  The Lorentz contraction of bound states in field theory is often appealed to
in qualitative descriptions of high energy particle collisions. Surprisingly,
the contraction has not been demonstrated explicitly even in simple cases such
as the hydrogen atom. It requires a calculation of wave functions evaluated at
equal (ordinary) time for bound states in motion. Such wave functions are not
obtained by kinematic boosts from the rest frame. Starting from the exact
Bethe-Salpeter equation we derive the equal-time wave function of a
fermion-antifermion bound state in QED, i.e., positronium or the hydrogen atom,
in any frame to leading order in alpha. We show explicitly that the bound state
energy transforms as the fourth component of a vector and that the wave
function of the fermion-antifermion Fock state contracts as expected.
Transverse photon exchange contributes at leading order to the binding energy
of the bound state in motion. We study the general features of the
corresponding fermion-antifermion-photon Fock states, and show that they do not
transform by simply contracting. We verify that the wave function reduces to
the light-front one in the infinite momentum frame.
