Gauge Theory

All known interactions in nature are gauge theories. This just means that the theoretical description employs fields that are not observable. The best known example is Maxwell theory which uses four fields in the Lagrangian but of which two are redundant: In vacuum an electromagnetic wave has only two independent polarization degrees of freedom. Mathematically this is encoded into the Lagrangian through a symmetry which depends on arbitrary functions which can be given the structure of a group acting everywhere in spacetime. The relevant group for QCD is $\mathrm{SU}(3)$ , for the electroweak interaction it is $\mathrm{SU}(2) \times \mathrm{U}(1)$ and for GR on a spacetime manifold $M$ it is its group of diffeomorphisms. Gauge theory is therefore a unifying mathematical framework that encompasses all known interactions and presents a beautiful interface between geometry and physics, in particular differential geometry and the theory of fibre bundles.