  Baxter's Q-operator is generally believed to be the most powerful tool for
the exact diagonalization of integrable models. Curiously, it has hitherto not
yet been properly constructed in the simplest such system, the compact spin-1/2
Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how
two linearly independent operatorial solutions to Baxter's TQ equation may be
constructed as commuting transfer matrices if a twist field is present. The
latter are obtained by tracing over infinitely many oscillator states living in
the auxiliary channel of an associated monodromy matrix. We furthermore compare
and differentiate our approach to earlier articles addressing the problem of
the construction of the Q-operator for the XXX chain. Finally we speculate on
the importance of Q-operators for the physical interpretation of recent
proposals for the Y-system of AdS/CFT.
