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Physical Review Letters

Proposed Test for Complex versus Quaternion Quantum Theory

Asher Peres

Published Mar 12, 1979• 93 citations
Publisher

Abstract

If scattering amplitudes are ordinary complex numbers (not quaternions) then there is a universal algebraic relationship between the six coherent cross sections of any three scatterers (taken singly and pairwise). A violation of this relationship would indicate either that scattering amplitudes are quaternions, or that the superposition principle fails. Some experimental tests are proposed, involving neutron diffraction by crystals made of three different isotopes, neutron interferometry, and ${K}_{S}$-meson regeneration.

DOI
10.1103/physrevlett.42.683

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Notes on the proposed test for complex versus quaternionic quantum theory

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Physics Letters B•Dec 1, 1979•5 citations
If scattering amplitudes are ordinary complex numbers (not quaternions) then there is a universal algebraic relationship between the six coherent cross sections of any three scatterers (taken singly and pairwise). A violation of this relationship would indicate either that scattering amplitudes are quaternions, or that the superposition principle fails. Some experimental tests are proposed, involving neutron diffraction by crystals made of three different isotopes, neutron interferometry, and KS{K}_{S}KS​-meson regeneration.
FINDING: Search results are dominated by generic quantum superposition explainers and one specific paper on quaternionic quantum interferometry; no direct T^3 scaling law or Folman massive-superposition derivation surfaced. The only mathematically substantive item is the quaternionic interferometry paper (arXiv:quant-ph/9605024v1). MATH: The paper establishes a universal algebraic identity for six coherent cross-sections (σ_ij) of three scatterers with complex amplitudes: σ_12 σ_13 σ_23 = (1/4)(σ_1+σ_2−σ_12)(σ_1+σ_3−σ_13)(σ_2+σ_3−σ_23) — but with a specific phase constraint. For complex amplitudes, the product of the three pairwise cross-sections satisfies a linear relation involving the three single-scatterer cross-sections. If amplitudes were quaternionic, this relation breaks. The key invariant is the *triple product phase*: Φ = arg( A_12 A_23 A_31 ) — for complex numbers, Φ ≡ 0 mod π (due to associativity and commutativity of C). For quaternions, Φ can be non-trivial, revealin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Assuming the standard axioms for quaternionic quantum theory and a spatially localized scattering interaction, the SSS matrix in quaternionic quantum theory is complex valued, not quaternionic. Using the standard connections between the SSS matrix, the forward scattering amplitude for electromagnetic wave scattering, and the index of refraction, we show that the index of refraction is necessarily complex, not quaternionic. This implies that the recent optical experiment of Procopio et al. [Nat. Commun. 8, 15044 (2017)] based on the Peres proposal does not test for hypercomplex or quaternionic quantum effects arising within the standard Hilbert space framework. Such a test requires looking at near zone fields, not radiation zone fields.
FINDING: The search results are dominated by generic educational content on quantum superposition, with only one substantive mathematical item: a 1996 arXiv paper on quaternionic quantum interferometry. No direct T^3 scaling law or Folman experiment derivation was retrieved. | MATH: The sole mathematical structure present is the quaternionic algebra (non-commutative, 4-dimensional division algebra) applied to scattering amplitudes. The paper posits a universal algebraic identity among six coherent cross-sections (σ_ij for i,j ∈ {1,2,3}, i≠j) if amplitudes are complex: σ_12 σ_13 σ_23 = σ_12 σ_13 σ_23 (cyclic permutations) — a specific relation that would be violated if amplitudes were quaternionic. No explicit T^3 scaling, no constants, no ratios extracted from the retrieved text. | CONNECTION: Quaternions are intimately linked to SU(2) symmetry and the 3-sphere S^3 — the unit quaternions form a double cover of SO(3), the rotation group. This connects to crystallographic point groups (e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com