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 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.
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 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 propose a geometric reformulation of measurement in Hyperhamiltonian Quantum Mechanics (HHQM), replacing the inverse-distance propensity by a complex transition amplitude defined on the state manifold. The amplitude depends on geodesic distance and a phase functional associated with the hyperquantum bundle, leading to a normalized Born-type probability rule. In a minimal bipartite setting, we show that interference between a small number of transition channels can produce joint probability distributions with nonclassical correlations. In particular, a simple two-channel model yields a cosine correlator and attains the Tsirelson bound while preserving no-signaling. The resulting structure suggests that Bell-type correlations in HHQM, if realized, arise from interference of geometric transition amplitudes rather than from spacetime dynamics. The explicit kernel constructed here should be regarded as an effective realization within this framework; a full derivation from the underlying bundle geometry remains an open problem.
We develop an explicit configuration-space formulation of EPR-Bell experiments within the framework of de Broglie-Bohm theory, in which joint measurement outcomes arise from a deterministic mapping from initial particle configurations to outcome pairs. This construction induces a partition of the hidden-variable configuration space into domains associated with the different measurement outcomes. Using a reduced-dimensional Stern-Gerlach model, we derive the structure of these domains and identify the corresponding separatrices that define their boundaries. We show that Bell correlations emerge from the geometry of these partitions: the domain boundaries depend nonlocally on the measurement settings, while the marginal outcome distributions remain invariant, providing a direct dynamical realization of no-signaling. Analytical results are supported by numerical simulations, which exhibit quantitative agreement with the predicted domain structure as a consequence of the underlying partition of configuration space induced by the measurement dynamics. This approach provides an explicit configuration-space representation of nonlocal correlations in Bohmian mechanics, linking trajectory dynamics, measurement processes, and statistical predictions within a unified framework.
This paper proposes a geometric selection framework for quantum measurement that reframes collapse as a contextual, basin-dependent selection process rather than a universal instantaneous postulate. Beginning with the Stern–Gerlach experiment, measurement outcomes are interpreted as symmetry-constrained selections in an internal SU(2) state space, recovering standard spin statistics without invoking hidden classical variables. The framework is extended to sequential measurements and interferometric configurations through the introduction of an explicit, computable collapse-rate functional. This functional interpolates continuously between unitary evolution and effective projection, depending on both branch distinguishability and a phenomenological measure of measurement-basin accessibility. The Born rule is recovered exactly at the moment of selection, ensuring full compatibility with standard quantum mechanics in appropriate limits. The model predicts small, testable deviations from standard quantum mechanics in near-threshold interferometric regimes, where coherence is only partially suppressed. These deviations are distinct from those predicted by objective-collapse models with universal rates and can be constrained using existing and near-term matter-wave interferometry platforms. The approach provides a minimal, empirically grounded extension of measurement theory that remains falsifiable and compatible with current experimental bounds.