Introduction and Orientation

Abstract

Einstein’s theory of general relativity identifies spacetime curvature with a gravitational interaction. The theory is formulated primarily in a geometrical language, while the group-theoretical concepts remain more in the background. In an operational spacetime approach, the hierarchy will be reversed.

A reversal of the priorities also for a mathematical treatment of manifolds is announced by S. Helgason in the preface to his book Groups and Geometric Analysis: The role of group theory in elementary classical analysis is a rather subdued one, the motion group of $\mathcal{R}^3$ enters rather implicitly in standard vector analysis,… In contrast our point of view here is to place a natural transformation group of a given space in the foreground. We use this group as a guide for the principal concepts.

I could not agree more with such a program, here for the operational treatment of the spacetime manifold. In the following, the basic structures of physics will be defined and considered, rather restrictively, via the representations of group and Lie algebra operations, which come in two forms; external, acting on spacetimelike degrees of freedom, and internal, acting on chargelike ones. The degrees of freedom are given by the dimensions of the representation spaces. The representations are characterized by invariants which will be collected into the concepts of interactions and objects (particles).