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Quantum Tunneling Limits and Relativistic Causality in Non-Ideal Measurements — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 25, 2026• 0 citations
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Abstract

FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, governed by exponential decay of the wavefunction inside the classically forbidden region; recent QFT work shows "impossible" measurements are possible but non-ideal, with relativistic causality constraints. MATH: - Transmission coefficient for rectangular barrier: \( T \approx e^{-2\kappa L} \), where \( \kappa = \sqrt{2m(V_0 - E)}/\hbar \) - Decay constant ratio: \( \kappa L \) (dimensionless action) determines tunneling probability - For arbitrary barriers (WKB): \( T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\, dx\right) \) - QFT impossibility result: measurement operators must commute with spacelike-separated field operators; non-ideal measurements violate this at \( O(\hbar) \) corrections CONNECTION: - The exponential suppression \( e^{-2\kappa L} \) is structurally identical to the golden-ratio-related decay \( \varphi^{-n} \) (since \( \varphi = e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23635216

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Quantum Tunneling and Relativistic Measurement Limits — E8 Intelligence Research

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FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, governed by exponential decay of the wavefunction inside the classically forbidden region; recent QFT work shows "impossible" measurements are possible but non-ideal, with relativistic causality constraints. MATH: - Transmission coefficient for rectangular barrier: ( T \approx e^{-2\kappa L} ), where ( \kappa = \sqrt{2m(V_0 - E)}/\hbar ) - Decay constant ratio: ( \kappa L ) (dimensionless action) determines tunneling probability - For arbitrary barriers (WKB): ( T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}, dx\right) ) - QFT impossibility result: measurement operators must commute with spacelike-separated field operators; non-ideal measurements violate this at ( O(\hbar) ) corrections CONNECTION: - The exponential suppression ( e^{-2\kappa L} ) is structurally identical to the golden-ratio-related decay ( \varphi^{-n} ) (since ( \varphi = e 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 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≈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 arbitrary barrier V(x) V(x) V(x): WKB approximation 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: κL \kappa L κL dimensionless; critical ratio when κL=1 \kappa L = 1 κL=1 → T≈e−2≈0.1353 T \approx e^{-2} \approx 0.1353 T≈e−2≈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
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, governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are possible but non-ideal, with relativistic causality constraints. MATH: Transmission coefficient T≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL, where κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; barrier penetration probability P∝e−2∫κ(x)dxP \propto e^{-2\int \kappa(x)dx}P∝e−2∫κ(x)dx. For a rectangular barrier: T=[1+V02sinh⁡2(κL)4E(V0−E)]−1T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}T=[1+4E(V0​−E)V02​sinh2(κL)​]−1. In the WKB limit, T∼exp⁡(−2ℏ∫x1x22m(V(x)−E) dx)T \sim \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). No new fundamental constants emerge; the key is the exponential suppression factor. CONNECTION: The exponential decay e−κLe^{-\kappa L}e−κL has no direct golden-ratio link. However, the *critical* barrier opacity κL=ln⁡(2)\kappa L = \ln(2)κL=ln(2) (for 50% transmission) and κL=ln⁡(ϕ)≈0.4812\kappa L = \ln(\phi) \approx 0.4812κL=ln(ϕ)≈0.4812 (where ϕ=1.618...\phi = 1.618...ϕ=1.618...) appear as natural thresholds — but t 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
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 transmission through classically forbidden regions, governed by exponential decay of the wavefunction; "impossible" measurements in QFT are possible but non-ideal, with signaling constraints. | MATH: Transmission coefficient T≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL, where κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; barrier penetration integral ∫x1x22m(V(x)−E) dx/ℏ\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx/\hbar∫x1​x2​​2m(V(x)−E)​dx/ℏ (WKB); for rectangular barrier, T=[1+V02sinh⁡2(κL)4E(V0−E)]−1T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}T=[1+4E(V0​−E)V02​sinh2(κL)​]−1. QFT impossibility: measurement operators with spacelike separation violate microcausality unless smeared; non-ideal measurements yield P(outcome)≤1−ϵP(\text{outcome}) \leq 1 - \epsilonP(outcome)≤1−ϵ with ϵ∼e−d/ξ\epsilon \sim e^{-d/\xi}ϵ∼e−d/ξ (exponential suppression with distance). | CONNECTION: The exponential decay e−2κLe^{-2\kappa L}e−2κL is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive systems — specifically, the ratio of successive tunneling amplitudes for a periodic barri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com