  We study conformal boundary conditions for the theory of a single real scalar
to investigate whether the known Dirichlet and Neumann conditions are the only
possibilities. For this free bulk theory there are strong restrictions on the
possible boundary dynamics. In particular, we find that the bulk-to-boundary
operator expansion of the bulk field involves at most a `shadow pair' of
boundary fields, irrespective of the conformal boundary condition. We
numerically analyze the four-point crossing equations for this shadow pair in
the case of a three-dimensional boundary (so a four-dimensional scalar field)
and find that large ranges of parameter space are excluded. However a `kink' in
the numerical bounds obeys all our consistency checks and might be an
indication of a new conformal boundary condition.
