The Imaginary Component of the Weak Momentum: Spectral Vanishing and Gauge Invariance
The paper develops a functional-analytic framework to analyze complex weak momentum $p_Q = p_R + ip_I$. It proves that the imaginary component, $p_I$, vanishes identically for any wavefunction in Sobolev space $H^1(\Omega)$ because it is determined by boundary data of the probability density $\rho=|\psi|^2$. This decomposition separates transport ($p_R$) from amplitude deformation ($p_I$), establishing a gauge-invariant language for quantum observables and linking the quantum potential to Fisher information.