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A contact form on an odd dimensional manifold determines Reeb dynamics which encapsulates generic classical mechanics. This can be quantized by generalizing Fedosov's approach to symplectic structures. The result is a formulation of quantum mechanics generally covariant with respect to choices of time, generalized position and momentum. Just as many structures in general relativity are effectvely studied by considering causal structures--i.e. conformal classes of metrics--it is also useful to consider conformal classes of contact forms, or in other words contact structures (odd dimensional manifolds whose tangent bundle is equipped with a maximally non-integrable hyperplane distribution). In the case of general relativity (GR), key machinery for the study of conformal manifolds is the so-called tractor calculus. This theory is part of general notion of a parabolic geometry, and thus extends nicely to contact structures and, as will be explained, their quantization. Key notions are parallel sections with respect to distinguished connections; just as these provide the link between conformal and Einstein structures in GR, here they relate contact structures to quantum mechanics. The general theory will be illustrated for the example of S^3 with its standard contact structure.

Joint work with Roger Casals and Gabe Herczeg