Does the Peres experiment using photons test for hyper-complex (quaternionic) quantum theories?

TL;DR

The paper demonstrates that the Peres experiment with photons cannot detect quaternionic quantum effects within standard Hilbert space quantum theory, as the index of refraction remains complex, not quaternionic.

Contribution

It shows that under standard assumptions, quaternionic quantum theory predicts a complex index of refraction, indicating the experiment does not test for quaternionic effects.

Findings

The $S$-matrix in quaternionic quantum theory is complex-valued.

The index of refraction is necessarily complex, not quaternionic.

The experiment by Procopio et al. does not test for quaternionic quantum effects.

Abstract

Assuming the standard axioms for quaternionic quantum theory and a spatially localized scattering interaction, the $S$-matrix in quaternionic quantum theory is complex valued, not quaternionic. Using the standard connections between the $S$-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. based on the Peres proposal does not test for hyper-complex or quaternionic quantum effects arising within the standard Hilbert space framework. Such a test requires looking at near zone fields, not radiation zone fields.