The Imaginary Component of the Weak Momentum: Spectral Vanishing and Gauge Invariance
We establish a rigorous functional-analytic framework for the imaginary component $p_I$ of the complex weak momentum $p_Q = p_R + ip_I = \frac{\hbar}{i}\nabla\lnψ$. For every wavefunction in the Sobolev space $H^1(Ω)$, including those with nodal sets, we prove that the expectation value of $p_I$ vanishes identically, as a consequence of the regularity of $ρ= |ψ|^2$ and without appeal to the polar form $ψ= Re^{iS/\hbar}$. We further show that \textnormal{(i)} the Bohm quantum potential arises entirely from $p_I$ and satisfies $\langle V_{\mathrm{qu}}\rangle = \langle p_I^2\rangle/2m$, from which the known Fisher information relation follows as a corollary; \textnormal{(ii)} the group velocity of a Bloch band is the density-weighted average of $p_R/m$, with no contribution from $p_I$; \textnormal{(iii)} the decomposition is gauge-invariant under magnetic minimal coupling, the vector potential acting only on $p_R$; and \textnormal{(iv)} the results extend to curved Riemannian manifolds.