If you prefer Hogwarts, Oxford and MWI: sPNP is compatible

Many Worlds and the Geometry of Information: A New Take on Quantum Dynamics

Still no collapse: How evolving curvature in probability space guides “particles” without hidden variables

Rethinking Quantum Motion with MWI + sPNP

In the Many‑Worlds Interpretation (MWI), every quantum outcome truly occurs in its own branch. Yet we still ask: What directs an observer’s path through these branches? Traditional Bohmian mechanics introduces hidden particles steered by a mysterious quantum potential. Instead, the sPNP (sPaceNPilottime) framework shows that geometry alone, the curvature of an information manifold, does all the work.

1. From Branch Densities to a Curved Space

In MWI, the universal wavefunction splits into decohered branches Ψ𝑏(𝑥,𝑡). We form the total branch density: P(x, t) = Σ_b |Ψ_b(x, t)|^2

Rather than treating 𝑃 as mere probability, sPNP uses it to define a conformally flat metric on configuration space: g_ij(x, t) = P(x, t) * δ_ij

This metric field 𝑔𝑖𝑗 endows the 3N‑dimensional space of particle configurations with an information geometry shaped by all branches.

1. Scalar Curvature as the “Quantum Force”

From the metric, we extract the scalar curvature R Fisher (x,t) using the well‑known conformal shortcut in d=3N dimensions:

R_Fisher(x, t) = – [4(d–1) / Λ^(d+2)/2] ∇^2 (Λ^(2–d)/2) where Λ(x,t) = P(x,t), d = 3N

Up to fixed numerical conventions, this curvature reproduces the Bohm quantum potential:

Q_sPNP(x, t) = – (ħ^2 / 8m) * R_Fisher(x, t)

But crucially, in sPNP this curvature evolves in time, unlike a static potential.

1. Dynamics via an Information‑Ricci Flow

Because the quantum state—and thus P—changes with time, the metric 𝑔𝑖𝑗 must flow. sPNP proposes a Ricci‑flow–style equation:

∂_t g_ij(x, t) = –2 * Ric_ij[g] + κ1 * (∂_i J_j + ∂_j J_i) + κ2 * (∇·J) * δ_ij

Ric_ij[g] is the Fisher–Rao Ricci tensor of the conformal metric.

J(x,t) = P(x,t) * v(x,t) is the total branch probability current.

κ1, κ2 are coefficients fixed by matching the detailed ∂ₜg derivation.

Inside a potential barrier, J is nonzero yet attenuated. The source terms (∂𝑖𝐽𝑗+∂𝑗𝐽𝑖) and (∇⋅𝐽) compete with curvature smoothing. This dynamic interplay ensures finite “dwell times” without invoking hidden trajectories.

1. Why sPNP + MWI Is Compelling

No Hidden Variables The only “guiding force” is the time‑dependent curvature of information space. Observers follow geodesics in 𝑔𝑖𝑗(𝑡), not mysterious particle paths.

Unified Decoherence Picture Decoherence naturally smooths curvature peaks via the Ricci term, causing branches to decouple into quasi‑classical regions.

Numerical Accessibility For small 𝑁, one discretizes 𝑃 on a grid and computes Laplacians of 𝑃 to get R Fisher. Simulations can compare geodesic flow in 𝑔𝑖𝑗 with standard Bohmian trajectories.

Conceptual Clarity Quantum phenomena, from interference fringes to tunneling, emerge as geometric effects on an evolving manifold of probabilities.

1. Next Steps and Frontiers

Infinite‑Dimensional Rigour: Formalize the Sobolev‑manifold structure of 𝑃(𝑥) and its tangent spaces, linking to the Hellinger and Fubini–Study metrics.

Relativistic & Field Extensions: Generalize to quantum fields by defining Fisher metrics on infinite‑dimensional field configurations, carefully addressing gauge symmetries.

Experimental Proposals: Identify regimes (partial decoherence, weak measurements) where the rate of curvature flow predicts subtle deviations from standard quantum statistics.

Connections to Optimal Transport: Contrast sPNP’s Fisher–Rao geometry with Wasserstein flows to see which better captures quantum vs. classical transport.

Conclusion

By marrying MWI with an information‑geometric, dynamically curved configuration space, sPNP offers a principle‑driven account of quantum dynamics. This version of sPNP is a venture into the fantasical worlds of MWI, where both Oxford and Hogwarts may be one in the same. But if we ignore the lore, sPNP with MWI is simple and straightforward: no collapse, no hidden particles, just geometry sculpted by information. The evolving curvature does the guiding, and observers simply follow the natural geodesic flow through the multiverse.