Abstract We develop a directional-amplitude representation of spin-3/2 quantum measurement based on three dipolar directional amplitudes, each retaining a binary hemispherical response to analyzer orientation. The three amplitudes form a rigid three-dimensional structure with 60-degree mutual separation, symmetrically arranged on a cone about a derived collective direction. For a pristine isotropic ensemble, this geometry produces equal occupation of the four Stern–Gerlach projection channels. Preparation is represented by a signed distribution of collective-axis orientations relative to the preparation axis, while the internal three-amplitude geometry of each particle remains fixed. Coherent combination of the three amplitudes and rotational completion of the rigid triad produce the collective response required for subsequent analyzer rotation. The resulting preparation distributions contain harmonic orders only through rank 3. Convolution of these prepared distributions with the collective response reproduces the complete standard spin-3/2 rotational transition matrix for arbitrary relative analyzer orientation, including the non-binomial intermediate-state probabilities. The construction additionally satisfies normalization, immediate repeatability, analyzer reversal, rotational covariance, isotropic recombination, and sequential re-preparation. A resolved intermediate output behaves subsequently as the corresponding freshly prepared state without requiring alteration of the internal 60-degree geometry. Together with the corresponding spin-1/2 and spin-1 constructions, the survival of the same elementary binary hemispherical response through the complete four-channel spin-3/2 rotational structure provides further evidence for a common fine-grained directional-amplitude representation underlying quantum spin measurement.
Abstract We develop a directional-amplitude representation of spin-3/2 quantum measurement based on three dipolar directional amplitudes, each retaining a binary hemispherical response to analyzer orientation. The three amplitudes form a rigid three-dimensional structure with 60-degree mutual separation, symmetrically arranged on a cone about a derived collective direction. For a pristine isotropic ensemble, this geometry produces equal occupation of the four Stern–Gerlach projection channels. Preparation is represented by a signed distribution of collective-axis orientations relative to the preparation axis, while the internal three-amplitude geometry of each particle remains fixed. Coherent combination of the three amplitudes and rotational completion of the rigid triad produce the collective response required for subsequent analyzer rotation. The resulting preparation distributions contain harmonic orders only through rank 3. Convolution of these prepared distributions with the collective response reproduces the complete standard spin-3/2 rotational transition matrix for arbitrary relative analyzer orientation, including the non-binomial intermediate-state probabilities. The construction additionally satisfies normalization, immediate repeatability, analyzer reversal, rotational covariance, isotropic recombination, and sequential re-preparation. A resolved intermediate output behaves subsequently as the corresponding freshly prepared state without requiring alteration of the internal 60-degree geometry. Together with the corresponding spin-1/2 and spin-1 constructions, the survival of the same elementary binary hemispherical response through the complete four-channel spin-3/2 rotational structure provides further evidence for a common fine-grained directional-amplitude representation underlying quantum spin measurement.
We develop a unified geometric representation of spin-½ Stern–Gerlach measurement and linear photon polarization using nonnegative distributions of individual directional amplitudes with deterministic binary responses. For spin-½, hemispherical classification of individual orientations reproduces the complete rotational transition law, P₊(α) = cos²(α/2). For linear photon polarization, an unoriented transverse axis and deterministic ±45° acceptance sectors reproduce Malus' law, P(β) = cos²β, through an exact angular overlap integral. The principal mathematical results are two support-constrained uniqueness theorems. In both geometries, requiring a normalized nonnegative distribution confined to the preparation acceptance region, together with agreement with the complete family of rotational transition probabilities, uniquely determines the corresponding cosine-weighted prepared-state density. The spherical and planar proofs establish how geometric support conditions eliminate ambiguities otherwise left unresolved by binary filtering. A systematic consistency analysis examines pristine populations, complete transition matrices, analyzer reversal, rotational covariance, immediate repeatability, operational isotropy of equal mixtures, and sequential re-preparation. The constructions reproduce the standard ideal measurement statistics while distinguishing definite individual directional amplitudes from their statistical distributions across prepared ensembles. Together, these results establish a common mathematical framework for two physically different binary measurement geometries, identifying the positive directional distributions compatible with their complete quantum transition statistics and providing a basis for investigating higher-spin measurement structures.
Abstract Recent work introduced a phase-coherent directional amplitude framework for quantum measurement in which the physical state of an individual particle is represented by a signed directional amplitude distribution on an orientation manifold. In that framework, Stern–Gerlach statistics, Malus’ law, and bipartite correlations saturating the Tsirelson bound were reproduced by integrating amplitude distributions over analyzer-defined geometric regions. These results suggested that the amplitude distributions used in such calculations are not arbitrary modeling choices but arise from symmetry and measurement geometry. In the present work this framework is extended to spin-1 systems. Sequential Stern–Gerlach measurements for spin-1 particles produce three possible outcomes and exhibit quadratic angular dependence. Using harmonic analysis on the sphere together with a geometric three-channel filtering model of the Stern–Gerlach analyzer, we derive the minimal directional amplitude distribution capable of reproducing the spin-1 transition laws. The analysis shows that the accessible harmonic sector is restricted to , and that both the distribution coefficients and analyzer thresholds are uniquely fixed by symmetry and operational transition rules. We then show that this result generalizes: for a spin- system, the directional amplitude distribution must lie within the harmonic sector . This establishes a structural connection between quantum measurement statistics and the representation theory of rotations. Within this perspective, quantum measurement acts as harmonic filtering of orientation-dependent amplitude fields, and state reduction emerges as geometric marginalization imposed by the measurement interaction.
Abstract. In a recent work, we established a phase-coherent directional amplitude framework that natively reproduces the full spectrum of bipartite quantum correlations, exactly saturating the Tsirelson bound . While that work demonstrated the predictive power of retaining continuous geometric phase information, the structural origin and uniqueness of the underlying signed amplitude distributions remained implicit. In the present paper we show that these distributions are uniquely determined by symmetry and measurement geometry. We propose a geometric interpretation of quantum measurement in which state reduction arises from marginalization over operationally inaccessible degrees of freedom. An arriving particle and a measurement apparatus uniquely determine a plane of relevance, while orthogonal degrees of freedom are physically filtered and marginalized from the effective description. Using harmonic analysis on the sphere, we prove that this geometric marginalization—together with rotational covariance and normalization—uniquely fixes the post-selection amplitude distributions. For spin-½ systems, rotational symmetry restricts the Legendre expansion to its lowest harmonics, forcing the exact signed distribution previously employed. Extended to bipartite product spaces , global rotational invariance uniquely determines the joint amplitude structure that reproduces the standard singlet correlation law. These results provide a kinematic explanation of the amplitude distributions underlying quantum spin correlations and offer a geometric account of quantum state reduction as marginalization over degrees of freedom that are operationally inaccessible to the measurement interaction.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
Complementary Phase Geometry (CPG) contains an intrinsic three-dimensional complementary rotational sector. Here this structure is examined as a geometric carrier of spin and polarization. A normalized complementary rotation bivector defines an oriented plane whose dual axis is identified with polarization. Its spinorial lift yields the standard double-cover structure Spin(3)≃SU(2) and the half-phase relation χ2=Υ. In the fundamental representation,S^a=2bσa,where b is the CPG action scale; with the empirical identification b=ℏ, the standard spin-1/2 algebra, projections, and Casimir are recovered. Combining the polarization--plane identification with the pre-existing mixed observable--complementary dynamics of CPG yields a further conditional prediction. Both the free-flight timing variance and the transverse-position variance contain the common dimensionless dependencesin2ϑsin2(2vωML),while their dimensional prefactors are different. When the rapid propagation phase is unresolved, the common angular dependence reduces to sin2ϑ. The absolute magnitudes remain controlled by presently undetermined complementary and mixed-coupling parameters. Phenomenological magnetic coupling reproduces standard Stern--Gerlach eigenchannels and analyzer-overlap weights. Magnetic moments, microscopic branch selection, and Born-rule statistics remain open.
A canonical formulation of projective quantum measurement is developed from the selected order-two Taub–NUT/Hopf–Spinᶜ primitive. Its complete normalized geometric trace also yields the infrared fine-structure constant under the electromagnetic coupling identification, providing a common foundation for coupling normalization and measurement evaluation. The ordered chiral carrier supplies an inherited Hermitian pairing and a preparation representation with fixed normalization. Naturality under the unspecified outgoing frame action exhausts the incoming record endomorphisms, while positivity and eigenstate calibration uniquely determine the Born evaluation. For canonically projected pure preparations, rank-one compression fixes the normalized complete-functional insertion ratio before determinant and collective integration. Canonical normalized restriction yields the Lüders-form state transformation. Conserved-current identities connect represented spatial records to the same evaluation, and a reduced Maxwell–Dirac calculation establishes two-channel separation with a quantified finite-resolution correction. Independent repeated preparations yield frequency concentration and, under the corresponding record probability law, convergence almost surely to the Born weights. Within the canonical incoming-record representation, the Born weights and normalized projection rule are therefore fixed by the same primitive geometry, without independent outcome weights or an additional probability exponent.