On the Duality of Momentum in Dirac's Energy-Momentum Space

This work resolves a foundational tension in Dirac's formulation of relativistic quantum mechanics: momentum appears in two distinct but undifferentiated roles—as a dispersion-locked parameter generating unitary evolution in the relativistic wave equation, and as a free phase variable subject to symplectic flow in the constraint formalism. We demonstrate that this duality reflects an intrinsic Lorentzian structure in energy-momentum space, characterized by the interval dΣ² = dE²/c² − dp². This geometry divides naturally into three sectors: symmetric (dispersion-locked), null (massless, E/|p| = c), and antisymmetric (free-momentum) regimes, each governed by different geometric constraints. Dispersion emerges as the global integral of an exact action one-form, while the (E,p) symplectic structure arises locally as the exterior derivative of the action's inexact form θ_E = E dp. Energy quantization appears as a topological stability condition requiring closed spectral loops to satisfy ∮ E dp = nh, defining "proto-eigenstates" whose Fourier encodings yield standard Hilbert-space eigenstates. We recover both the Bohr spectrum and Dirac fine-structure energies from spectral phase closure, revealing α = e²/(ℏc) as a geometric action-normalization ratio between Maxwell U(1) curvature and Lorentz spinor action—not a coupling parameter from QED perturbation theory. The framework synthesizes unitary flow and symplectic uncertainty into a single bilinear Lorentzian structure, showing quantum states as objective in their symmetric unitary existence but path-dependent in their antisymmetric observed manifestation. This geometric reconstruction bridges Dirac's wave mechanics and constraint formalism through native (E,p) phase space, offering a foundation for quantum theory that treats energy-momentum as primary coordinates rather than conjugate variables derived from spacetime position.