Branching stochastic mechanics: Relative localization and collective poles from Bohm/Fisher feedback
Abstract
Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger--Nagasawa pair. Its centered forward--backward kernel CFB=𝔼ω[ψFψB] resolves an organized connected sector, with weight ρBSM=−CFB(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⋆. 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⋆ and the projected gap matches mc⋆2/ℏ, 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.
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BibTeX
@misc{bischoff2026branching,
title = {{Branching stochastic mechanics: Relative localization and collective poles from Bohm/Fisher feedback}},
author = {Benoit Bischoff and Eric Dumonteil},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02520},
eprint = {2609.02520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02520},
}
RIS
TY - UNPB
TI - Branching stochastic mechanics: Relative localization and collective poles from Bohm/Fisher feedback
AU - Benoit Bischoff
AU - Eric Dumonteil
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.02520
UR - https://arxiv.org/abs/2609.02520
AB - Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger--Nagasawa pair. Its centered forward--backward kernel C_FB=𝔼_ω[ψ_Fψ_B] resolves an organized connected sector, with weight ρ_BSM=−C_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_⋆. 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_⋆ and the projected gap matches mc_⋆^2/ℏ, 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.
ER -