The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer

TL;DR

This paper shows that relativity of simultaneity constrains two-level quantum systems to standard complex quantum theory, but allows quaternionic quantum theory if relational assumptions are relaxed, impacting the design of interference experiments.

Contribution

It demonstrates how relativity of simultaneity restricts the state space of two-level systems to complex quantum theory, with an exception for quaternionic quantum theory under relaxed assumptions.

Findings

Relativity of simultaneity limits interference phenomena to complex quantum theory.

Quaternionic quantum theory remains compatible if relational assumptions are relaxed.

Experimental designs must consider relativity effects to detect beyond-quantum phenomena.

Abstract

The patterns of fringes produced by an interferometer have long been important testbeds for our best contemporary theories of physics. Historically, interference has been used to contrast quantum mechanics to classical physics, but recently experiments have been performed that test quantum theory against even more exotic alternatives. A physically motivated family of theories are those where the state space of a two-level system is given by a sphere of arbitrary dimension. This includes classical bits, and real, complex and quaternionic quantum theory. In this paper, we consider relativity of simultaneity (that observers may disagree about the order of events at different locations) as applied to a two-armed interferometer, and show that this forbids most interference phenomena more complicated than those of complex quantum theory. If interference must depend on some relational property of the setting (such as path difference), then relativity of simultaneity will limit state spaces to standard complex quantum theory, or a subspace thereof. If this relational assumption is relaxed, we find one additional theory compatible with relativity of simultaneity: quaternionic quantum theory. Our results have consequences for current laboratory interference experiments: they have to be designed carefully to avoid rendering beyond-quantum effects invisible by relativity of simultaneity.