I'll discuss an unexpected connection between the symmetries of the
classical Kepler problem, and a special four-dimensional quantum field
theory. The Kepler problem possess a non-obvious integral of motion, the
Runge-Lenz vector, which ensures closure of the planetary orbits in the
two-body approximation. The same symmetry in quantum mechanics accounts
for the approximate degeneracies of the hydrogen atom, which are however
broken by relativistic effects. In this talk I will show how the
recently discovered integrability of the so-called N=4 super Yang-Mills
model originates precisely from the Runge Lenz vector, providing a fully
consistent quantum field theory model in which it is conserved. I will
review some of its implications, in particular how it has led to the
exact computation of the four- and five-particles amplitudes in this
model to all values of the coupling, and how it severely constrains the
higher-point amplitudes. Finally I will discuss recent developments
concerning our understanding of collinear and Regge limits in this model.