The imaginary component of the weak momentum: spectral vanishing and gauge invariance
Year: 2026
Type: article
Abstract: We establish a rigorous functional-analytic framework for the imaginary component p I of the complex weak momentum p Q = p R + i p 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 ) satisfying standard boundary conditions, 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 = R e i S / ℏ and holds for any ψ . 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 I 2 ⟩ /
Author: A N Khondker
Institution: Clarkson University
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,325
OpenAlex ID: W7221160453