Deformation and Contraction of Symmetries in Special Relativity

Abstract

This dissertation gives an account of the fundamental principles underlying two conceptionally different ways of embedding Special Relativity into a wider context. Both of them root in the attempt to explore the full scope of the Relativity Postulate. The first approach uses Lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifiable sense near to those in Special Relativity, while being rather far away in a qualitative way. The corresponding models for spacetime are seen to be four-dimensional versions of the prototypical planar geometries associated with the work of Cayley and Klein. The close relationship between algebraic and geometric methods displayed by these considerations is being substantialized in terms of light-like spacetime extensions. The second direction of departures from Special Relativity stresses and develops the algebraic view on spacetime by considering Hopf instead of Lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modifications of the laws of Special Relativity at length scales comparable to the Planck length. The twofold character of this work, and the presentation of an example for the fully geometric character of a specific Hopf algebraic deformation of the PoincareI algebra, enable a conclusion that speculates on a possible relationship between the two developed viewpoints via the technique of nonlinear realizations. A non-perturbative approach to the latter is given which generalizes to all the considered geometries.