  The classical Kepler problem, as well as its quantum mechanical version, the
Hydrogen atom, enjoy a well-known hidden symmetry, the conservation of the
Laplace-Runge-Lenz vector, which makes these problems superintegrable. Is there
a relativistic quantum field theory extension that preserves this symmetry? In
this Letter we show that the answer is positive: in the non-relativistic limit,
we identify the dual conformal symmetry of planar $\mathcal{N}=4$ super
Yang-Mills with the well-known symmetries of the Hydrogen atom. We point out
that the dual conformal symmetry offers a novel way to compute the spectrum of
bound states of massive $W$ bosons in the theory. We perform nontrivial tests
of this setup at weak and strong coupling, and comment on the possible
extension to arbitrary values of the coupling.
