Partial Phase Coherence and Emergent Observables in Layered Systems

Abstract We propose a minimal phase-based framework in which observable quantities emerge from partial coherent summation over a large number of underlying phase degrees of freedom. The construction is motivated by many-body systems with distributed phase variables, where only coarse-grained or partial information is accessible. Observable quantities are defined as sums over subsets of microscopic phase variables, naturally leading to interference-based measures of the form |O|2. This provides a structural perspective on the emergence of probabilistic behavior from phase coherence. In a coarse-grained limit, the system admits a continuum phase field description. A scalar field Lagrangian provides a natural minimal effective description, analogous to the continuum limit of XY-type models. Furthermore, promoting global phase symmetry to a local one suggests a natural route toward an effective U(1)-like gauge structure. The framework remains agnostic about microscopic details and is intended as a minimal, internally consistent structure connecting phase coherence, observables, and effective field descriptions in many-body systems.