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Quantum Tunneling and the Limits of Nonlocal Measurements in QFT — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 27, 2026• 0 citations
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Abstract

FINDING: Quantum tunneling is a wave-mechanical transmission through classically forbidden barriers, governed by exponential decay of the wavefunction; "impossible" measurements in QFT are possible but non-ideal, revealing fundamental limits on joint nonlocal observables. | MATH: Transmission coefficient \(T \approx e^{-2\kappa L}\), where \(\kappa = \sqrt{2m(V_0-E)}/\hbar\) for a rectangular barrier of height \(V_0\), width \(L\). For arbitrary barriers, WKB: \(T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right)\). The QFT "impossible measurement" result (arXiv:2311.13644) shows that nonlocal joint measurements violate causality unless they are non-ideal — mathematically, the measurement map is not a completely positive trace-preserving (CPTP) map on the joint algebra; it requires a non-unital, non-CP extension, with the deviation quantified by a signaling parameter \(\epsilon > 0\). | CONNECTION: The exponential decay \(e^{-2\kappa L}\) is a pure exponen Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22121797

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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 \(T \approx e^{-2\kappa L}\), where \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); barrier penetration integral \(\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx/\hbar\) (WKB); for rectangular barrier, \(T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}\). QFT impossibility: measurement operators with spacelike separation violate microcausality unless smeared; non-ideal measurements yield \(P(\text{outcome}) \leq 1 - \epsilon\) with \(\epsilon \sim e^{-d/\xi}\) (exponential suppression with distance). | CONNECTION: The exponential decay \(e^{-2\kappa L}\) 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

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FINDING: Quantum tunneling is a wave-mechanical transmission through classically forbidden barriers, governed by exponential decay of the wavefunction; "impossible" measurements in QFT are possible but non-ideal, revealing fundamental limits on joint nonlocal observables. | MATH: Transmission coefficient (T \approx e^{-2\kappa L}), where (\kappa = \sqrt{2m(V_0-E)}/\hbar) for a rectangular barrier of height (V_0), width (L). For arbitrary barriers, WKB: (T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)},dx\right)). The QFT "impossible measurement" result (arXiv:2311.13644) shows that nonlocal joint measurements violate causality unless they are non-ideal — mathematically, the measurement map is not a completely positive trace-preserving (CPTP) map on the joint algebra; it requires a non-unital, non-CP extension, with the deviation quantified by a signaling parameter (\epsilon > 0). | CONNECTION: The exponential decay (e^{-2\kappa L}) is a pure exponen Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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FINDING: Quantum tunneling is a wave-mechanical penetration of finite barriers, with transmission probability governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are shown to be possible but non-ideal, revealing a fundamental tension between relativity and joint nonlocal measurements. 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 - For a rectangular barrier of height V0V_0V0​ and width aaa: T=11+V02sinh⁡2(κa)4E(V0−E) T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}} 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 - Exponential suppression factor: e−2κa e^{-2\kappa a} e−2κa — the core "impossibility" barrier. - QFT "impossible measurements" (arXiv:2311.13644v2): signaling between space-like separated regions is avoided only if measurement operators are strictly local; nonlocal joint measurements violate microcausalit 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 exponentially suppressed by barrier width and height; "impossible" QFT measurements are possible but non-ideal, revealing fundamental limits on joint nonlocal observables. 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. 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 semiclassical (WKB) limit: 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). The "impossible measurement" result (arXiv:2311.13644) shows that ideal joint nonlocal measurements in non-relativistic QM would violate no-signaling; only approximate (non-ideal) measurements are consistent, with error bounds scaling as ϵ∼ℏ/Δt\epsilon \sim \hbar / \Delta tϵ∼ℏ/Δt for spacelike separations. CONNECTION: The exponential decay e−2κLe^{-2\kappa L}e−2κL is structurally identica 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 barriers, with the transmission coefficient governed by exponential decay of the wavefunction inside the classically forbidden region; "impossible" relativistic measurements are possible but non-ideal, revealing fundamental limits on joint nonlocal observables. MATH: - Transmission coefficient for rectangular barrier: 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. - For arbitrary barrier V(x) V(x) V(x): WKB tunneling amplitude T≈exp⁡(−2∫x1x22mℏ2(V(x)−E) dx) T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2}(V(x)-E)}\,dx\right) T≈exp(−2∫x1​x2​​ℏ22m​(V(x)−E)​dx). - Reflection/transmission amplitudes satisfy ∣R∣2+∣T∣2=1 |R|^2 + |T|^2 = 1 ∣R∣2+∣T∣2=1 (unitarity). - In QFT "impossible measurements": signaling violation arises unless measurement operators are strictly local; nonlocal joint measurements require smearing or finite resolution, leading to ΔtΔE≳ℏ \Delta t \Delta E \gtrsim \hbar ΔtΔE≳ℏ constraints. CONNECTION: - The exponential decay factor 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; "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 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