*Unified Field Theories

Abstract

This chapter touches – symmetry-inspired – theoretical considerations which are mathematically far advanced, but so far are not at all required by deviations between the predictions of the Standard Model or general relativity and experiment.

The success of the electro-weak unification within the Glashow-Salam-Weinberg model motivated the search for “Grand Unified Theories”. The most natural larger symmetry group is SU(5), but its predictions are already ruled out by experiment. It is nevertheless treated in this chapter since on the SU(5) gauge theory all essential features and necessary requirements of a grand unification program can be made visible. The other favorite is based on a local SO(10) symmetry group, whose capabilities for hosting known and further hypothetical gauge bosons in appropriate multiplets is also discussed in the first section of this chapter.

The next section deals with models of deriving Yang-Mills and gravity in four dimensions from gravity formulated in higher dimensions, and commonly called Kaluza-Klein models. The different contributions by T. Kaluza and by O. Klein are carefully disentangled, in order to see the merits and the problems of this program. The important feature of “compactification” of the higher dimension beyond 4 is outlined in more detail, and specifically a model of spontaneous compactification is illustrated. Further the question of how the Standard Model can be hosted by a 4 + 7 gravitational theory is addressed.

Another topic is supersymmetry and supergravity, an approach with high aspirations in the 1970’s, but still en vogue today in the string models. The merits of placing fermions and bosons, that is “matter” particles and “force particles”, into multiplets of a larger symmetry group with Grassmann even and odd generators are described. In the spirit of “internal” symmetries there are global and local variants of supersymmetry. The globally supersymmetric Wess-Zumino model and the supersymmetric Yang-Mills theory serve to motivate the structure of the super-Poincaré algebra, and its representation in turn organizes the irreducible multiplets. It is shown why it is the case that localized supersymmetry leads to supergravity, and how one can get supergravity from gauged supergroups.

The final section of this already speculative chapter touches topical ultra-speculative ideas, amongst others the string model.

Further progress lies in the direction of making our equations invariant under wider and still wider transformations.