发表机构
Université Paris-Saclay, CEA Institut de Recherche sur les Lois Fondamentales de l’Univers; Université Paris-Saclay, École Normale Supérieure Paris-Saclay（巴黎萨克雷大学，CEA基础定律研究所; 巴黎萨克雷大学，巴黎萨克雷高等师范学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究发展分支随机力学（BSM）的玻姆/费舍尔反馈随机场论，经MSRJD形式化与2PI闭合分析，得到定域部分、饱和信息速度$c_\star$及两类极点族等结果。
AI 中文摘要
论文I通过将薛定谔-长泽（Schrödinger-Nagasawa）对提升为互易的前向与后向分支场，引入了分支随机力学（Branching Stochastic Mechanics, BSM）。其中心连通核$C_{\rm FB}=\mathbb E_\omega[\psi_F\psi_B]$承载着有序的互易部分，其中$\mathbb E_\omega$表示对分支噪声实现的期望，反相关分支上的$\rho_{\rm BSM}=-C_{\rm FB}(x,x)$。本文在此基础上，发展了作用于该连通部分的玻姆/费舍尔反馈的随机场论。从BSM的乘性分支协方差出发，采用马丁-西吉亚-罗斯-扬森-德多米尼西（Martin-Siggia-Rose-Janssen-de Dominicis, MSRJD）形式化与因果双环双粒子不可约（two-particle-irreducible, 2PI）闭合，自洽地确定响应函数与关联函数。自由连通理论呈现久期增长与紫外积累，而修饰理论则形成有限的相对屏蔽长度。简化数值演化显示该定域部分的有界形成，自相似费舍尔构造定义了饱和信息速度$c_\star$。随后，玻恩-奥本海默分离将内部相对组织与集体传播区分开。恢复完整频率结构得到两个固定-$q$极点族：无隙差分支与有隙和分支。差分支的红外速度在饱和时趋近于$c_\star$，公共锥与投影和部分隙则被表述为集体理论的额外不动点匹配条件。
英文摘要
Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger--Nagasawa pair. Its centered forward--backward kernel $C_{\rm FB}=\mathbb E_ω[ψ_Fψ_B]$ resolves an organized connected sector, with weight $ρ_{\rm BSM}=-C_{\rm FB}(x,x)$ on the anticorrelated branch. Here we investigate how this sector forms, localizes, and propagates under Bohm/Fisher feedback. We retain the branching covariance on the prescribed background as the bare noise kernel and truncate the nonlinear interaction to Bohm/Fisher drift vertices. A Martin--Siggia--Rose--Janssen--de Dominicis formulation and a causal two-loop two-particle-irreducible (2PI) closure determine response and correlation functions self-consistently. While the free connected theory exhibits secular growth and ultraviolet accumulation, the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of the localized sector, and a self-similar Fisher construction defines a saturated information velocity $c_\star$. A Born--Oppenheimer separation connects this internal organization to collective propagation. The instantaneous adiabatic kernel has two pole families at fixed internal momentum: a gapless difference branch and a gapped sum branch. Projecting both inverse response kernels onto the same localized internal profile defines their collective coefficients. If both propagation speeds match $c_\star$ and the projected gap matches $mc_\star^2/\hbar$, the gapped branch takes the infrared Klein--Gordon form. These matching conditions define a candidate relativistic fixed point whose dynamical realization remains to be tested.
Comments35 pages, 6 figures
