The Imaginary Component of the Weak Momentum: Spectral Vanishing and Gauge Invariance

Abstract:We establish a rigorous functional-analytic framework for the imaginary component $p_I$ of the complex weak momentum $p_Q = p_R + ip_I = \frac{\hbar}{i}\nabla\ln\psi$. For every wavefunction in the Sobolev space $H^1(\Omega)$, including those with nodal sets, we prove that the expectation value of $p_I$ vanishes identically, as a consequence of the regularity of $\rho = |\psi|^2$ and without appeal to the polar form $\psi = Re^{iS/\hbar}$. 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.

arXiv:2610.12112 [quant-ph]

(or arXiv:2610.12112v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.12112 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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