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

Abstract

Paper I introduced branching stochastic mechanics (BSM) by lifting the Schrödinger-Nagasawa pair to reciprocal forward and backward branching fields. Their centered connected kernel $C_{\rm FB}=\mathbb E_ω[ψ_Fψ_B]$ carries the organized reciprocal sector, where $\mathbb E_ω$ denotes expectation over branching-noise realizations, with $ρ_{\rm BSM}=-C_{\rm FB}(x,x)$ on the anticorrelated branch. Here we develop the stochastic field theory of the Bohm/Fisher feedback that acts on this connected sector. Starting from the multiplicative branching covariance of BSM, a Martin-Siggia-Rose-Janssen-de~Dominicis (MSRJD) formulation and a causal two-loop two-particle-irreducible (2PI) closure are used to determine response and correlation functions self-consistently. The free connected theory exhibits secular growth and ultraviolet accumulation, whereas the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of this localized sector, and a self-similar Fisher construction defines the saturated information velocity $c_\star$. A Born-Oppenheimer separation then distinguishes internal relative organization from collective propagation. Restoring the complete frequency structure gives two fixed-$q$ pole families: a gapless difference branch and a gapped sum branch. The infrared velocity of the difference branch approaches $c_\star$ at saturation. The common cone and the projected sum-sector gap are then formulated as additional fixed-point matching conditions for the collective theory.