52,717 research outputs found

### 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

### 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

### 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

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