#### Date of Award

Spring 5-22-1981

#### Document Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Asim O. Barut

#### Second Advisor

Homer G. Ellis

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

#### Abstract

The Poincairé group was derived from actions on a four-dimensional space-time continuum, Minkowsi space. In this work, we logically reverse this and take the group itself as physically fundamental, then explore the possibility of the group action in an arbitrary number of finite dimensions. The intention is to study what dimensions different from four reveal about physics, under the assumption of Lorentz invariance. We induce real irreducible representations of the covering group, SL(2,C), and report all the little groups and the dimensions and orbits of the fixed vectors. One significant result is that for a group to have non-trivial vectors fixed by rotations or boosts, the number of dimensions must be the square of an integer. Thus, four-dimensional reality is the smallest non-trivial space-time with the expected non-trivial action of rotations and boosts. We also report the general form of the second-order invariant tensors, the generalization of the metric of Special Relativity, as well as all third-order invariants in all finite-dimensional spaces. Finally, we explore some of the properties of the nine-dimensional space that is the next larger space that has the familiar action of rotations and boosts, i.e. non-trivial fixed vectors. We report specifics of the geometry implied by the group action in this nine-dimensional space, as well as the form of two fourth-order invariants.

#### Recommended Citation

Weiss, Marc, "A Family of Groups Generalizing the Poincaré Group and Associated Physical Systems" (1981). *Mathematics Graduate Theses & Dissertations*. 66.

https://scholar.colorado.edu/math_gradetds/66

#### Included in

Algebra Commons, Elementary Particles and Fields and String Theory Commons, Other Mathematics Commons, Other Physics Commons, Quantum Physics Commons