The space-time symmetry group of a spin 1/2 elementary particle

The space-time symmetry group of a model of a relativistic spin 1/2 elementary particle, which satisfies Dirac's equation when quantized, is analyzed. It is shown that this group, larger than the Poincare group, also contains space-time dilations and local rotations. It has two Casimir operators, one is the spin and the other is the spin projection on the body frame. Its similarities with the standard model are discussed. If we consider this last spin observable as describing isospin, then, this Dirac particle represents a massive system of spin 1/2 and isospin 1/2. There are two possible irreducible representations of this kind of particles, a colourless or a coloured one, where the colour observable is also another spin contribution related to the zitterbewegung. It is the spin, with its twofold structure, the only intrinsic property of this Dirac elementary particle.Comment: Contribution to the JINR(Dubna) SPIN05 workshop, 18 pages, 1 figure. Abstract and minor changes of sections 2 to

An interaction Lagrangian for two spin 1/2 elementary Dirac particles

The kinematical formalism for describing spinning particles developped by the author is based upon the idea that an elementary particle is a physical system with no excited states. It can be annihilated by the interaction with its antiparticle but, if not destroyed, its internal structure can never be modified. All possible states of the particle are just kinematical modifications of any one of them. The kinematical state space of the variational formalism of an elementary particle is necessarily a homogeneous space of the kinematical group of spacetime symmetries. By assuming Poincare invariance we have already described a model of a classical spinning particle which satisfies Dirac's equation when quantized. We have recently shown that the spacetime symmetry group of this Dirac particle is larger than the Poincare group. It also contains spacetime dilations and local rotations. In this work we obtain an interaction Lagrangian for two Dirac particles, which is invariant under this enlarged spacetime group. It describes a short- and long-range interaction such that when averaged, to supress the spin content of the particles, describes the instantaneous Coulomb interaction between them. As an application, we analyse the interaction between two spinning particles, and show that it is possible the existence of metastable bound states for two particles of the same charge, when the spins are parallel and provided some initial conditions are fulfilled. The possibility of formation of bound pairs is due to the zitterbewegung spin structure of the particles because when the spin is neglected, the bound states vanish

Is General Relativity a simpler theory?

Gravity is understood as a geometrization of spacetime. But spacetime is also the manifold of the boundary values of the spinless point particle in a variational approach. Since all known matter, baryons, leptons and gauge bosons are spinning objects, it means that the manifold, which we call the kinematical space, where we play the game of the variational formalism of an elementary particle is greater than spacetime. This manifold for any mechanical system is a Finsler metric space such that the variational formalism can always be interpreted as a geodesic problem on this space. This manifold is just the flat Minkowski space for the free spinless particle. Any interaction modifies its flat Finsler metric as gravitation does. The same thing happens for the spinning objects but now the Finsler metric space has more dimensions and its metric is modified by any interaction, so that to reduce gravity to the modification only of the spacetime metric is to make a simpler theory, the gravitational theory of spinless matter. Even the usual assumption that the modification of the metric only involves dependence of the metric coefficients on the spacetime variables is also a restriction because in general these coefficients are dependent on the velocities. In the spirit of unification of all forces, gravity cannot produce, in principle, a different and simpler geometrization than any other interaction.Comment: 10 pages 1 figure, several Finsler metric examples and a conclusion section added. Minor correction

Left-orderings on free products of groups

We show that no left-ordering on a free product of (left-orderable) groups is isolated. In particular, we show that the space of left-orderings of free product of finitely generated groups is homeomorphic to the Cantor set. With the same techniques, we also give a new and constructive proof of the fact that the natural conjugation action of the free group (on two or more generators) on its space of left-orderings has a dense orbit.Comment: 13 page

On spaces of Conradian group orderings

We classify $C$-orderable groups admitting only finitely many $C$-orderings. We show that if a $C$-orderable group has infinitely many $C$-orderings, then it actually has uncountably many $C$-orderings, and none of these is isolated in the space of $C$-orderings. As a relevant example, we carefully study the case of Baumslag-Solitar's group B(1,2). We show that B(1,2) has four $C$-orderings, each of which is bi-invariant, but its space of left-orderings is homeomorphic to the Cantor set