Chapter 59
Tunnelling as Phase Re-alignment
Quantum Coherence and Boundary Systems
Author: Mark Anthony Higgins
Revised Date: 23 February 2026
Chapter 59
Overview
Quantum tunnelling is not the passage of a particle through a wall — it is phase continuity across a suppressed curvature zone.
Where classical energy halts, coherence continues; where amplitude decays, phase re-aligns.
In the HST framework, tunnelling is not a violation of energy conservation, but a preservation of boundary coherence — the field re-establishing its curvature memory on the far side of a potential barrier.
🔹 1. Classical Form — The Barrier Paradox
In classical mechanics, a particle of energy less than the potential barrier cannot pass.
In wave mechanics, however, the de Broglie curvature persists into the forbidden region with an exponentially decaying amplitude:
\[ \psi(x) \sim e^{-\kappa x}, \qquad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar} \]
The probability of transmission through a barrier of width becomes:
\[ T \approx e^{-2\kappa a} \]
The amplitude falls, but the phase remains defined — and where phase persists, curvature coherence can re-form beyond the boundary.
What emerges is not a particle that “passed through,” but a field that never fully disconnected.
🔸 2. Coherence Interpretation — The Continuity of Curvature
In the HST view, the wavefunction’s phase embodies the memory of curvature alignment across space.
When the wave meets a barrier, the field compresses its curvature — energy density rises locally, amplitude shrinks, but phase continuity remains non-zero.
The field therefore seeks equilibrium by re-aligning its curvature across the boundary.
The transmission coefficient is not the “chance of passage,” but the degree of successful phase inheritance across the barrier.
The coherence index expresses how intact the phase relationship remains:
\[ I_B = e^{-2\kappa a} \quad \text{(idealised form)} \]
Perfect coherence () allows full phase alignment and amplitude restoration.
Reduced corresponds to phase mismatch, yielding partial transmission and reflection.
🔹 3. Barrier Geometry — Phase Suppression, Not Stopping
Inside the barrier, the field does not vanish; it transforms.
Amplitude attenuates exponentially, but its curvature continues to evolve under imaginary wavenumber .
This region acts as a phase compressor — converting oscillatory curvature into stored potential curvature.
At the exit, phase continuity requires that the re-emerging field adopt a curvature consistent with both regions — effectively realigning itself into a stable outgoing wave.
This re-emergence is not a traversal in space-time, but a restoration of coherence geometry across a boundary discontinuity.
🔸 4. Decoherence and Partial Reflection
If environmental noise or internal friction reduces coherence, falls below 1.
The feedback law,
\[ \dot{I_B} = -\lambda (1 - I_B) \]
governs the degradation of phase integrity inside the barrier.
Low increases effective barrier width — phase misalignment accumulates, transmission falls exponentially, and reflection dominates.
In this regime, the barrier behaves as a phase-diffusive interface, redirecting curvature back into the incident region.
Tunnelling thus becomes incomplete re-alignment rather than failure of motion.
🔹 5. The Re-aligned State — Memory Restored Beyond the Barrier
On the far side, the emerging field carries a modified phase — slightly delayed, amplitude-reduced, but still coherent.
This new wave is a continuation of the original field, having re-established curvature continuity after temporary suppression.
The system’s coherence loop remains closed, albeit with reduced amplitude:
\[ \psi_{\text{out}} = I_B^{1/2} e^{i\phi_{\text{shift}}} \psi_{\text{in}} \]
Tunnelling is thus re-coherence, not transport.
🔸 6. Unified Coherence Framework — The Barrier as Phase Mirror
In HST–Flow logic:
Thus, tunnelling is the field’s way of maintaining coherence conservation when energy constraints disallow classical continuity.
It proves that the wavefunction is not a description of probability, but of phase structure resilience — the persistence of curvature memory even through apparent separation.
🧠 Summary
Tunnelling is phase re-alignment through suppressed curvature.
It is the field’s ability to maintain coherence continuity across boundaries that classical mechanics deems impassable
In the HST–Flow interpretation:
A barrier never stops coherence — it only tests how faithfully the universe can remember its own phase.
