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
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
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(θ)=I0cos2θ (Malus, 1809). Quantum spin-1/2: for a spin state ∣+⟩n measured along axis m, probability P=cos2(θ/2), where θ is the angle between axes n and m. Equivalently, P=21(1+cosθ)=21(1+n⋅m). The quantum Malus law is formally identical to classical if one uses a Wigner-like quasi-distribution W(θ) that can be negative; the classical case has W(θ)=δ(θ−θ0) (positive). Generalization to spin-j: Pj(θ)=[cos2j(θ/2)] for extreme eigenstates, with intermediate proj Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
For nearly a century, quantum mechanics (QM) has established itself as an exceptionally effective framework for predicting the collective movements and statistical distributions of particles. However, its predictive success comes with a significant conceptual cost: the apparent abandonment of local realism. When one attempts to apply the standard quantum formalism at the scale of an individual particle, paradoxes inevitably arise. This article argues that these contradictions are not intrinsic properties of nature, but rather stem from a fundamental category error: the misapplication of ensemble statistical laws to individual events, which unrightfully promotes statistical averages to instantaneous physical realities. To address the root cause of this conceptual deviation, we must establish a foundational axiom: the Principle of Particle Reality. This principle posits that if a particle exists as a real entity, it must inherently possess definite physical properties—such as a precise polarization, velocity, and trajectory—independent of our capacity to detect them. Ignoring this ontology has led the standard model to treat physical realities as abstract paradoxes. To resolve them, this work proposes a unified reconsideration of three main pillars of quantum interpretation. First, regarding locality, we demonstrate that the passage probabilities through two polarizers for a particle within an unpolarized beam remain identical regardless of their order. Governed by Malus's law and the strict mathematical symmetry \cos^2(a-b) = \cos^2(b-a), this establishes that polarizers can be arranged in any sequential order or even in parallel for statistically identical pairs without altering the total passage probabilities. By establishing a physical equivalence, we show that exact quantum correlations emerge directly from a particle-by-particle application of the strictly positive classical Malus's law (0 \le p \le 1). This successfully explains the violation of Bell’s inequalities within a strictly local framework, bypassing the non-physical negative quasi-probabilities encountered in previous angular hidden-variable models. Secondly, on the illusion of state superposition, we show that failing to recognize that an interferometer physically and deterministically alters the polarization orientation of a particle led to an abstract formulation; in reality, this involves a deterministic geometric transformation of probability distributions. Finally, addressing wave-particle duality, we challenge the standard postulate of artificial simultaneity in the double-slit configuration, which caused a deep confusion between the collective behavior of a group and the sequential passage of an individual entity. We introduce an ``event-by-event'' formulation where interference fringes emerge instead from temporal correlations preserved by the non-commutativity of averaging. By restoring the strict distinction between statistical ensemble laws and individual reality through the Principle of Particle Reality, this work aims to definitively end ``quantum geocentrism'' and to lay the foundations for a coherent, local, and realistic sub-quantum physics.
For nearly a century, quantum mechanics (QM) has established itself as an exceptionally effective framework for predicting the collective movements and statistical distributions of particles. However, its predictive success comes with a significant conceptual cost: the apparent abandonment of local realism. When one attempts to apply the standard quantum formalism at the scale of an individual particle, paradoxes inevitably arise. This article argues that these contradictions are not intrinsic properties of nature, but rather stem from a fundamental category error: the misapplication of ensemble statistical laws to individual events, which unrightfully promotes statistical averages to instantaneous physical realities. To address the root cause of this conceptual deviation, we must establish a foundational axiom: the Principle of Particle Reality. This principle posits that if a particle exists as a real entity, it must inherently possess definite physical properties—such as a precise polarization, velocity, and trajectory—independent of our capacity to detect them. Ignoring this ontology has led the standard model to treat physical realities as abstract paradoxes. To resolve them, this work proposes a unified reconsideration of three main pillars of quantum interpretation. First, regarding locality, we demonstrate that a polarizer does not create a state but merely reveals a pre-existing polarization. The concrete proof lies in the reduction of a beam’s intensity, which is fundamentally a physical drop in the number of transmitted particles. By establishing a physical equivalence between parallel and sequential measurements via the principle of Statistical Identity, we show that exact quantum correlations emerge directly from a particle-by-particle application of the strictly positive classical Malus’s law (0 ≤ p ≤ 1). This successfully explains the violation of Bell’s inequalities within a strictly local framework, bypassing the non-physical negative quasi-probabilities encountered in previous angular hidden-variable models. Secondly, on the illusion of state superposition, we show that failing to recognize that an interferometer physically and deterministically alters the polarization orientation of a particle led to an abstract formulation; in reality, this involves a deterministic geometric transformation of probability distributions. Finally, addressing wave-particle duality, we challenge the standard postulate of artificial simultaneity in the double-slit configuration, which caused a deep confusion between the collective behavior of a group and the sequential passage of an individual entity. We introduce an “event-by-event” formulation where interference fringes emerge instead from temporal correlations preserved by the non-commutativity of averaging. By restoring the strict distinction between statistical ensemble laws and individual reality through the Principle of Particle Reality, this work aims to definitively end “quantum geocentrism” and to lay the foundations for a coherent, local, and realistic sub-quantum physics.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
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.com