Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation

Although describing very different physical systems, both the Klein–Gordon equation for tachyons (m2<0) and Helmholtz share a remarkable property: unitary irreducible representation of corresponding invariance group on suitable subspace solutions is only achieved if non-local scalar product defined. Then, subset oscillatory supports unirrep Euclidean group, with m2<0 tachyonic Poincaré group. As consequence, these systems also similar structures, such as certain singularized projectors spaces, but they must be treated carefully in each case. We analyze differences analogies, compare equations conventional m2>0 equation, provide unified framework products three equations.