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Quantum Tunneling and Relativistic Measurement Limits — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 21, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with the transmission coefficient governed by exponential decay of the wavefunction inside the classically forbidden region; recent work shows "impossible" relativistic measurements are mathematically possible but non-ideal. MATH: - Transmission coefficient: \( T \approx e^{-2\kappa L} \), where \( \kappa = \sqrt{2m(V_0 - E)}/\hbar \) (rectangular barrier, \( E < V_0 \)). - For arbitrary barrier \( V(x) \): WKB approximation \( T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right] \). - Decay constant ratio: \( \kappa L \) dimensionless; critical ratio when \( \kappa L = 1 \) → \( T \approx e^{-2} \approx 0.1353 \). - QFT impossibility result (arXiv:2311.13644): nonlocal joint measurements in relativistic QFT produce superluminal signaling unless measurement operators are restricted; the "impossible" case corresponds to a mathematical idealization with zero-wi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23525808

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FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with the transmission coefficient governed by exponential decay of the wavefunction inside the classically forbidden region; "impossible" quantum measurements in QFT are shown to be possible but non-ideal, revealing a fundamental tension between relativity and joint nonlocal measurements. MATH: - Transmission coefficient for a rectangular barrier (height V0V_0V0​, width aaa, particle energy E<V0E < V_0E<V0​): T≈16E(V0−E)V02e−2κa,κ=2m(V0−E)ℏ T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa a}, \quad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar} T≈V02​16E(V0​−E)​e−2κa,κ=ℏ2m(V0​−E)​​ (Exact: T=[1+V02sinh⁡2(κa)4E(V0−E)]−1T = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}T=[1+4E(V0​−E)V02​sinh2(κa)​]−1) - WKB approximation for arbitrary barrier V(x)V(x)V(x): T≈exp⁡(−2ℏ∫x1x22m(V(x)−E) dx) T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\, dx\right) T≈exp(−ℏ2​∫x1​x2​​2m(V(x)−E)​dx) - For the QFT "impossible measurements" result: the non-ideal measurement operators are positive operator-valued measures (POVMs) with finite norm, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with recent QFT work showing "impossible" measurements are possible but non-ideal, revealing a fundamental limit on measurement locality. MATH: - Tunneling transmission coefficient (WKB approximation): T≈e−2γ T \approx e^{-2\gamma} T≈e−2γ, where γ=1ℏ∫x1x22m(V(x)−E) dx \gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)} \, dx γ=ℏ1​∫x1​x2​​2m(V(x)−E)​dx — exponential decay in barrier width. - For a rectangular barrier of height V0 V_0 V0​ and width a a a: T=[1+V02sinh⁡2(κa)4E(V0−E)]−1 T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)} \right]^{-1} T=[1+4E(V0​−E)V02​sinh2(κa)​]−1, with κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ. - QFT impossibility result (arXiv:2311.13644): nonlocal joint measurements in non-relativistic QM produce superluminal signaling unless the measurement is "ideal" (projective, with vanishing variance in the measured observable). The mathematical constraint is that the measurement operator must commute with all spacelike-separated observables — a condition that fails Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with a transmission coefficient governed by exponential decay of the wavefunction inside the barrier; recent QFT work shows "impossible" measurements are possible but non-ideal, revealing fundamental limits on relativistic measurement. MATH: - Transmission coefficient for rectangular barrier: T≈e−2κa T \approx e^{-2\kappa a} T≈e−2κa, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0 - E)}/\hbar κ=2m(V0​−E)​/ℏ, a a a = barrier width. - General WKB tunneling probability: T≈exp⁡(−2ℏ∫x1x22m(V(x)−E) dx) T \approx \exp\left( -\frac{2}{\hbar} \int_{x_1}^{x_2} \sqrt{2m(V(x) - E)} \, dx \right) T≈exp(−ℏ2​∫x1​x2​​2m(V(x)−E)​dx). - Decay constant ratio: κa \kappa a κa dimensionless — no new fundamental constant emerges. - QFT impossibility result (arXiv:2311.13644): nonlocal joint measurements in QFT lead to superluminal signaling unless measurement is "non-ideal" — formalized via operator algebra constraints, not a simple equation. CONNECTION: - The exponential decay \( e^{-2\kappa a} Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with the transmission probability governed by exponential decay of the wavefunction inside the barrier; recent QFT work shows "impossible" measurements are mathematically possible but non-ideal. | MATH: Transmission coefficient T≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL for rectangular barrier, where κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; decay length δ=1/κ=ℏ/2m(V0−E)\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}δ=1/κ=ℏ/2m(V0​−E)​. For arbitrary barrier V(x)V(x)V(x), WKB approximation: T≈exp⁡(−2∫x1x22m(V(x)−E)/ℏ dx)T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)T≈exp(−2∫x1​x2​​2m(V(x)−E)​/ℏdx). QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor e−2κLe^{-2\kappa L}e−2κL is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for NNN identical barriers, transmission resonances occur when kL=nπkL = n\pikL=nπ, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with a transmission coefficient governed by exponential decay of the wavefunction inside the barrier; "impossible" relativistic measurements are shown to be possible but non-ideal, revealing a fundamental tension between QFT and measurement theory. MATH: - Transmission coefficient T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0 - E)}/\hbar κ=2m(V0​−E)​/ℏ (rectangular barrier, E<V0 E < V_0 E<V0​). - For a general barrier V(x) V(x) V(x): T≈exp⁡[−2ℏ∫x1x22m(V(x)−E) dx] T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\, dx\right] T≈exp[−ℏ2​∫x1​x2​​2m(V(x)−E)​dx] (WKB approximation). - Reflection coefficient R=1−T R = 1 - T R=1−T (unitarity). - In QFT impossibility: signaling between spacelike-separated regions is avoided only if measurement operators commute — the paper (arXiv:2311.13644) shows nonlocal joint measurements in non-relativistic QM produce the same paradox, implying a constraint on measurement algebra: \( [\hat{O}_A, \hat{O}_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com