sPNP’s Resolution of the Tunneling Speed Anomaly for a Bohmian Practitioner
sPNP extends Bohmian mechanics into a self‑consistent, curvature‑driven trajectory theory that reproduces v(E) ∝ |E − V0|^½ for any smooth barrier, in contrast to the standard Bohmian law.
- The Bohmian Shortfall
Standard Bohmian velocity: v_B(x) = (∇S(x)) / m* with Ψ(x) = R(x)·exp[i S(x)/ħ], ρ = R². Under a deep barrier (V > E), R is real and ∇S→0 ⇒ v_B→0. No mechanism in v_B(x) gives an increasing speed as E drops.
Quantum potential: Q_B(x) = −(ħ²/2m*) · (ΔR)/R, but Q_B only modifies the force, not the velocity law directly.
- sPNP’s Geometric Upgrade
2.1 Fisher–Rao Metric on Configuration Space
Define the kernel‑regularized density ρ̃(x) = ∫ dy K(x,y)·ρ(y), K(x,y) = (4πQ0²)^(n/2)·exp[−|x−y|²/(4Q0²)], with Q0 a fixed length cutoff. Then the Fisher–Rao metric is g_ij(x) = m*·δ_ij + (ħ²/Q0²)·∂_i∂_j ln ρ̃(x)
ρ̃(x) is a Gaussian smoothing of the probability density over a length scale Q₀.
2.2 Jacobi–Fisher Geodesics
Trajectories follow geodesics on (x, g): g_ij(x)·(dx^i/dτ)(dx^j/dτ) = 2[E − V(x)], d²x^i/dτ² + Γ^i_jk(x)·(dx^j/dτ)(dx^k/dτ) = 0, Γ^i_jk = ½·g^il(∂_j g_lk + ∂_k g_lj − ∂_l g_jk).
In the barrier region, ∂² ln ρ̃ is large and negative ⇒ g_ij develops “valleys.” Geodesics accelerate through valleys, giving a nonzero under‑barrier speed.
Here τ is chosen so that g_ij ẋ^i ẋ^j = 2[E−V(x)], reproducing the Jacobi–Maupertuis action ∫√[2(E−V)g_ij ẋ^i ẋ^j] dτ.
- Universal Square‑Root Law
For any smooth, symmetric barrier with turning points x₁(E), x₂(E), WKB yields near those points x₂ − x₁ ∝ |E − V0|^½. The dominant curvature contribution to the geodesic “speed” is v_sPNP(E) ∼ κ/Q0, κ = √[2m*(V0 − E)]/ħ ∝ |E − V0|^½, so v_sPNP(E) ∝ |E − V0|^½.
This holds for Eckart, double‑Gaussian, smoothed‑step, etc., without retuning Q0.
Christoffel symbol in the x‑direction scales like ∂ₓ³ ln ρ̃ ∼ κ / Q0² so the acceleration scales like κ / Q0² and the integrated velocity gain across a distance ∼ Q0 is (κ / Q0²) × Q0 = κ / Q0
- Why “Bohm + Local Curvature” Fails
Consider a hybrid law v_hybrid(x) = (∇S)/m* + β·∇[Δ ln R]. For smooth barriers, beyond boundary layers Δ ln R is O(1) ⇒ ∇[Δ ln R] ∝ |E − V0|⁰ ⇒ α=0. Only by retuning β per barrier could one match α=½. No single β can reproduce α=½ for both Eckart and double‑Gaussian profiles.
- Self‑Consistent Back‑Reaction
sPNP includes the fixed‑point iteration. Ψ^(n) → g[Ψ^(n)] → Q_sPNP → V_eff → Ψ^(n+1), and one proves the map is a contraction in H¹ norm for realistic Q0 and E ranges. Thus the geodesic law remains stable under quantum back‑reaction.
- Is sPNP what Bohm was searching for?
David Bohm famously interpreted the quantum potential as an “information potential” that informs the particle of the environment’s configuration and "acted on by a still more subtle kind of information". Bohm believed there was something deeper going on. Bohm introduced the concept of "active information" within his ontological interpretation of QM. In this framework, information is not merely passive data but plays an active role in shaping the behavior of quantum systems, suggesting that information has a formative influence on physical processes. Bohm said, "current quantum mechanical laws are only simplifications and abstractions from a vast totality, of which we are only 'scratching the surface'. That is to say, in physical experiments and observations carried out thus far, deeper levels of this totality have not yet revealed themselves." Is sPNP this deeper level (curvature geodesics) based on active information (Fisher Information Matrix)?
- Conclusion for a Bohmian
Retains deterministic trajectories and quantum potential logic.
Upgrades the guidance law to geodesic flow on a nonlocal, nonlinear information‑metric manifold.
Explains the universal square‑root tunneling speed with no extra tuning, unlike any local Bohmian extension.
sPNP is therefore the natural next step for a trajectory‑based quantum theory in light of the Sharoglazova et al. tunneling experiment.
