Operational Time, Inertia, and Phase Coherence

This work proposes an operational answer to the question of what time is: time is the process by which a physical system produces mutually distinguishable states, a quantity made precise in quantum mechanics through Quantum Fisher Information and the Mandelstam–Tamm bound. Building on this definition, the paper develops a framework in which inertia and gravitation are read as different derivatives of a single underlying quantity — the quantum phase. The framework points to a natural acceleration scale a⋆ = mc³/ℏ, constructible solely from m, c, and ℏ, and shows that for the electron this coincides algebraically with the acceleration in the Schwinger field. It further explores a phase-coherence factor C(a), its kinematic relation to a finite "phase-proper-time" window (via the Gudermann function), a possible Unruh-related spectral signature, and a scalar/vector decomposition of the four-force under an effective-momentum ansatz. The study positions itself explicitly not as a new theory of physics, but as a consistent rereading of standard physics and a research program that formulates the relevant open problems sharply. Throughout the text, results that are algebraically exact are transparently separated from those that hold open-problem / speculative status. Among the algebraically exact results are the four-force rereading, the recovery of Newtonian gravity from flowing space, the Schwinger-scale coincidence, and the kinematic identity for C(a); among the open problems are the physical mechanism behind the Schwinger coincidence, the status of C(a) as a genuine QFI modulation, and the derivation of the internal tick duration at the level of the QFI or the Wightman function. The framework is compatible with the relational-time and quantum-reference-frame programs (Page–Wootters, Rovelli) and shares conceptual ground with the flowing-space / river-model tradition (Painlevé–Gullstrand, Hamilton–Lisle), while preserving Lorentz invariance and treating the natural acceleration scale as a characteristic scale rather than a maximum-acceleration law.