Abstract
We fuse the Immutable-Past (IP) entropy-gradient ontology with a hydrodynamic, de Broglie–Bohm (dBB) pilot-wave metaphor that distinguishes three domains: an “air” phase encoding unrealised possibilities, a “surface” phase where history is written, and a “water” bulk that stores the zero-entropy record. By mapping Bohm’s quantum potential onto the IP gradient and treating decoherence as surface-wave damping, we obtain a logically consistent, Lorentz-respecting model in which particles (beads) deterministically trace world-lines guided by a non-collapsing wavefunction while empty-wave branches appear as transient eddies. The construction preserves the SU(2) ℏ⁄2 quantisation floor, conserves energy, and honours the single axiom that the past is immutable.
1 Introduction
The IP programme reimagines reality as a low-dimensional surface descending from high entropy (future potential) to zero entropy (Immutable Past). Standard quantum mechanics explains measurement outcomes via stochastic collapse; dBB offers deterministic guidance but leaves the physical status of empty waves ambiguous. We propose a two-phase hydrodynamic metaphor that (i) visualises Bohmian guidance on the IP surface, (ii) assigns empty-wave energy to observable “eddies,” and (iii) avoids conflicts with relativity and thermodynamics.
2 Ontological Partition
| Phase | Physical role | Mathematical object | Entropic status |
|---|---|---|---|
| Air | Unrealised futures | Full many-body wavefunction ψ | High |
| Surface | Eternal Now | 3-D hypersurface Σ(τ) for each observer | Intermediate |
| Water | Immutable Past | Portion of trajectory r(τ′ ≤ τ) | Zero |
The projection ψ → Σ is epistemic: it compresses 3N-D configuration space onto the observer’s 3-D slice without altering non-local correlations.
3 Guidance as Gradient Descent
For a spin-½ particle the Pauli-Bohm velocity field is:
v(r,t) = (ℏ/m) · Im[ψ†∇ψ]/(ψ†ψ)
+ (ℏ/2m) · (∇×(ψ†σψ))/(ψ†ψ).We reinterpret −∇Q (with Q the quantum potential) as the local slope of the IP entropy gradient. Motion along this slope writes bits into Σ; motion orthogonal to Σ is forbidden, enforcing determinism once initial conditions are fixed.
4 Empty Waves as Eddies
When ψ splits into disjoint lobes, the occupied lobe remains coupled to the bead; the unoccupied lobe forms a basin in Q. In the surface-wave picture this basin is an eddy characterised by:
- Local minimum of Q – trapped phase energy.
- No bead inside – no further history written.
Decoherence erodes eddy walls by transferring phase information to environmental modes (air turbulence), rendering the eddy dynamically inert without violating unitarity.
5 Guard-Rail Conditions
5.1 Dimensional Consistency
Projection to Σ is explanatory; underlying dynamics remain 3N-D and non-local. Correlations predicted by Bell-type experiments are unaltered.
5.2 Unitary Evolution
Eddy erosion is effective, not fundamental: ψ evolves unitarily; probability is conserved. Surface-wave damping models the coarse-graining that renders interference unobservable.
5.3 Lorentz Covariance
Each inertial observer carries their own Σ(τ). Transformations between observers relabel slices but do not alter invariant world-lines.
5.4 Energy Conservation
Energy tied up in empty-wave eddies disperses into higher-entropy degrees of freedom (micro-ripples). Total energy-momentum is conserved.
5.5 SU(2) Quantisation Floor
The Pauli algebra permits exactly two orthogonal projectors; no additional surface basins arise unless the internal symmetry group is enlarged (e.g., F=1,2). Metrological advances cannot break the ℏ⁄2 limit.
5.6 Entropy Arrow
Local entropy drops when a bead selects a basin, yet global entropy rises as unused lobes decohere. This reconciles the IP “down-gradient” narrative with the conventional second law along world-lines.
6 Applications
- Stern–Gerlach full-loop interferometry – trajectory bundles reconverge, verifying that eddies can re-engage before environmental damping completes.
- Walking-droplet analogues – millimetre-scale bead-wave systems exhibit discrete orbit modes, visualising the ℏ⁄2 floor.
- Quantum-gravity table-top tests – mapping phase-shift gradients onto surface curvature suggests interferometric probes of metric fluctuations without invoking collapse.
7 Discussion
The two-phase picture provides intuitive leverage over measurement without multiplying postulates. It demystifies empty waves, locates irreversibility in decoherence, and retains deterministic guidance—all while preserving the immutability of the recorded past. Remaining open problems include a fully relativistic Bohmian field theory compatible with the IP surface foliation and experimental schemes to detect residual eddy energy.
8 Conclusion
By layering air (potential), surface (present), and water (past) atop Bohm’s pilot-wave mechanics, we embed the IP entropy-gradient axiom into a deterministic yet non-local quantum narrative. Logical consistency with SU(2) algebra, energy conservation, Lorentz symmetry, and the second law is maintained, and the metaphor yields concrete experimental analogues. The approach therefore clears all major conceptual roadblocks and offers a fertile platform for further theoretical and empirical exploration.
References
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