Real-vector-space quantum theory with a universal quantum bit

We explore a model of the world based on real-vector-space quantum theory. In our model the familiar complex phase appearing in quantum states is replaced with a single binary object that we call the ubit, which is not localized and which can interact with any object in the world. Ordinary complex-vector-space quantum theory can be recovered from this model if we simply impose a certain restriction on the sets of allowed measurements and transformations (Stueckelberg's rule), but in this paper we try to obtain the standard theory, or a close approximation to it, without invoking such a restriction. We look particularly at the effective theory that applies to a subsystem when the ubit is interacting with a much larger environment. In a certain limit it turns out that the ubit-environment interaction has the effect of enforcing Stueckelberg's rule automatically, and we obtain a one-parameter family of effective theories—modifications of standard quantum theory—that all satisfy this rule. The one parameter is the ratio s/ω, where s quantifies the strength of the ubit's interaction with the rest of the world and ω is the ubit's rotation rate. We find that when this parameter is small but not zero, the effective theory is similar to standard quantum theory but is characterized by spontaneous decoherence of isolated systems.