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Quantum Tunneling Enables Non-Ideal Relativistic Measurements — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 17, 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 barrier; "impossible" relativistic measurements are shown to be possible but non-ideal, revealing a fundamental constraint on measurement theory. | MATH: Transmission coefficient \(T \approx e^{-2\kappa L}\) where \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); barrier penetration probability \(P \propto e^{-2\int \kappa(x)dx}\); for rectangular barrier, \(T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}\). 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 \(e^{-2\kappa L}\) 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

DOI
10.5281/zenodo.22806065

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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 barrier; "impossible" relativistic measurements are shown to be possible but non-ideal, revealing a fundamental constraint on measurement theory. | MATH: Transmission coefficient (T \approx e^{-2\kappa L}) where (\kappa = \sqrt{2m(V_0-E)}/\hbar); barrier penetration probability (P \propto e^{-2\int \kappa(x)dx}); for rectangular barrier, (T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-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^{-2\kappa L}) 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
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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≈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 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; 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 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 barriers, not a classical "jump"; recent QFT work shows "impossible" nonlocal measurements are possible but non-ideal, revealing a fundamental trade-off in measurement theory. | MATH: Transmission coefficient T≈e−2κLT \approx e^{-2\kappa L}T≈e−2κL, 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 = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}T=[1+4E(V0​−E)V02​sinh2(κL)​]−1. In QFT, the "impossible measurement" problem is resolved by recognizing that ideal projective measurements violate microcausality; the trade-off is quantified by a non-ideal measurement parameter ϵ\epsilonϵ such that signaling error ∝ϵ2\propto \epsilon^2∝ϵ2, with ϵ→0\epsilon \to 0ϵ→0 only in the non-relativistic limit. | CONNECTION: The exponential decay e−2κLe^{-2\kappa L}e−2κL is a pure decay constant — no golden ratio appears in the standard barrier solution. However, the *critical* barrier width where tunneling probability drops to 1/ϕ1/\phi1/ϕ (≈0.618) occurs at \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com