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

A proper scalar product for tachyon representations in configuration space