Directional-Amplitude Geometry of Spin-½ and Linear Photon Polarization: Deterministic Filtering, Uniqueness, and Consistency
- 1Independent Researcher
Abstract
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.
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BibTeX
@misc{karlsen2026directionalamplitude,
title = {{Directional-Amplitude Geometry of Spin-½ and Linear Photon Polarization: Deterministic Filtering, Uniqueness, and Consistency}},
author = {Jorgen Karlsen},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23603945},
url = {https://zenodo.org/records/23603945},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Directional-Amplitude Geometry of Spin-½ and Linear Photon Polarization: Deterministic Filtering, Uniqueness, and Consistency
AU - Jorgen Karlsen
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23603945
UR - https://zenodo.org/records/23603945
AB - 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.
ER -