Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations

We construct sesquilinear forms which are invariant under the similarity groups of the Klein–Gordon and Helmholtz equations. These give rise to positive definite inner products on subspaces of solutions of these equations. For the Klein–Gordon case, the known Poincaré-invariant inner product for positive-energy solutions is recovered. For the Helmholtz case, a new Euclidean-invariant inner product is presented which involves the function and its normal derivative to a line, integrated nonlocally over that line.