Macroscopically distinguishable superposition in infinitely many degrees of freedom

We investigate the concept of macroscopically distinguishable superpositions within an infinite array of boson sites. Our approach is rigorous within the frame of Hilbert space theory. In this context, it is natural to differentiate between states -- and corresponding dynamics -- that involve only finitely many degrees of freedom, referred to as local, and those that are inherently nonlocal. Previous studies have shown that such systems can support nonlocal coherent states (NCS). In this work, we demonstrate that NCS can dynamically evolve into nonlocal cat states under the influence of a nonlocal Hamiltonian -- specifically, the square of the total number operator. Crucially, the resulting dynamics cannot be decomposed into local factors. Furthermore, we explore broader mathematical implications of these phenomena within the framework of generalized bosons. Our findings highlight that the concepts of coherent states and nonlocal cat states are not inherently bound together; rather, their fusion is a distinctive feature of standard bosons. Finally, we propose that if the generalized boson framework can be physically realized in engineered quantum systems, the phenomena described here may hold significant relevance for both physics and materials science.