In a local theory of branching within the Many Worlds Interpretation (MWI), the notion of spatially separated systems is interesting to consider. The MWI posits that whenever a quantum measurement occurs, the universe branches into multiple distinct branches, each representing a different outcome of the measurement. These branches are considered to be separate, but they coexist in a larger "multiverse" or "universal wavefunction."
In a local interpretation of branching, it is suggested that the branching process occurs independently and simultaneously for different spatially separated systems. This means that when a measurement is made on one system, it branches into multiple outcomes, while simultaneously, another spatially separated system undergoes its own branching process.
From this perspective, spatially separated systems within the MWI are not causally connected, and the branching in one system does not affect the branching in another system. Each system follows its own set of branching possibilities based on its local interactions and measurements.
This viewpoint is in contrast to non-local interpretations, such as the original MWI proposed by Hugh Everett, which suggests that all branches of the wavefunction are connected and influence each other, even across large distances. In a non-local interpretation, spatially separated systems are considered to be entangled and share a common wavefunction.
However, it's important to note that the Many Worlds Interpretation is a subject of ongoing debate and discussion within the field of quantum foundations. Different physicists have proposed variations and alternative formulations of the MWI, leading to different interpretations of how spatially separated systems behave within the framework.
Ultimately, the precise nature of spatially separated systems within a local theory of branching in the Many Worlds Interpretation is still a topic of investigation and open to interpretation and debate among physicists.