Book: Kinematical Theory of Elementary Spinning Particles
By Martin Rivas
Summary:
It is a classical theory based on three fundamental principles:
- Restricted Relativity Principle (A Kinematical group of space-time symmetries)
- Variational Formalism (Lagrangian description of dynamics)
- Atomic Principle (Definition of elementary particle)
For the description of the free motion of an elementary particle it is not necessary the use of some explicit free Lagrangian. The analysis of the Noether's constants of the motion supplies the dynamics in any reference frame. The Atomic Principle states that an elementary particle has no excited states and this leads for the external interaction Lagrangian to the minimal coupling.
The group parameters of the kinematical group represent the maximum number and kind of variables we need to describe the initial and final states of an elementary particle in a variational formulation. This manifold of the boundary variables of an elementary particle in a variational formalism is a homogeneous space of the kinematical group. This manifold is always a metric Finsler space and the metric is determined by the Lagrangian. When the particle interacts, the Lagrangian, and therefore the metric, changes. Euler-Lagrange equations are geodesic equations in this manifold.
An elementary particle is a localized and orientable mechanical system. It is a system of six degrees of freedom: three represent the location of a point (the center of charge CC). The other three represent the orientation of a comoving Cartesian frame attached to that point. Particles move and rotate. As described by Frenet-Serret (1847) equations, the point satisfies fourth order differential equations and in the free motion it describes a helix. Zitterbewegung is already contained in F-S equations.
If this moton satisfies the Relativity Principle, the motion of the CC has to be at an unreachable velocity for the inertial observers. The CC is moving at the speed c, and the kinematical group must be, at least, the Poincaré group.