Fisher-Informational Time: A Causal-Geometric Framework for Emergent Clock Time Physical Distinguishability

TL;DR

This paper proposes a Fisher-informational framework where physical time emerges from the distinguishability of states, rather than being fundamental, connecting classical and quantum systems through information geometry.

Contribution

It introduces a causal-informational parameter based on Fisher information to reconstruct clock time from physical state distinguishability, bridging classical and quantum perspectives.

Findings

Defined a Fisher-geometric distance along causal trajectories.

Connected Fisher information to clock quality and time reconstruction.

Illustrated concepts with a qubit clock and decay process examples.

Abstract

We develop a Fisher-informational reformulation of physical time in which clock time is not regarded as a fundamental ontological substance, but as an emergent calibration of causally ordered distinguishability among physical states. The operational starting point is that clocks do not measure time itself; rather, they instantiate reproducible physical processes whose distinguishable states are correlated with other events. We introduce a causal-informational parameter, denoted by Lambda_F, defined as an accumulated Fisher-geometric distance along a causally admissible trajectory in state space. In classical statistical systems, this parameter is generated by the Fisher information metric; in quantum systems, the corresponding construction is associated with quantum Fisher information, the Bures metric, and the Fubini-Study geometry of projective Hilbert space. The manuscript distinguishes model-dependent Fisher information from quantum Fisher information, clarifies the reparameterization of Schrodinger dynamics, and gives explicit examples involving a qubit clock, an exponential decay process, and a Fisher characterization of clock quality. The proposal is positioned relative to relational time, the Page-Wootters mechanism, thermal time, quantum speed-limit relations, information geometry, and the problem of time in quantum gravity. We do not claim that relational or emergent time is new. The specific contribution is the use of Fisher distinguishability as an operational precursor from which ordinary clock time can be reconstructed. In this sense, the central statement of the paper is: time is not measured by clocks; clock time is reconstructed from the Fisher distinguishability accumulated along causally ordered physical changes.