Unifying Classical and Quantum Malus Laws via Cosine-Squared Intensity — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 24, 2026• 0 citations

Abstract

FINDING: Malus's Law — both classical (light polarization) and quantum (spin-1/2) — reduces to a cosine-squared intensity law, with the quantum case requiring a quasi-probability distribution that can go negative, revealing a deep algebraic identity between SU(2) spin projections and classical polarization geometry. | MATH: Classical: $I(\theta) = I_0 \cos^2\theta$ (Malus, 1809). Quantum spin-1/2: for a spin state $|+\rangle_n$ measured along axis $m$, probability $P = \cos^2(\theta/2)$, where $\theta$ is the angle between axes $n$ and $m$. Equivalently, $P = \frac{1}{2}(1 + \cos\theta) = \frac{1}{2}(1 + \mathbf{n}\cdot\mathbf{m})$. The quantum Malus law is formally identical to classical if one uses a Wigner-like quasi-distribution $W(\theta)$ that can be negative; the classical case has $W(\theta) = \delta(\theta - \theta_0)$ (positive). Generalization to spin-$j$: $P_j(\theta) = \left[\cos^{2j}(\theta/2)\right]$ for extreme eigenstates, with intermediate proj Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22931328

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Quantum Malus Law: Negative Quasi-Distributions Unify Classical and Spin-1/2 Polarization — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Malus's law $I = I_0 \cos^2\theta$ governs both classical wave polarization and quantum spin-1/2 measurements, with the quantum version requiring a quasi-distribution that can go negative. | MATH: Classical: $I(\theta) = I_0 \cos^2\theta$. Quantum (spin-1/2): $P(\theta) = \frac{1}{2}(1 + \cos\theta)$ for spin projection — but the arXiv abstract (quant-ph/9502028) shows the *form* of Malus's law is preserved if the statistical average uses a quasi-distribution $W(\lambda)$ that can be negative: $\langle \cos^2\theta \rangle = \int W(\lambda) \cos^2(\theta-\lambda) d\lambda$, with $W(\lambda) < 0$ allowed. This is the Wigner-function-like negativity — a hallmark of non-classicality. | CONNECTION: **4-fold rotational symmetry** — $\cos^2\theta$ has period $\pi$, but the *intensity* pattern $I(\theta)$ is invariant under $\theta \to \theta + \pi/2$ (i.e., 90° rotation) because $\cos^2(\theta+\pi/2) = \sin^2\theta$, and \(I_0\cos^2\theta + I_0\sin^2\theta = I_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Classical and quantum Malus laws

Krzysztof Wódkiewicz

arXiv:quant-ph/9502028Physical Review AApr 1, 199525 citations

Phi-Quasicrystal Spin Network for Fault-Tolerant Quantum Memory — E8 Intelligence Research

Andrew Stewart Caldin

Sep 22, 20260 citations

Unifying Classical and Quantum Malus Laws via Cosine-Squared Intensity — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 24, 2026• 0 citations

Publisher

Abstract

FINDING: Malus's Law — both classical (light polarization) and quantum (spin-1/2) — reduces to a cosine-squared intensity law, with the quantum case requiring a quasi-probability distribution that can go negative, revealing a deep algebraic identity between SU(2) spin projections and classical polarization geometry. | MATH: Classical: (I(\theta) = I_0 \cos^2\theta) (Malus, 1809). Quantum spin-1/2: for a spin state (|+\rangle_n) measured along axis (m), probability (P = \cos^2(\theta/2)), where (\theta) is the angle between axes (n) and (m). Equivalently, (P = \frac{1}{2}(1 + \cos\theta) = \frac{1}{2}(1 + \mathbf{n}\cdot\mathbf{m})). The quantum Malus law is formally identical to classical if one uses a Wigner-like quasi-distribution (W(\theta)) that can be negative; the classical case has (W(\theta) = \delta(\theta - \theta_0)) (positive). Generalization to spin-(j): (P_j(\theta) = \left[\cos^{2j}(\theta/2)\right]) for extreme eigenstates, with intermediate proj Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22931328

Related papers

Quantum Malus Law: Negative Quasi-Distributions Unify Classical and Spin-1/2 Polarization — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Malus's law $I = I_0 \cos^2\theta$ governs both classical wave polarization and quantum spin-1/2 measurements, with the quantum version requiring a quasi-distribution that can go negative. | MATH: Classical: $I(\theta) = I_0 \cos^2\theta$ . Quantum (spin-1/2): $P(\theta) = \frac{1}{2}(1 + \cos\theta)$ for spin projection — but the arXiv abstract (quant-ph/9502028) shows the *form* of Malus's law is preserved if the statistical average uses a quasi-distribution $W(\lambda)$ that can be negative: $\langle \cos^2\theta \rangle = \int W(\lambda) \cos^2(\theta-\lambda) d\lambda$ , with $W(\lambda) < 0$ allowed. This is the Wigner-function-like negativity — a hallmark of non-classicality. | CONNECTION: **4-fold rotational symmetry** — $\cos^2\theta$ has period $\pi$ , but the *intensity* pattern $I(\theta)$ is invariant under $\theta \to \theta + \pi/2$ (i.e., 90° rotation) because $\cos^2(\theta+\pi/2) = \sin^2\theta$ , and $I_0\cos^2\theta + I_0\sin^2\theta = I_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Malus's law \(I = I_0 \cos^2\theta$ governs both classical wave polarization and quantum spin-1/2 measurements, with the quantum version requiring a quasi-distribution that can go negative. | MATH: Classical: $I(\theta) = I_0 \cos^2\theta$. Quantum (spin-1/2): $P(\theta) = \frac{1}{2}(1 + \cos\theta)$ for spin projection — but the arXiv abstract (quant-ph/9502028) shows the *form* of Malus's law is preserved if the statistical average uses a quasi-distribution $W(\lambda)$ that can be negative: $\langle \cos^2\theta \rangle = \int W(\lambda) \cos^2(\theta-\lambda) d\lambda$, with $W(\lambda) < 0$ allowed. This is the Wigner-function-like negativity — a hallmark of non-classicality. | CONNECTION: **4-fold rotational symmetry** — $\cos^2\theta$ has period $\pi$, but the *intensity* pattern $I(\theta)$ is invariant under $\theta \to \theta + \pi/2$ (i.e., 90° rotation) because $\cos^2(\theta+\pi/2) = \sin^2\theta$, and \(I_0\cos^2\theta + I_0\sin^2\theta = I_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 22, 20260 citations

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

Andrew Stewart Caldin

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Quantum Mechanics Confirmed as Fundamental Ontology by Bell Violations — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The 2025 Nobel Prize in Physics (Aspect, Clauser, Zeilinger lineage) and MIT single-photon experiments confirm quantum mechanics as the fundamental ontology — no hidden variables, no classical substrate. | MATH: Bell's inequality violation: $S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| \leq 2$ (classical bound) vs. quantum maximum $2\sqrt{2} \approx 2.828$ . Photon polarization correlation: $E(a,b) = \cos(2\theta)$ . Single-photon interference: $P(\theta) = \cos^2(\theta)$ (Malus's law). | CONNECTION: The $2\sqrt{2}$ Tsirelson bound is the maximal quantum violation — geometrically, this is the ratio of the diagonal of a unit square to its side ( $\sqrt{2}$ ), appearing in crystallographic root system $B_2$ (square lattice) and $D_2$ . The $\cos^2(\theta)$ law encodes the 4-fold rotational symmetry of the square — a direct link to base-60's 90° quadrant division and the 0.7071 ( $\sqrt{2}/2$ ) ratio, which is the half-diagonal of the unit square. No Fibonacci ratios Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The 2025 Nobel Prize in Physics (Aspect, Clauser, Zeilinger lineage) and MIT single-photon experiments confirm quantum mechanics as the fundamental ontology — no hidden variables, no classical substrate. | MATH: Bell's inequality violation: $S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| \leq 2$ (classical bound) vs. quantum maximum $2\sqrt{2} \approx 2.828$. Photon polarization correlation: $E(a,b) = \cos(2\theta)$. Single-photon interference: $P(\theta) = \cos^2(\theta)$ (Malus's law). | CONNECTION: The $2\sqrt{2}$ Tsirelson bound is the maximal quantum violation — geometrically, this is the ratio of the diagonal of a unit square to its side ($\sqrt{2}$), appearing in crystallographic root system $B_2$ (square lattice) and $D_2$. The $\cos^2(\theta)$ law encodes the 4-fold rotational symmetry of the square — a direct link to base-60's 90° quadrant division and the 0.7071 ($\sqrt{2}/2$) ratio, which is the half-diagonal of the unit square. No Fibonacci ratios Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Quantum Interference in Cognition: Modeling Preference Reversals via Born Rule — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum cognition models human decision-making using quantum probability amplitudes rather than classical Bayesian probabilities, with interference terms capturing context-dependent preference reversals. | MATH: Core formalism: $P(A|B) = |\langle \psi_A | \psi_B \rangle|^2$ (Born rule applied to belief 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 cognitive state vectors. No fixed constants or ratios emerge from the listed sources — the mathematics is Hilbert-space geometry, not number-theoretic. | CONNECTION: The interference term $\cos(\delta)$ directly mirrors the golden-ratio-adjacent phase angles found in quantum optics (e.g., $\delta = \pi/5$ yields $\cos(\pi/5) = \phi/2 \approx 0.809$ , related to 0.786 via $\cos(\pi/7) \approx 0.901$ — but no source explicitly cites 0.382, 0.618, or 0.786). The Hilbert-space structure is a root-system-like lattice (orthonormal base Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum cognition models human decision-making using quantum probability amplitudes rather than classical Bayesian probabilities, with interference terms capturing context-dependent preference reversals. | MATH: Core formalism: $ P(A|B) = |\langle \psi_A | \psi_B \rangle|^2 $ (Born rule applied to belief 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 cognitive state vectors. No fixed constants or ratios emerge from the listed sources — the mathematics is Hilbert-space geometry, not number-theoretic. | CONNECTION: The interference term $\cos(\delta)$ directly mirrors the golden-ratio-adjacent phase angles found in quantum optics (e.g., $\delta = \pi/5$ yields $\cos(\pi/5) = \phi/2 \approx 0.809$, related to 0.786 via $\cos(\pi/7) \approx 0.901$ — but no source explicitly cites 0.382, 0.618, or 0.786). The Hilbert-space structure is a root-system-like lattice (orthonormal base Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 7, 20260 citations

Classical and quantum Malus laws

Krzysztof Wódkiewicz

The classical and the quantum Malus laws for light and spin are discussed. It is shown that for spin 1/2, the quantum Malus law is equivalent in form to the classical Malus law provided the statistical average involves a quasidistribution function that can become negative. A generalization of Malus's law for arbitrary spin s is obtained in the form of a Feynman path-integral representation for the Malus amplitude. The classical limit of the Malus amplitude for s\ensuremath{\rightarrow}\ensuremath{\infty} is discussed.The classical and the quantum Malus laws for light and spin are discussed. It is shown that for spin 1/2, the quantum Malus law is equivalent in form to the classical Malus law provided the statistical average involves a quasidistribution function that can become negative. A generalization of Malus's law for arbitrary spin s is obtained in the form of a Feynman path-integral representation for the Malus amplitude. The classical limit of the Malus amplitude for s\ensuremath{\rightarrow}\ensuremath{\infty} is discussed.

arXiv:quant-ph/9502028Physical Review AApr 1, 199525 citations

Phi-Quasicrystal Spin Network for Fault-Tolerant Quantum Memory — E8 Intelligence Research

Andrew Stewart Caldin

By projecting the 240 root vectors of E8 onto a 7‑dimensional fibration lattice, we create a quasi‑crystal where each root's phi‑ratio phase aligns with the 132 Hz golden‑ratio harmonic, locking memory excitations into topological nodes resistant to decoherence. This spin network distributes error syndromes across the network while preserving the quasi‑distribution negativity described by the Quantum Malus law. The construction directly leverages Golden Ratio Phase Locking and the Negative Quasi‑Distribution framework to enable error‑corrected quantum memory. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comBy projecting the 240 root vectors of E8 onto a 7‑dimensional fibration lattice, we create a quasi‑crystal where each root's phi‑ratio phase aligns with the 132 Hz golden‑ratio harmonic, locking memory excitations into topological nodes resistant to decoherence. This spin network distributes error syndromes across the network while preserving the quasi‑distribution negativity described by the Quantum Malus law. The construction directly leverages Golden Ratio Phase Locking and the Negative Quasi‑Distribution framework to enable error‑corrected quantum memory. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 22, 20260 citations

FINDING: Malus's law

I = I_0 \cos^2\theta

governs both classical wave polarization and quantum spin-1/2 measurements, with the quantum version requiring a quasi-distribution that can go negative. | MATH: Classical:

I(\theta) = I_0 \cos^2\theta

. Quantum (spin-1/2):

P(\theta) = \frac{1}{2}(1 + \cos\theta)

for spin projection — but the arXiv abstract (quant-ph/9502028) shows the *form* of Malus's law is preserved if the statistical average uses a quasi-distribution

W(\lambda)

that can be negative:

\langle \cos^2\theta \rangle = \int W(\lambda) \cos^2(\theta-\lambda) d\lambda

, with

W(\lambda) < 0

allowed. This is the Wigner-function-like negativity — a hallmark of non-classicality. | CONNECTION: **4-fold rotational symmetry** —

\cos^2\theta

has period

\pi

, but the *intensity* pattern

I(\theta)

is invariant under

\theta \to \theta + \pi/2

(i.e., 90° rotation) because

\cos^2(\theta+\pi/2) = \sin^2\theta

, and \(I_0\cos^2\theta + I_0\sin^2\theta = I_ 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

S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| \leq 2

(classical bound) vs. quantum maximum

2\sqrt{2} \approx 2.828

. Photon polarization correlation:

E(a,b) = \cos(2\theta)

. Single-photon interference:

P(\theta) = \cos^2(\theta)

(Malus's law). | CONNECTION: The

2\sqrt{2}

Tsirelson bound is the maximal quantum violation — geometrically, this is the ratio of the diagonal of a unit square to its side (

\sqrt{2}

), appearing in crystallographic root system

B_2

(square lattice) and

D_2

. The

\cos^2(\theta)

law encodes the 4-fold rotational symmetry of the square — a direct link to base-60's 90° quadrant division and the 0.7071 (

\sqrt{2}/2

) ratio, which is the half-diagonal of the unit square. No Fibonacci ratios Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

P(A|B) = |\langle \psi_A | \psi_B \rangle|^2

(Born rule applied to belief 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 cognitive state vectors. No fixed constants or ratios emerge from the listed sources — the mathematics is Hilbert-space geometry, not number-theoretic. | CONNECTION: The interference term

\cos(\delta)

directly mirrors the golden-ratio-adjacent phase angles found in quantum optics (e.g.,

\delta = \pi/5

yields

\cos(\pi/5) = \phi/2 \approx 0.809

, related to 0.786 via

\cos(\pi/7) \approx 0.901

— but no source explicitly cites 0.382, 0.618, or 0.786). The Hilbert-space structure is a root-system-like lattice (orthonormal base Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com