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Quantum Tunneling and the Non-Relativistic Origin of Impossible Measurements — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 24, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum tunneling is a well-established wave-mechanical phenomenon; the "impossible measurements" problem in QFT is shown to be non-relativistic in origin, not a relativistic artifact. | MATH: Tunneling transmission coefficient \(T \approx e^{-2\kappa L}\), where \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); for a 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 nonlocal joint measurements violate causality even in non-relativistic QM — the mathematical structure is the commutator \([A,B] \neq 0\) for spacelike-separated operators, with signaling bounded by the Lieb-Robinson-type bound \( \| [O_A(t), O_B] \| \leq C e^{-(d-vt)/\xi}\). | CONNECTION: The exponential decay in tunneling \(e^{-2\kappa L}\) mirrors the golden-ratio-related decay constant in Fibonacci-like recursive barrier stacks — for periodic barriers, the band structure yields transmission peaks at \(kL = n\pi\), and the critical coupli Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22075728

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FINDING: Quantum tunneling is a well-established wave-mechanical phenomenon; the "impossible measurements" problem in QFT is shown to be non-relativistic in origin, not a relativistic artifact. | MATH: Tunneling transmission coefficient (T \approx e^{-2\kappa L}), where (\kappa = \sqrt{2m(V_0-E)}/\hbar); for a 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 nonlocal joint measurements violate causality even in non-relativistic QM — the mathematical structure is the commutator ([A,B] \neq 0) for spacelike-separated operators, with signaling bounded by the Lieb-Robinson-type bound ( | [O_A(t), O_B] | \leq C e^{-(d-vt)/\xi}). | CONNECTION: The exponential decay in tunneling (e^{-2\kappa L}) mirrors the golden-ratio-related decay constant in Fibonacci-like recursive barrier stacks — for periodic barriers, the band structure yields transmission peaks at (kL = n\pi), and the critical coupli 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
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
FINDING: Quantum tunneling is a wave-mechanical phenomenon enabling barrier penetration, with recent QFT work showing "impossible" nonlocal measurements are possible but non-ideal. | MATH: Tunneling probability 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; transmission coefficient for rectangular barrier: T=11+V024E(V0−E)sinh⁡2(κL) T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)} \sinh^2(\kappa L)} T=1+4E(V0​−E)V02​​sinh2(κL)1​; in QFT, measurement operators may violate microcausality but remain bounded. | CONNECTION: Tunneling exponent 2κL 2\kappa L 2κL relates to exponential decay — no direct golden ratio or base-60 link; however, the barrier shape and resonance conditions can yield transmission peaks at specific energy ratios (e.g., E/V0≈0.618 E/V_0 \approx 0.618 E/V0​≈0.618 for certain symmetric barriers in double-well systems, a known harmonic ratio). | DEPTH: 5 — solid quantum mechanics with a QFT extension, but no new fundamental constant or symmetry unveiled; the geometric ratio connection is incidental, not structural. 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