hausdorff dimension D=2 exhibit both an uncertainty principle and a de Broglie relation. The incorporation of fractal time with D=2 places an upper bound on the macroscopic velocities of 'fractalons', which in turn requires that the macroscopic physics be Lorentz covariant. On a microscopic scale, the presence of fractal time is interpreted in terms of the appearance of particle-antiparticle pairs when observation energies become of the order of mc2. The author proposes two field equation descriptions of fractalons based on random walk space-time trajectories and subsequently relates these equations to the free particle Klein-Gordon and Dirac equations respectively.

http://iopscience.iop.org/article/10.1088/0305-4470/16/9/012/pdf

Fractal space-time: a geometric analogue of relativistic quantum mechanics

This article is divided in five parts, the first one having nothing to do with tachyons. In fact, to prepare the ground, in part I (sect.2) we shall merely show that special relativity - even without tachyons - can be given a form such to describe both particles and antiparticles. Part II is the largest one: initially, after some historical remarks and having revisited the postulates of special relativity, we presnt a review of the elegant ''model theory'' of tachyons in two dimensions; passing then to four dimensions, we review the main results of the classical theory of tachyons that do not depend on the existence of Superluminal reference frames (or that are at least independent of the explicit form of the Superluminal Lorentz ''transformations''). In particular, we discuss how tachyons would look like, i.e. their apparent ''shape''. Last but not least, all the common causality problems are thoroughly solved, on the basis of the previously reviewed tachyon kinematics. Part III deals with tachyons in general relativity; in particular the question of the apparent Superluminal expansions in astrophysics is reviewed. Part IV shows the interesting, possible role of tachyons in elementary-particle physcis and in quantum theory. In part V, the last one, we face the (still open) problem of the Superluminal Lorentz ''transformations'' in four dimensions, by introducing, for instance, an auxiliary six-dimensional space-time, and finally present the electromagnetic theory of tachyons: a theory that can be relevant also from the ''experimental'' point of view.

Classical tachyons and possible applications

SummaryIt is shown that supersymmetry can be formulated on a space-time lattice. The transformation properties of superfields are demonstrated and the simplest action that gives the Wess-Zumino model in the continuous limit is discussed.RiassuntoSi mostra che la supersimmetria può essere formulata in un reticolo dello spazio-tempo. Si dimostrano le proprietà di trasformazione dei supercampi e si discute l’azione più semplice che dà il modello di Wess-Zumino nel limite continuo.РеэюмеПокаэывается, что суперсимметрия может быть сформулирована на пространственно-вре менной рещетке. Демонстрируются свойства преобраэований суперполей. Обсуждается простейщий способ, который дает модель Весса-Зумино в непрерывном пределе.

Lattice supersymmetry@@@Суперсимметрия рещетки

Quantum mechanics in a discrete space-time

β-Function in a non-covariant Yang-Mills theory

In order to adequately discuss the transformations between the space-time co-ordinates of two inertial frames moving superluminally relative to each other, it is necessary to use either four complex space-time co-ordinates or six real space-time co-ordinates—three of space and three of time. A major difficulty then arises in the physical understanding of the extra parameters. In these two cases, considered separately, specific transformations are found for both the superluminal and the subluminal cases. It is found that in the subluminal cases the extra co-ordinates are completely uncoupled from the original ones, while in the superluminal cases there is a necessary coupling.

Superluminal transformations using either complex space-time or real space-time symmetry

Special relativity is developed using a real six-dimensional space-time. It is shown that the imposition of parallel time displacements in each frame necessarily implies subluminal transformations. Conservation of six-momentum yields the usual three-momentum equation together with a vector equation, which replaces the scalar energy equation of the corresponding four-dimensional theory. On energetic grounds alone it is shown that a particle may decay into a set of particles whose total rest mass is greater than that of the original particle, and that particles may be produced from the vacuum state. When virtual particle production is considered, this suggests a tentative link between the uncertainty principle and a six-dimensional theory with a restriction to small deviations from a constant time direction.

Particle decay in six-dimensional relativity

A real six-dimensional space-time (three space and three time dimensions) scheme is used to discuss both subluminal and superluminal transformations between the inertial frames of observers. Properties of the linear transformations are derived by using the relative matrix velocities between frames. A link with the usual four-dimensional subluminal scheme is obtained by averaging over all possible orientations of the three time axes of each observer. An insight into the effect of working with three time co-ordinates is gained by considering an example which is the subluminal six-dimensional equivalent of the four-dimensional case of two frames which are in standard configuration.

Subluminal and superluminal transformations in six-dimensional special relativity

Comments on the use of three time dimensions in relativity

This paper is concerned with a discrete space and time description of physical events. This involves transforming partial differentials with respect to space and time into finite differences evaluated between neighbouring space-time points. By considering a set of such finite-difference forms of the Euler equations for classical fields, a rule for such a transformation is obtained. It is found that each partial differential must be replaced by the corresponding central difference. Energy and momentum eigenvalues are found for particles moving in one dimension along which the separation between adjacent space points is ξ. It is found that these quantities have maximum values proportional to ξ−2 and ξ−1 respectively, and that there are finite numbers of distinct eigenvalues of each sort. The Schrodinger equation is transformed into a difference equation and is solved formally to give theS-matrix. It is found that the wave function does not tend to a limit at infinite times.

E. A. B. Cole

Papers

Superluminal transformations using either complex space-time or real space-time symmetry

Particle decay in six-dimensional relativity

Subluminal and superluminal transformations in six-dimensional special relativity

Comments on the use of three time dimensions in relativity

Transition from a continuous to a discrete space--time scheme.