Harmonic Structure of Spin-1 Quantum Measurement: Geometric Marginalization of Directional Amplitude Distributions

This third paper in the directional-amplitude series asks whether the spin-½  construction developed in Papers 1 and 2 can be extended to higher spin without introducing a new elementary measurement rule.

The central result is that the complete spin-1 measurement structure can be represented by the same directional amplitudes with binary hemispherical response, while recovering the exact rotational formalism and statistics of conventional spin-1 quantum mechanics.

Main results and significance

• Spin-1 measurement is represented using two connected dipolar directional amplitudes, each having a binary hemispherical response to analyzer orientation, rather than introducing a primitive three-valued response.

·        Requiring the pristine isotropic ensemble to split equally into the three Stern–Gerlach channels, so that   which fixes the mutual angle of the two amplitudes at exactly .

• The same fixed particle geometry supports three distinct prepared directional-amplitude distributions, , , and . Their equal mixture reconstructs the original isotropic ensemble.
• Convolution of these prepared distributions with the fixed-pair response reproduces the complete standard spin-1 rotational probabilities for a subsequently rotated analyzer.
• The  state retains nontrivial second-rank directional structure even though its first-rank polar contribution vanishes: zero measured projection does not imply absence of fine-grained directional-amplitude structure.
• The prepared distributions terminate at , consistent with the standard tensor-rank structure of spin 1; the quadrupolar sector emerges from the organization of the two dipolar amplitudes rather than from an additional primitive quadrupolar amplitude.
• The construction additionally satisfies immediate repeatability, analyzer reversal, rotational covariance, isotropic recombination, and sequential re-preparation. A selected intermediate channel behaves subsequently as the same prepared state obtained by direct preparation in that channel.
• The standard rotational probabilities are recovered independently through conventional coherent composition of two spin-½ amplitudes, providing a direct connection between the directional-amplitude geometry and the conventional spin-1 representation.

The two-amplitude particle geometry with a fixed angle survives a set of independent constraints: the pristine  split, all three prepared states, arbitrary analyzer rotation, the structurally distinct  state, repeatability, reversal, rotational covariance, isotropic recombination, and sequential measurement. Preparation changes the ensemble organization of the pair directions, while the internal angle between the amplitudes remains unchanged.