  We obtain analytic expressions for the time correlation functions of a liquid
of spherical particles, exact in the limit of high dimensions $d$. The
derivation is long but straightforward: a dynamic virial expansion for which
only the first two terms survive, followed by a change to generalized spherical
coordinates in the dynamic variables leading to saddle-point evaluation of
integrals for large $d$. The problem is thus mapped onto a one-dimensional
diffusion in a perturbed harmonic potential with colored noise. At high
density, an ergodicity-breaking glass transition is found. In this regime, our
results agree with thermodynamics, consistently with the general Random First
Order Transition scenario. The glass transition density is higher than the best
known lower bound for hard sphere packings in large $d$. Because our
calculation is, if not rigorous, elementary, an improvement in the bound for
sphere packings in large dimensions is at hand.
