Quantum Tunneling and the Fundamental Trade-Off in Nonlocal Measurements — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 23, 2026• 0 citations

Abstract

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 $T \approx e^{-2\kappa L}$, $\kappa = \sqrt{2m(V_0-E)}/\hbar$; for a rectangular barrier, $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$. 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 $\epsilon$ such that signaling error $\propto \epsilon^2$, with $\epsilon \to 0$ only in the non-relativistic limit. | CONNECTION: The exponential decay $e^{-2\kappa L}$ is a pure decay constant — no golden ratio appears in the standard barrier solution. However, the *critical* barrier width where tunneling probability drops to $1/\phi$ (≈0.618) occurs at \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22910090

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Publisher

Abstract

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 \approx e^{-2\kappa L}), (\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}). 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 (\epsilon) such that signaling error (\propto \epsilon^2), with (\epsilon \to 0) only in the non-relativistic limit. | CONNECTION: The exponential decay (e^{-2\kappa L}) 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/\phi) (≈0.618) occurs at \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22910090

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Quantum Tunneling and Nonlocal QFT Measurements: Barrier Penetration Meets Microcausality — E8 Intelligence Research

Andrew Stewart Caldin

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 $T \approx e^{-2\kappa L}$ where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$ ; transmission coefficient for rectangular barrier: $T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)} \sinh^2(\kappa L)}$ ; in QFT, measurement operators may violate microcausality but remain bounded. | CONNECTION: Tunneling exponent $2\kappa L$ 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., $E/V_0 \approx 0.618$ 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.comFINDING: 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 \approx e^{-2\kappa L} $ where $ \kappa = \sqrt{2m(V_0 - E)}/\hbar $; transmission coefficient for rectangular barrier: $ T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)} \sinh^2(\kappa L)} $; in QFT, measurement operators may violate microcausality but remain bounded. | CONNECTION: Tunneling exponent $ 2\kappa L $ 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/V_0 \approx 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

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

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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 $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$ for a rectangular barrier of height $V_0$ , width $L$ , particle energy $E$ . For arbitrary barriers, WKB approximation: $T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m(V(x)-E)}{\hbar^2}} \, dx\right)$ . 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 $e^{-2\kappa L}$ is a pure exponential — no direct golden-r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: 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 \approx e^{-2\kappa L} $, where $ \kappa = \sqrt{2m(V_0 - E)}/\hbar $ for a rectangular barrier of height $ V_0 $, width $ L $, particle energy $ E $. For arbitrary barriers, WKB approximation: $ T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m(V(x)-E)}{\hbar^2}} \, dx\right) $. 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^{-2\kappa L} $ 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

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Quantum Tunneling Limits on Joint Nonlocal Quantum Measurements — E8 Intelligence Research

Andrew Stewart Caldin

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 $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ . For a rectangular barrier: $T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}$ . In the semiclassical (WKB) limit: $T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right)$ . 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 $\epsilon \sim \hbar / \Delta t$ for spacelike separations. CONNECTION: The exponential decay $e^{-2\kappa L}$ is structurally identica Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: 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 $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$. For a rectangular barrier: $T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}$. In the semiclassical (WKB) limit: $T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right)$. 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 $\epsilon \sim \hbar / \Delta t$ for spacelike separations. CONNECTION: The exponential decay $e^{-2\kappa L}$ is structurally identica Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

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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.comFINDING: 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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Quantum Tunneling Limits Measurement Locality via Non-Ideal Barrier Penetration — E8 Intelligence Research

Andrew Stewart Caldin

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): $T \approx e^{-2\gamma}$ , where $\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)} \, dx$ — exponential decay in barrier width. - For a rectangular barrier of height $V_0$ and width $a$ : $T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)} \right]^{-1}$ , with $\kappa = \sqrt{2m(V_0-E)}/\hbar$ . - 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.comFINDING: 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 \approx e^{-2\gamma} $, where $ \gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)} \, dx $ — exponential decay in barrier width. - For a rectangular barrier of height $ V_0 $ and width $ a $: $ T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)} \right]^{-1} $, with $ \kappa = \sqrt{2m(V_0-E)}/\hbar $. - 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

Oct 19, 20260 citations

Quantum Tunneling: Exponential Barrier Penetration and Non-Ideal Measurements in QFT — E8 Intelligence Research

Andrew Stewart Caldin

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 $T \approx e^{-2\kappa L}$ for rectangular barrier, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; decay length $\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}$ . For arbitrary barrier $V(x)$ , WKB approximation: $T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)$ . QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor $e^{-2\kappa L}$ is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for $N$ identical barriers, transmission resonances occur when $kL = n\pi$ , a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: 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 $T \approx e^{-2\kappa L}$ for rectangular barrier, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; decay length $\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}$. For arbitrary barrier $V(x)$, WKB approximation: $T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)$. QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor $e^{-2\kappa L}$ is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for $N$ identical barriers, transmission resonances occur when $kL = n\pi$, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 17, 20260 citations

T \approx e^{-2\kappa L}

where

\kappa = \sqrt{2m(V_0 - E)}/\hbar

; transmission coefficient for rectangular barrier:

T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)} \sinh^2(\kappa L)}

; in QFT, measurement operators may violate microcausality but remain bounded. | CONNECTION: Tunneling exponent

2\kappa L

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/V_0 \approx 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

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0 - E)}/\hbar

for a rectangular barrier of height

V_0

, width

L

, particle energy

E

. For arbitrary barriers, WKB approximation:

T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m(V(x)-E)}{\hbar^2}} \, dx\right)

. 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^{-2\kappa L}

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

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

. For a rectangular barrier:

T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}

. In the semiclassical (WKB) limit:

T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\,dx\right)

. 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

\epsilon \sim \hbar / \Delta t

for spacelike separations. CONNECTION: The exponential decay

e^{-2\kappa L}

is structurally identica Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

T \approx e^{-2\gamma}

, where

\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)} \, dx

— exponential decay in barrier width. - For a rectangular barrier of height

V_0

and width

a

T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)} \right]^{-1}

, with

\kappa = \sqrt{2m(V_0-E)}/\hbar

. - 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

T \approx e^{-2\kappa L}

for rectangular barrier, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; decay length

\delta = 1/\kappa = \hbar/\sqrt{2m(V_0-E)}

. For arbitrary barrier

V(x)

, WKB approximation:

T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}/\hbar \, dx\right)

. QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor

e^{-2\kappa L}

is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for

N

identical barriers, transmission resonances occur when

kL = n\pi

, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com