Branching stochastic mechanics. I. Clustering and connected correlations within a branching-process representation of the Schr\"odinger equation
Can finite-range correlations hide in the statistics of an extended quantum state? The Schr\"odinger-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, \(\Phi_F\) and \(\Phi_B\): 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 Schr\"odinger 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 \(\Phi_F\Phi_B\). In the reciprocal basis, the fields equal a smooth reference \(R\) plus centered fluctuations \(\psi_F,\psi_B\), defining the signed kernel \(C_{\rm FB}(x,y)=\mathbb E_\omega[\psi_F(x)\psi_B(y)]\). On the anticorrelated branch, \(\rho_{\rm BSM}(x)=-C_{\rm FB}(x,x)\) is the positive paired density. Stationary recovery requires its diagonal to match the Born profile, 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 \(D_{\rm eff}\) and positive relaxation rate \(\mu_{\rm FB}\), correlations have screening length \(\xi_{\rm FB}=\sqrt{D_{\rm eff}/\mu_{\rm FB}}\). Thus an extended Born density and a finite correlation range can coexist in one stochastic kernel, suggesting particle-like organization.