Mathematical and physical aspects of gauge theories

Conclusion

It is now a broadly-admitted fact that gauge theories provide intense research stimulation for both mathematicians and physicists. We tried to show that, as far as bundles and connections are the adequate minimal concepts to formalize the very act of beings which produce their own space in a consistent way, or, which is the same, the very act of informing a basis world B through a parametrized space of operations E, gauge theories realize a true translation between geometrical and physical information.

In Section I.A, we tried to show how the concept of communication via experimentation finds this equivalence and makes a necessary collision between mathematics and physics.

We gave some striking theoretical and practical facts: the obtaining of interaction Lagrangians, Higgs phenomenon and good energy behaviour of graphs arising from broken symmetries, the ‘miraculous’ encounter between connection and path integral formalism. We could have added the relation with twistor theory, the theory of strong interactions, super gravity theories where the basis world B is not even a manifold. Some authors even speak of geometrodynamics and ‘pregeometry’ [21] and would like to understand physics as a cosmogony.

In a way, this makes us understand Einstein's unsatisfied mind when he wrote down his famous equation. The equivalence between a curvature tensor and an impulsion-energy tensor requires a careful analysis of the act of setting consistent spaces (via experimentation) when one doesn't reduce material beings (monads) to material points ruled by the universal Newton's observer.

So, as far as gauge theories pretend to found and explain the process of translation between Geometry and Physics, they provide an extremely solid epistemological basement for the so-called mathematical Physics.