  We propose a new inner product for scalar fields that are solutions of the
Klein-Gordon equation with $m^2<0$. This inner product is non-local, bearing an
integral kernel including Bessel functions of the second kind, and the
associated norm proves to be positive definite in the subspace of oscillatory
solutions, as opposed to the conventional one. Poincar\'e transformations are
unitarily implemented on this subspace, which is the support of a unitary and
irreducible representation of the proper orthochronous Poincar\'e group. We
also provide a new Fourier Transform between configuration and momentum spaces
which is unitary, and recover the projection onto the representation space.
This new scenario suggests a revision of the corresponding quantum field
theory.
