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Tractor Calculus is a powerful tool for analyzing Weyl invariance; althoughfundamentally linked to the Cartan connection, it may also be arrived atgeometrically by viewing a conformal manifold as the space of null rays in aLorentzian ambient space. For dimension d conformally flat manifolds we showthat the (d+2)-dimensional Fefferman--Graham ambient space corresponds to themomentum space of a massless scalar field. Hence on the one hand the null coneparameterizes physical momentum excitations, while on the other hand, null rayscorrespond to points in the underlying conformal manifold. This allows us toidentify a fundamental set of tractor operators with the generators ofconformal symmetries of a scalar field theory in a momentum representation.Moreover, these constitute the minimal representation of the non-compactconformal Lie symmetry algebra of the scalar field with Kostant--Kirillovdimension d+1. Relaxing the conformally flat requirement, we find that whilethe conformal Lie algebra of tractor operators is deformed by curvaturecorrections, higher relations in the enveloping algebra corresponding to theminimal representation persist. We also discuss potential applications of theseresults to physics and conformal geometry.

The so(d+2,2) Minimal Representation and Ambient Tractors: the Conformal Geometry of Momentum Space