  Tractor Calculus is a powerful tool for analyzing Weyl invariance; although
fundamentally linked to the Cartan connection, it may also be arrived at
geometrically by viewing a conformal manifold as the space of null rays in a
Lorentzian ambient space. For dimension d conformally flat manifolds we show
that the (d+2)-dimensional Fefferman--Graham ambient space corresponds to the
momentum space of a massless scalar field. Hence on the one hand the null cone
parameterizes physical momentum excitations, while on the other hand, null rays
correspond to points in the underlying conformal manifold. This allows us to
identify a fundamental set of tractor operators with the generators of
conformal symmetries of a scalar field theory in a momentum representation.
Moreover, these constitute the minimal representation of the non-compact
conformal Lie symmetry algebra of the scalar field with Kostant--Kirillov
dimension d+1. Relaxing the conformally flat requirement, we find that while
the conformal Lie algebra of tractor operators is deformed by curvature
corrections, higher relations in the enveloping algebra corresponding to the
minimal representation persist. We also discuss potential applications of these
results to physics and conformal geometry.
