The Quantum Geometry of Supersymmetry and the Generalized Group Extension Problem

Abstract

We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the conventional concept of symmetry with that of exchange statistics and the spin-statistics relation. We show how this quantum group symmetry is reconstructed from the traditional (super) group symmetry, statistics and spin-statistics relation. The old question of extending the Poincare group to unify external and internal symmetries (solved by supersymmetry) is reexamined in the new framework. The reason why we should allow supergroups in this case becomes completely transparent. However, the true symmetries are not expressed by groups or supergroups here but by ordinary (not super) quantum groups. We show in this generalized framework that supersymmetry remains the most general unification of internal and space-time symmetries provided that all particles are either bosons or fermions. Finally, we demonstrate with some examples how quantum geometry provides a natural setting for the construction of super-extensions, super-spaces, super-derivatives etc.