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The historical Klein-Gordon transformation of complex-valued first-order intime Schroedinger equations iterates these in a naively straightforward waywhich changes them into complex-valued second-order in time equations that havea plethora of extraneous solutions -- the transformation is anoperator-calculus analogue of the squaring of both sides of an algebraicequation. The real and imaginary parts of a Schroedinger equation, however, arewell known to be precisely the dynamical equation pair of the real-valuedclassical Hamiltonian functional which is numerically equal to the expectationvalue of that Schroedinger equation's Hermitian Hamiltonian operator. Thepurely real-valued second-order in time Euler-Lagrange equation of thecorresponding classical Lagrangian functional is also isomorphic to thatSchroedinger equation, and for symmetric Hamiltonians has exactly the sameformal appearance as the corresponding naive complex-valued Klein-Gordonequation, but none of the latter's extraneous solutions. These quantumSchroedinger-equation isomorphisms to classical Euler-Lagrange equations arethe technical manifestation of a key theoretical aspect of the principle ofcomplementarity, one which is elegantly illustrated by the isomorphicfree-photon wave-function complement to the vector potential of source-freeclassical electrodynamics.

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