Klein-Gordon Transformation sans Extraneous Insertions: the Isomorphic Classical Complement to a Quantum System

The historical Klein-Gordon transformation of complex-valued first-order in time Schroedinger equations iterates these in a naively straightforward way which changes them into complex-valued second-order in time equations that have a plethora of extraneous solutions -- the transformation is an operator-calculus analogue of the squaring of both sides of an algebraic equation. The real and imaginary parts of a Schroedinger equation, however, are well known to be precisely the dynamical equation pair of the real-valued classical Hamiltonian functional which is numerically equal to the expectation value of that Schroedinger equation's Hermitian Hamiltonian operator. The purely real-valued second-order in time Euler-Lagrange equation of the corresponding classical Lagrangian functional is also isomorphic to that Schroedinger equation, and for symmetric Hamiltonians has exactly the same formal appearance as the corresponding naive complex-valued Klein-Gordon equation, but none of the latter's extraneous solutions. These quantum Schroedinger-equation isomorphisms to classical Euler-Lagrange equations are the technical manifestation of a key theoretical aspect of the principle of complementarity, one which is elegantly illustrated by the isomorphic free-photon wave-function complement to the vector potential of source-free classical electrodynamics.