Symmetric Spaces in Relativity and Quantum Theories

Abstract

The following is an attempt to fill in the gap between some modern mathematical concepts and physics. The mathematics basically used is the differential geometry of symmetric spaces, which was formulated in a new way by O. Loos in his two books. He has shown that the old definition of a symmetric space coincides with his axioms (S1) to (S4) for a symmetric space as a manifold with multiplication. This definition makes the analogy with Lie groups (more general with arbitrary groups) obvious, where only the multiplication is changed to a (Lie) group multiplication. One can ask nearly all questions, which are solved for Lie groups, in the same way for symmetric spaces, and one can answer most of them! The most important concepts, related to a Lie group is its Lie algebra. In the same sense, there is a linear structure on the tangent space of a symmetric space, the “Lie triple system”. A third example of this kind is discussed, where the manifold is a “domain of positivity”, its tangent space carrying a “formal real Jordan algebra” structure.