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

Andrew Stewart Caldin

Published Oct 9, 2026• 0 citations

Abstract

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.com

DOI: 10.5281/zenodo.23255428

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

Andrew Stewart Caldin

Published Oct 9, 2026• 0 citations

Publisher

Abstract

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.com

DOI: 10.5281/zenodo.23255428

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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 Joint Nonlocal Measurements — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling is a wave-mechanical penetration of finite barriers, with transmission probability governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are shown to be possible but non-ideal, revealing a fundamental tension between relativity and joint nonlocal measurements. MATH: - Tunneling transmission coefficient (WKB approximation): $T \approx e^{-2\gamma}$ , where $\gamma = \frac{1}{\hbar} \int_{x_1}^{x_2} \sqrt{2m(V(x)-E)} \, dx$ - For a rectangular barrier of height $V_0$ and width $a$ : $T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}}$ , with $\kappa = \sqrt{2m(V_0-E)}/\hbar$ - Exponential suppression factor: $e^{-2\kappa a}$ — the core "impossibility" barrier. - QFT "impossible measurements" (arXiv:2311.13644v2): signaling between space-like separated regions is avoided only if measurement operators are strictly local; nonlocal joint measurements violate microcausalit 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 barriers, with transmission probability governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are shown to be possible but non-ideal, revealing a fundamental tension between relativity and joint nonlocal measurements. 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 $ - For a rectangular barrier of height $V_0$ and width $a$: $ T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}} $, with $ \kappa = \sqrt{2m(V_0-E)}/\hbar $ - Exponential suppression factor: $ e^{-2\kappa a} $ — the core "impossibility" barrier. - QFT "impossible measurements" (arXiv:2311.13644v2): signaling between space-like separated regions is avoided only if measurement operators are strictly local; nonlocal joint measurements violate microcausalit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

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

Andrew Stewart Caldin

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

Aug 27, 20260 citations

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 Limits on Relativistic Nonlocal Measurements — E8 Intelligence Research

Andrew Stewart Caldin

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: $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$ . - For arbitrary barrier $V(x)$ : WKB tunneling amplitude $T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2}(V(x)-E)}\,dx\right)$ . - Reflection/transmission amplitudes satisfy $|R|^2 + |T|^2 = 1$ (unitarity). - In QFT "impossible measurements": signaling violation arises unless measurement operators are strictly local; nonlocal joint measurements require smearing or finite resolution, leading to $\Delta t \Delta E \gtrsim \hbar$ constraints. CONNECTION: - The exponential decay factor 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 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 \approx e^{-2\kappa L} $, where $ \kappa = \sqrt{2m(V_0 - E)}/\hbar $. - For arbitrary barrier $ V(x) $: WKB tunneling amplitude $ T \approx \exp\left(-2\int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2}(V(x)-E)}\,dx\right) $. - Reflection/transmission amplitudes satisfy $ |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 $ \Delta t \Delta E \gtrsim \hbar $ constraints. CONNECTION: - The exponential decay factor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 10, 20260 citations

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\gamma}

, where

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

- For a rectangular barrier of height

V_0

and width

a

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

, with

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

- Exponential suppression factor:

e^{-2\kappa a}

— the core "impossibility" barrier. - QFT "impossible measurements" (arXiv:2311.13644v2): signaling between space-like separated regions is avoided only if measurement operators are strictly local; nonlocal joint measurements violate microcausalit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

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}

, where

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

. - For arbitrary barrier

V(x)

: WKB tunneling amplitude

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

. - Reflection/transmission amplitudes satisfy

|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

\Delta t \Delta E \gtrsim \hbar

constraints. CONNECTION: - The exponential decay factor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com