On the emergence of quantum mechanics from stochastic processes

• URL: http://arxiv.org/abs/2602.22095v1
• Date: Wed, 25 Feb 2026 16:43:04 GMT
• Title: On the emergence of quantum mechanics from stochastic processes
• Abstract summary: We generalize the correspondence by lifting an arbitrary kernel $$ in finite dimension to a map $$ on $B(mathcal H)$.<n>We isolate Chapman--Kolmogorov divisibility of the lifted family as the decisive additional constraint.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The stochastic--quantum correspondence reinterprets quantum dynamics as arising from an underlying stochastic process on a configuration space. We generalize the correspondence by lifting an arbitrary stochastic kernel $Γ$ in finite dimension to a map $φ$ on $B(\mathcal H)$, formulating the associated lift-compatibility relation, and giving an explicit dictionary between $Γ$ and CPTP (Kraus) maps. We isolate Chapman--Kolmogorov divisibility of the lifted family as the decisive additional constraint: when a CK-consistent CPTP family exists, the lift admits a Lindblad master equation form. In this picture, off-diagonal (phase) degrees of freedom act as a compressed carrier of history dependence not fixed by transition kernels alone; conversely, the apparent emergence of quantum phase information from a phase-blind stochastic description is explained as a memory effect. Finally, we state and prove a divisibility criterion for the underlying stochastic kernels, expressed as a condition involving divisibility of the lifted map together with a diagonality requirement on the density operator.

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