Branching stochastic mechanics: Relative localization and collective poles from Bohm/Fisher feedback

Abstract

Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schr\"odinger--Nagasawa pair. Its centered forward--backward kernel $C_{\rm FB}=\mathbb E_\omega[\psi_F\psi_B]$ resolves an organized connected sector, with weight $\rho_{\rm BSM}=-C_{\rm FB}(x,x)$ 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.