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Cartan waywiser geometry and its application to the standard model

Jun 05
3:00 pm o'clock to 4:30 pm o'clock
SR 02.729

We provide a pedagogical introduction to Cartan geometry using the concept of idealized waywisers and embedded 2D manifolds. Waywisers are devices traditionally used to measure distance. By stripping away irrelevant features we arrive at the mathematical notion of idealized waywisers. These are mathematical “platonic” constructs which consist of a symmetric space (the “wheel” of the waywiser) which can be rolled along paths on the manifold and a contact point between the symmetric space and the manifold. To this end we introduce two mathematical variables: (i) a point of contact V^A and (ii) a so(3)-valued gauge connection A^{AB} that determines how much the idealized waywiser has rotated when rolled along a path. From the pair (V^A,A^{AB}) we can now construct all the familiar objects of standard differential geometry: metric tensor, affine connection, tetrad and co-tetrad, spin connection, torsion, Riemannian curvature, and so on. All these objects are different characterizations of the change of contact point. The generalization to higher dimensions and relativistic spacetimes is immediate and we show how to formulate general relativity in terms of the waywiser variables (V^A,A^{AB}) including the Holst action. We take the connection A^{AB} to either valued in so(1,4) or so(2,3) and show how we can either treat the contact point V^A as a dynamical variable subject to field equations of its own or as a non-dynamical absolute object which is a priori postulated. Next we turn to the matter fields of the standard model. Building on the above picture, in which gravity is fundamentally about “rolling”, we show that coupling a matter field to gravity can be achieved by the gauge principle, i.e. we simply add a “rolling” A to a matter field and then write down a suitable action containing the associated gauge covariant derivative. For example, coupling a scalar field phi to gravity simply consists of the replacement phi –> phi^A and a suitable polynomial action. The same is true for all other matter fields including Yang-Mills fields. Time permitting we end by a discussion of aspects of the electroweak theory.