The Dirac sea of phase: Unifying phase paradoxes and Talbot revivals in multimode waveguides
The quantum mechanical description of phase remains a fundamental challenge, with theoretical efforts tracing from the early works of London and Dirac to discrete formalisms. In this work, we extend the action-angle formalism to the Helmholtz-Schr\"odinger equation by introducing a phase-dependent wavefunction $\phi(\theta, t)$ residing in the Hardy space $H^2(\mathbb{D})$. This mathematical structure, defined by functions analytic on the unit disk with square-integrable boundary values, naturally ensures the positivity of the energy spectrum while providing a rigorous framework for wave dynamics in photonic systems. We demonstrate that establishing a self-adjoint phase operator requires extending the Hilbert space to $L^2$, a procedure that necessitates the admission of negative energy states. We interpret these states through an analogy with the Dirac sea, where the existence of antiphase or antiphoton modes provides a conceptual framework for understanding the fundamental limits of phase localization and quantum uncertainty. This formalism is applied to light propagation in multimode waveguides characterized by anharmonic refractive index profiles. By mapping modal dispersion to our phase representation, we show that the deviation of propagation constants from linear spacing governs the spatial evolution of the optical field. This approach offers a clear mechanism for the emergence of periodic self-imaging known as the Talbot effect, the generation of fractional revivals, and the formation of complex fractal interference patterns, providing a robust toolkit for the characterization and design of multimode interference devices.