Probability Geometry in Quantum Mechanics: Recovering Amplitude–Phase Coupling Beyond the Born Norm

Standard quantum mechanics defines probability as the squared Euclidean norm of the wavefunction, ρ = |Ψ|2. While mathematically convenient, this choice enforces an orthogonality between amplitude and phase that is not dictated by physical necessity. In this article, we show that this convention eliminates a generic amplitude–phase coupling term that would otherwise appear when quantum dynamics is written in real variables. By replacing the Euclidean norm with a positive, non-orthogonal probability metric, we construct a minimal modification of quantum mechanics in which probability remains real, positive, and conserved, but no longersuppresses amplitude–phase cross terms. The resulting theory preserves linearity, superposition, and the Schrödinger spectrum, while predicting new background and vacuum-scale effects. We argue that standard quantum mechanics emerges as a special orthogonal limit of a more general probability geometry.