二维单位环面上来自多体量子吉布斯态的$\mathcal{P}(Φ)_2$理论
$\mathcal{P}(Φ)_2$ Theory from many-body quantum Gibbs states
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中文总结 AI 辅助
研究从多体量子吉布斯态推导二维单位环面上的$\mathcal{P}(Φ)_2$测度，构建对应量子模型，处理高阶相互作用时维克重整化产生低阶相互作用层次结构，利用对数稳定性估计控制低阶项。
中文摘要 AI 辅助
我们推导了二维单位环面上的$\mathcal{P}(Φ)_2$测度，作为多体量子吉布斯态的严格极限。特别地，对应于$\Phi^{2p}_2$测度的量子模型在巨正则系综中构建，具有一般对称的$p$体相互作用势，无需因式分解形式。与弗罗利希等人研究的$\Phi^4_2$情况不同，处理高阶相互作用时，维克重整化产生了完整的低阶相互作用层次结构，我们用图形形式系统组织。分析的一个关键要素是对非局部哈特里泛函和多体哈密顿量的对数稳定性估计，它在相互作用范围内是均匀的，使我们能用主导$p$体相互作用的正性控制维克反项产生的所有低阶项。
英文摘要
We derive the $\mathcal{P}(Φ)_2$ measure on the two-dimensional unit torus as the rigorous limit of many-body quantum Gibbs states. In particular, the quantum model corresponding to the $Φ^{2p}_2$ measure is formulated in the grand-canonical ensemble with a general symmetric $p$-body interaction potential which is not required to have a factorized form. Unlike in the $Φ^4_2$ case investigated by Fröhlich--Knowles--Schlein--Sohinger in \cite{FKSS25}, in the treatment of higher-order interactions, Wick renormalization generates a full hierarchy of lower-order interactions which we organize systematically using a graphical formalism. One key ingredient of our analysis is a logarithmic stability estimate for both the nonlocal Hartree functional and the many-body Hamiltonian that is uniform in the interaction range, allowing us to use the positivity of the leading $p$-body interaction to control all lower-order terms generated by the Wick counterterms.
