The role of conformal symmetry in gravity and the standard model
  In this paper we consider conformal symmetry in the context of manifolds with
general affine connection. We extend the conformal transformation law of the
metric to a general metric compatible affine connection, and find that it is a
symmetry of both the geodesic equation and the Riemann tensor. We derive the
generalised Jacobi equation and Raychaudhuri equation and show that they are
both conformally invariant. Using the geodesic deviation~(Jacobi) equation we
analyse the behaviour of geodesics in different conformal frames.
  Since we find that our version of conformal symmetry is exact in classical
pure Einstein's gravity, we ask whether one can extend it to the standard
model. We find that it is possible to write conformal invariant lagrangians in
any dimensions for vector, fermion and scalar fields, but that such lagrangians
are only gauge invariant in four dimensions. Provided one introduces a dilaton
field, gravity can be conformally coupled to matter.
