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PreprintarXiv licencearXiv

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

  • A. N. Khondker
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Abstract

We establish a rigorous functional-analytic framework for the imaginary component pI of the complex weak momentum pQ = pR + ipI = ℏ/i∇lnψ. For every wavefunction in the Sobolev space H1(Ω), including those with nodal sets, we prove that the expectation value of pI vanishes identically, as a consequence of the regularity of ρ= |ψ|2 and without appeal to the polar form ψ= ReiS/ℏ. We further show that \textnormal{(i)} the Bohm quantum potential arises entirely from pI and satisfies ⟨ Vqu⟩ = ⟨ pI2⟩/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 pR/m, with no contribution from pI; \textnormal{(iii)} the decomposition is gauge-invariant under magnetic minimal coupling, the vector potential acting only on pR; and \textnormal{(iv)} the results extend to curved Riemannian manifolds.

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Cite

BibTeXRIS
BibTeX
@misc{khondker2026imaginary,
  title = {{The Imaginary Component of the Weak Momentum: Spectral Vanishing and Gauge Invariance}},
  author = {A. N. Khondker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12112},
  eprint = {2610.12112},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12112},
}
RIS
TY  - UNPB
TI  - The Imaginary Component of the Weak Momentum: Spectral Vanishing and Gauge Invariance
AU  - A. N. Khondker
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12112
UR  - https://arxiv.org/abs/2610.12112
AB  - We establish a rigorous functional-analytic framework for the imaginary component p_I of the complex weak momentum p_Q = p_R + ip_I = ℏ/i∇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/ℏ. We further show that \textnormal{(i)} the Bohm quantum potential arises entirely from p_I and satisfies ⟨ V_qu⟩ = ⟨ p_I^2⟩/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.
ER  -