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Quantum Tunneling and Nonlocal Measurements: Relativity's Quantum Limits — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 30, 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; "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 \(V_0\), width \(a\), particle energy \(E < V_0\)): \[ T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa a}, \quad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar} \] (Exact: \(T = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}\)) - WKB approximation for arbitrary barrier \(V(x)\): \[ T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\, dx\right) \] - 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

DOI
10.5281/zenodo.23052227

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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 (V_0), width (a), particle energy (E < V_0)): [ T \approx \frac{16E(V_0-E)}{V_0^2} e^{-2\kappa a}, \quad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar} ] (Exact: (T = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1})) - WKB approximation for arbitrary barrier (V(x)): [ T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}, dx\right) ] - 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
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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 a transmission coefficient governed by the barrier's width, height, and particle energy; "impossible" measurements in QFT are shown to be possible but non-ideal, revealing a fundamental limit on joint nonlocal measurements. | 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 for a rectangular barrier of height V0 V_0 V0​, width L L L, particle energy E E E. For arbitrary barriers, WKB approximation: T≈exp⁡(−2∫x1x22m(V(x)−E)ℏ2 dx) T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m(V(x)-E)}{\hbar^2}} \, dx\right) T≈exp(−2∫x1​x2​​ℏ22m(V(x)−E)​​dx). The QFT result (arXiv:2311.13644) shows impossible measurements are possible but not ideal — the mathematical structure involves the breakdown of the projection postulate in relativistic settings, with signaling constraints tied to the light cone. | CONNECTION: The exponential decay e−2κL e^{-2\kappa L} e−2κL is a pure exponential — no direct golden-r 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 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≈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 of height V0V_0V0​, width LLL. For arbitrary barriers, WKB: 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 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 ϵ>0\epsilon > 0ϵ>0. | CONNECTION: The exponential decay e−2κLe^{-2\kappa L}e−2κL is a pure exponen 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