Branching stochastic mechanics: Clustering and connected correlations within a branching-process representation of the Schrödinger equation
Abstract
Can finite-range correlations hide in the statistics of an extended quantum state? The Schrödinger-Nagasawa transform represents the wave function by positive forward and backward diffusion fields whose product is the Born density. We promote them to branching superprocesses, ΦF and ΦB: diffusion samples stochastic paths, whereas Bohm/Fisher-controlled branching generates genealogies of alternative continuations. With rates evaluated on the prescribed Born density, their means reproduce Schrödinger dynamics exactly. The connected sector exhibits supercritical, critical, and subcritical clustering in confinement and has critical dimension dc=2 in free space. Stationary eigenmodes remain extended; subcritical clusters acquire the reduced de~Broglie scale. We then let the branching rate respond to the fluctuating product ΦFΦB. In the reciprocal basis, the fields equal a smooth reference R plus centered fluctuations ψF,ψB, defining the signed kernel CFB(x,y)=𝔼ω[ψF(x)ψB(y)]. On the anticorrelated branch, ρBSM(x)=−CFB(x,x) is the positive organized connected weight. Stationary matching of this weight to the Born profile is imposed as a closure condition, while off-diagonal decay defines the correlation range. The pair equation splits into collective and relative sectors: spectral cancellation selects the Born collective mode, while suppression of the leading density fluctuation selects the anticorrelated source channel. With relative diffusivity Deff and positive relaxation rate μFB, correlations have screening length ξFB=√(Deff/μFB). Within this reciprocal closure, an extended Born-shaped connected weight and a finite correlation range can coexist in one stochastic kernel.
Related
- Second-Order Rayleigh-Schrödinger Perturbation Theory for the GRASP2018 Package: Core Correlations
- Second-order Rayleigh-Schrödinger Perturbation Theory for the {\sc Grasp}2018 Package: Core-valence Correlations
- Second-Order Rayleigh-Schrödinger Perturbation Theory for the GRASP2018 Package: Three-Particle Feynman Diagram Contribution to Valence-Valence Correlations
- Second-Order Rayleigh-Schrödinger Perturbation Theory for the GRASP2018 Package: Three-Particle Feynman Diagram Contribution to Core-Valence Correlations
- Branching stochastic mechanics: Relative localization and collective poles from Bohm/Fisher feedback
BibTeX
@misc{dumonteil2026branching,
title = {{Branching stochastic mechanics: Clustering and connected correlations within a branching-process representation of the Schrödinger equation}},
author = {Eric Dumonteil and Benoît Bischoff and Alain Letourneau and Loïc Thulliez and Corentin Doutre},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29807},
eprint = {2608.29807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29807},
}
RIS
TY - UNPB
TI - Branching stochastic mechanics: Clustering and connected correlations within a branching-process representation of the Schrödinger equation
AU - Eric Dumonteil
AU - Benoît Bischoff
AU - Alain Letourneau
AU - Loïc Thulliez
AU - Corentin Doutre
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2608.29807
UR - https://arxiv.org/abs/2608.29807
AB - Can finite-range correlations hide in the statistics of an extended quantum state? The Schrödinger-Nagasawa transform represents the wave function by positive forward and backward diffusion fields whose product is the Born density. We promote them to branching superprocesses, Φ_F and Φ_B: diffusion samples stochastic paths, whereas Bohm/Fisher-controlled branching generates genealogies of alternative continuations. With rates evaluated on the prescribed Born density, their means reproduce Schrödinger dynamics exactly. The connected sector exhibits supercritical, critical, and subcritical clustering in confinement and has critical dimension d_c=2 in free space. Stationary eigenmodes remain extended; subcritical clusters acquire the reduced de~Broglie scale. We then let the branching rate respond to the fluctuating product Φ_FΦ_B. In the reciprocal basis, the fields equal a smooth reference R plus centered fluctuations ψ_F,ψ_B, defining the signed kernel C_FB(x,y)=𝔼_ω[ψ_F(x)ψ_B(y)]. On the anticorrelated branch, ρ_BSM(x)=−C_FB(x,x) is the positive organized connected weight. Stationary matching of this weight to the Born profile is imposed as a closure condition, while off-diagonal decay defines the correlation range. The pair equation splits into collective and relative sectors: spectral cancellation selects the Born collective mode, while suppression of the leading density fluctuation selects the anticorrelated source channel. With relative diffusivity D_eff and positive relaxation rate μ_FB, correlations have screening length ξ_FB=√(D_eff/μ_FB). Within this reciprocal closure, an extended Born-shaped connected weight and a finite correlation range can coexist in one stochastic kernel.
ER -