Cartan gravity, matter fields, and the gauge principle

Abstract Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang–Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: (1) a ‘contact vector’ V A which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being ‘rolled’ on top of it, and (2) a gauge connection A μ A B , here taken to be valued in the Lie algebra of S O ( 2 , 3 ) or S O ( 1 , 4 ) , which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan’s geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. In particular, the Hodge dual, which characterizes the structure of bosonic field equations, pops up automatically. Furthermore, the energy–momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy–momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang–Mills fields. This suggests a way to unify a U ( 1 ) gauge field with gravity into a S O ( 1 , 5 ) -valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to S O ( 1 , 3 ) × U ( 1 ) . The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions.