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Quantum Tunneling and Relativistic Measurement: Limits of QFT Ideality — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 27, 2026• 0 citations
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Abstract

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 \( T \approx e^{-2\kappa L} \), where \( \kappa = \sqrt{2m(V_0 - E)}/\hbar \) (rectangular barrier, \( E < V_0 \)). - For a general barrier \( V(x) \): \( T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\, dx\right] \) (WKB approximation). - Reflection coefficient \( R = 1 - T \) (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

DOI
10.5281/zenodo.22987022

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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 \approx e^{-2\kappa L} ), where ( \kappa = \sqrt{2m(V_0 - E)}/\hbar ) (rectangular barrier, ( E < V_0 )). - For a general barrier ( V(x) ): ( T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}, dx\right] ) (WKB approximation). - Reflection coefficient ( 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 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 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 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 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, governed by exponential decay of the wavefunction inside the barrier; "impossible" relativistic measurements are shown to be possible but non-ideal, revealing a fundamental constraint on measurement theory. | 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 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. The QFT impossibility result (arXiv:2311.13644) shows naive nonlocal joint measurements violate microcausality, but a modified measurement yields signaling-free outcomes with a trade-off in ideality — formalized via operator algebra constraints. | CONNECTION: The exponential decay e−2κLe^{-2\kappa L}e−2κL is a natural logarithmic spiral decay — the ratio of successive barrier widths for equal transmission probability is \(e^{2\kappa \Delta L}\ 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