Cite: A sharp obstruction to local $(\rho,T)$ Dirac interpolation and an exact fluctuation--current bridge
Citation formats for prexiv:260513.cj6fjm. Use /cite.bib or /cite.ris for the raw files (TODO).
BibTeX
@misc{bai2026_260513cj6fjm,
title = {A sharp obstruction to local $(\rho,T)$ Dirac interpolation and an exact fluctuation--current bridge},
author = {Dong Bai},
year = {2026},
note = {PreXiv id: prexiv:260513.cj6fjm},
doi = {10.99999/prexiv:260513.cj6fjm},
url = {https://prexiv.example/m/prexiv:260513.cj6fjm},
}
RIS
TY - GEN
TI - A sharp obstruction to local $(\rho,T)$ Dirac interpolation and an exact fluctuation--current bridge
AU - Dong Bai
PY - 2026
DO - 10.99999/prexiv:260513.cj6fjm
ID - prexiv:260513.cj6fjm
UR - https://prexiv.example/m/prexiv:260513.cj6fjm
AB - A long-standing question in the Lieb--Oxford programme asks whether the universal lower bound on the indirect Coulomb energy of fermions admits a pointwise local refinement whose low-kinetic-ratio limit reproduces the Dirac exchange constant. We show that the answer is no, in a sharp form. Let \(\rho_\Psi(x)\) and \(T_\Psi(x)\) denote the one-particle density and local kinetic energy density of a normalized antisymmetric \(N\)-fermion wavefunction~\(\Psi\), and let \(E_{\xc}(\Psi)\) be its indirect Coulomb (exchange-correlation) energy. Write \(C_{\mathrm{cell}}:=\tfrac{3}{5}(4\pi/3)^{1/3}\approx 0.96720\) and let \(C_{D}(q):=\tfrac{3}{4}(6/\pi)^{1/3}q^{-1/3}\) be the Dirac constant for spin multiplicity~\(q\). We prove that any bounded \(F:[0,\infty)\to[0,\infty)\) with \(\limsup_{r\downarrow 0}F(r)<C_{\mathrm{cell}}\) fails the natural local lower bound for some antisymmetric \(\Psi\) with~\(N\) arbitrarily large. Since \(C_{\mathrm{cell}}>C_{D}(q)\) for every \(q\ge 1\), no universal local lower bound depending only on \((\rho_\Psi,T_\Psi)\) can converge to \(C_{D}(q)\) in the low-\(\mathcal R\) regime. As a positive companion we establish a Gaussian scale-window identity that rewrites the indirect energy of an admissible kernel family exactly in terms of local number variances, and combine it with a canonical-commutator uncertainty inequality to bound it from below by a conjugate-current variance.
ER -
Plain text
Dong Bai (2026). A sharp obstruction to local $(\rho,T)$ Dirac interpolation and an exact fluctuation--current bridge. PreXiv prexiv:260513.cj6fjm, doi:10.99999/prexiv:260513.cj6fjm.