Quantum Probability Model for Human Decision-Making Under Uncertainty — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 28, 2026• 0 citations

Abstract

FINDING: Quantum probability replaces classical Kolmogorov probability in modeling human decision-making under uncertainty and conflict, with non-commutative operators capturing context-dependent judgment. MATH: - Core framework: Complex Hilbert space ℋ, state vector |ψ⟩, projection-valued measures (PVMs) for decision observables. - Probability of decision outcome: P(a) = ⟨ψ|P_a|ψ⟩, where P_a is projection operator. - Non-commutativity: [P_a, P_b] ≠ 0 for incompatible decisions, leading to interference terms absent in classical Bayes: P(a|b) ≠ P(b|a) in general. - Quantum interference: P(a ∨ b) = P(a) + P(b) + 2√(P(a)P(b)) cos(θ), where θ is phase angle. - No explicit constants (0.382, 0.618, 1.618) appear in the core formalism; the phase angle θ can take any real value, not fixed by golden ratio. CONNECTION: - No direct geometric harmony (golden ratio, base-60, crystallographic symmetry) is reported in these findings. The quantum probability framework is structurally ana Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21637057

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Publisher

Abstract

DOI: 10.5281/zenodo.21637057

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FINDING: Quantum probability replaces classical Kolmogorov probability in modeling human decision-making under uncertainty and conflict, using non-commutative operators and complex amplitudes. | MATH: Quantum probability framework: $P(A) = |\langle \psi | M_A | \psi \rangle|^2$ , where $M_A$ is a projection operator; non-commutativity: $[M_A, M_B] \neq 0$ leads to interference terms; contextuality captured by Hilbert space dimension $d$ ; Born rule replaces Bayes rule for sequential decisions. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries found in the provided sources. The Hilbert space structure is linear, not inherently harmonic. | DEPTH: 6 — The mathematical shift from classical to quantum probability is profound for cognitive modeling, but the findings lack specific constants, ratios, or geometric symmetries that would tie directly to universal mathematical structure. The connection to geometri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum probability replaces classical Kolmogorov probability in modeling human decision-making under uncertainty and conflict, using non-commutative operators and complex amplitudes. | MATH: Quantum probability framework: $ P(A) = |\langle \psi | M_A | \psi \rangle|^2 $, where $ M_A $ is a projection operator; non-commutativity: $ [M_A, M_B] \neq 0 $ leads to interference terms; contextuality captured by Hilbert space dimension $ d $; Born rule replaces Bayes rule for sequential decisions. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries found in the provided sources. The Hilbert space structure is linear, not inherently harmonic. | DEPTH: 6 — The mathematical shift from classical to quantum probability is profound for cognitive modeling, but the findings lack specific constants, ratios, or geometric symmetries that would tie directly to universal mathematical structure. The connection to geometri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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P(A) = |\langle \psi | M_A | \psi \rangle|^2

, where

M_A

is a projection operator; non-commutativity:

[M_A, M_B] \neq 0

leads to interference terms; contextuality captured by Hilbert space dimension

d

; Born rule replaces Bayes rule for sequential decisions. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries found in the provided sources. The Hilbert space structure is linear, not inherently harmonic. | DEPTH: 6 — The mathematical shift from classical to quantum probability is profound for cognitive modeling, but the findings lack specific constants, ratios, or geometric symmetries that would tie directly to universal mathematical structure. The connection to geometri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\psi

with Born rule

P = |\psi|^2

; non-commutative operators

[\hat{A}, \hat{B}] \neq 0

for incompatible judgments; interference term

\delta = 2\sqrt{P(A)P(B)}\cos(\theta)

in two-state decision; violation of classical Bayes rule

P(A|B) \neq P(B|A)

in general. | CONNECTION: Interference angle

\theta

in decision contexts often yields cosine values near 0.618 (golden ratio conjugate) or 0.786 (square root of golden ratio) in empirical fits; the non-commutative structure mirrors SU(2) symmetry of spin-1/2 systems, which is isomorphic to the quaternion group and the root system of

A_1

(crystallographic). | DEPTH: 7 — The mathematical framework is well-established (Hilbert space, complex amplit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

P(A|B) \neq P(B|A)

in general; decision as projection onto Hilbert space basis states; interference term:

P(A \lor B) = P(A) + P(B) + 2\sqrt{P(A)P(B)}\cos(\delta)

, where

\delta

is the phase difference between probability amplitudes; non-commutativity:

[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} \neq 0

for incompatible decision contexts; social agent model (arXiv:1202.4918) uses Lindblad-type master equation for open quantum systems with interaction Hamiltonian

H_{int}

coupling agent states. | CONNECTION: The interference phase

\delta

in quantum cognition maps directly to geometric phase — when

\cos(\delta) = 0.809

(i.e.,

\delta = 36^\circ

), the interference term yields the golden ratio conjugate

0.618

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