Real Classical Waves in Quantum Configuration Space

We show that the configuration-space formalism of quantum mechanics represents real classical waves exactly, with the temporal Hilbert transform as the imaginary unit, as in unified quantum mechanics. For a linear, nondissipative wave equation $\partial_t^2u+\Omega^2u=0$ whose positive frequency operator $\Omega$ has a spectral gap, the temporal Hilbert transform acts on solutions as $\mathcal H_t=-\Omega^{-1}\partial_t$. The completion $\Psi=u-i\mathcal H_t u$ is therefore an exact bijection onto solutions of $i\partial_t\Psi=\Omega\Psi$, and $\mathcal H_t$ coincides with the positive-frequency complex structure of canonical field quantization, inducing a Kahler structure on the space of real waves. A canonical normalization identifies the quantum norm with the wave action and the Hamiltonian expectation with the wave energy. For narrowband waves near cutoff, the Schrodinger equation follows with an explicit error bound. Symmetric tensor powers of the one-quantum wave function yield real configuration-space representations whose multi-time form retains each classical wave equation unchanged and whose amplitudes are classical joint-detection amplitudes. For Kerr media, the classical limit corresponds to the mean-field limit, with particle number equal to the wave action in units of $\hbar$. Simulations confirm the exact identities to machine precision, show that the error bound is sharp within a factor of two, and verify the mean-field convergence.