bibkey: tolias2025kineticmoments authors: P. Tolias; T. Dornheim; J. Vorberger year: 2025 title: “Kinetic contribution to the arbitrary order odd frequency moments of the dynamic structure factor” doi: 10.1002/ctpp.70090 url: https://arxiv.org/abs/2508.17810v1 claim: “Therefore, our conjecture states that the following result holds for the interacting uniform electron gas M_{\mathrm{S,KIN}}^{(2k+1)}(q)=\left(\frac{\hbar{q}^2}{2m}\right)^{2k+1}\frac{1}{2k+2}\sum_{i=0}^{k}\binom{2k+2}{2i+1}\left(\frac{2}{\hbar{q}}\right)^{2i}\langle{p}^{2i}\rangle_{0}\,, where $k$ is an arbitrary non-negative integer.” strata_touched:

• D5/S3/Quantum/KineticMoments/IsotropicAverage
• D5/S3/Quantum/KineticMoments/OddMoment license: citation-only triage: anchor

Kinetic odd frequency moments

Verified locator

DOI: https://doi.org/10.1002/ctpp.70090

Source: https://arxiv.org/abs/2508.17810v1

The locators refer to the arXiv v1 PDF: Sec. 2.2, Eq. (9), p. 3; Sec. 2.3, Eqs. (16)–(17), p. 5; Sec. 2.4, pp. 5–6. The journal locator is Contributions to Plasma Physics (2026), e70090.

Sec. 2.2, p. 3:

The pure kinetic contribution to the odd dynamic structure factor frequency moments of arbitrary order is obtained from Eq.(4) by considering only the kinetic part of the Hamiltonian, i.e. setting $\hat{H}\equiv \hat{K}$.

Equation (9) splits the $2k+1$ commutator nests between the density operator and its conjugate, with $\ell$ on the conjugate side. The kinetic Hamiltonian is $\hat{K}={\sum }_j|{\hat{\mathbf{p}}}_j{|}^2/(2m)$.

Sec. 2.3, p. 5:

Therefore, our conjecture states that the following result holds for the interacting uniform electron gas

$$M_{\mathrm{S},\mathrm{KIN}}^{(2k+1)}(q)={\left(\frac{\hslash q^2}{2m}\right)}^{2k+1}\frac{1}{2k+2}\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{2k+2}{2i+1}\right){\left(\frac{2}{\hslash q}\right)}^{2i}⟨p^{2i}{⟩}_0,$$

where $k$ is an arbitrary non-negative integer.

Equation (17) expresses the same assertion as a nested-commutator average. Sec. 2.4 evaluates the orders $k=0,1,2,3$ directly.

Momentum-space encoding and scope

The configuration carrier is Fin N → EuclideanSpace ℝ (Fin 3). The density operators sum pullbacks by the single-particle shifts ${\mathbf{p}}_j\mapsto {\mathbf{p}}_j\pm \hslash \mathbf{q}$; the kinetic operator multiplies by ${\sum }_j|{\mathbf{p}}_j{|}^2/(2m)$. The commutator average is the integral of its proved multiplier against one momentum-configuration probability measure. Its per-particle even moment is $N^{-1}{\sum }_j\int |{\mathbf{p}}_j{|}^{2i}d\nu$.

Simultaneous determinant-$+1$ rotations preserve $\nu$, and integrability of $|{\mathbf{p}}_j{|}^{2k}$ is a hypothesis for each particle. No independence, exchangeability, fermionic symmetry or Gaussian law is imposed. The statement constructs no thermodynamic-limit state, proves no finiteness of Coulomb moments and asserts nothing about the full non-kinetic moment.