Comment on Adler's "Does the Peres experiment using photons test for hyper-complex (quaternionic) quantum theories?"

TL;DR

This paper responds to Adler's critique of our experimental bounds on quaternionic quantum effects, providing a relativistic Klein-Gordon scattering example that demonstrates persistent quaternionic effects, thus clarifying the scope of our original findings.

Contribution

We present a relativistic Klein-Gordon scattering example showing quaternionic effects persist, addressing Adler's assumptions and clarifying the implications for quaternionic quantum mechanics.

Findings

Quaternionic effects can persist in relativistic scattering.

Adler's assumptions exclude certain relativistic effects.

Our experiment places bounds on post-quantum theories.

Abstract

In his recent article [arXiv:1604.04950], Adler questions the usefulness of the bound found in our experimental search for genuine effects of hyper-complex quantum mechanics [arXiv:1602.01624]. Our experiment was performed using a black-box (instrumentalist) approach to generalized probabilistic theories; therefore, it does not assume a priori any particular underlying mechanism. From that point of view our experimental results do indeed place meaningful bounds on possible effects of "post-quantum theories", including quaternionic quantum mechanics. In his article, Adler compares our experiment to non-relativistic and M\"oller formal scattering theory within quaternionic quantum mechanics. With a particular set of assumptions, he finds that quaternionic effects would likely not manifest themselves in general. Although these assumptions are justified in the non-relativistic case, a proper calculation for relativistic particles is still missing. Here, we provide a concrete relativistic example of Klein-Gordon scattering wherein the quaternionic effects persist. We note that when the Klein-Gordon equation is formulated using a Hamiltonian formalism it displays a so-called "indefinite metric", a characteristic feature of relativistic quantum wave equations. In Adler's example this is directly forbidden by his assumptions, and therefore our present example is not in contradiction to his work. In complex quantum mechanics this problem of an indefinite metric is solved in second quantization. Unfortunately, there is no known algorithm for canonical field quantization in quaternionic quantum mechanics.