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 , defined as the imaginary part of the local momentum field ℏ i ∇ ln ψ. For all wavefunctions in the Sobolev space H 1 (Ω), including those with nodal sets, we prove that the expectation value of p I vanishes identically -a structural consequence of the regularity of the probability density ρ = |ψ| 2 rather than a dynamical coincidence -so that the imaginary component carries no contribution to the physical momentum expectation value. The derivation makes no appeal to the polar form ψ = Re iS/ℏ and holds for any ψ ∈ H 1 (Ω). We further prove four structural results: (i) the quantum potential of Bohm arises entirely from p I , and its expectation value satisfies the exact identity ⟨V qu ⟩ = ⟨p 2 I ⟩/2m, proved via Sobolev integration by parts without invoking the polar form -from which the known Fisher information relation ⟨V qu ⟩ = ℏ 2 I[ρ]/8m [1, 2] follows as a corollary; (ii) the group velocity of a Bloch band is the density-weighted average of the local transport velocity p R /m with no contribution from p I ; (iii) the decomposition is gauge-invariant under magnetic minimal coupling, with the vector potential acting exclusively on the real component p R ; and (iv) the result extends to curved Riemannian manifolds, establishing the decomposition as an intrinsic property of the wavefunction independent of any particular representation. Together, these results provide a rigorous operator-theoretic framework for the imaginary weak momentum, while preserving the spectral integrity of linear quantum theory.