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.
We investigate the emergent time scenario in quantum cosmology based on the Page–Wotters approach. Using a quantum cosmological model with a qubit clock, it is demonstrated how the entanglement between the qubit clock and the geometry derives emergence of a time parameter, which defines evolution of the timeless quantum state of the universe. We show that the universe wave function conditioned by a qubit clock obeys the standard Schrödinger equation and the Fisher information for the clock state, which quantifies entanglement between the universe and the clock, and contributes as a negative energy density.
A companion letter established that four simple axioms on statistical comparison uniquely select the Bhattacharyya coefficient as a clock functional, and that its binary reversal sector yields the Lorentz time-dilation factor without presupposing a spacetime metric. The present paper asks whether that uniqueness structure can be extended from kinematics to dynamics. Five linked results are established. First, the continuum parity-reversal extension of the clock functional on the local celestial sphere S2 yields, in the weak-anisotropy dipole sector, a locally Minkowskian quadratic norm. Second, on the resulting spatial slices, a local action written in current form and constrained by probability conservation forces the transport current to be potential, j=ρ∇S/m. Third, an intrinsic second-order density geometry, taken as the local continuum descendant of the unique monotone Fisher-Rao metric identified by Čencov, fixes the density-gradient term uniquely up to overall scale as the Fisher information. Fourth, the resulting Euler-Lagrange system is exactly the Madelung form of quantum mechanics. Finally, by the standard substitution ψ=ρeiS/ℏ, the Schrödinger equation follows. The central result is not merely that Fisher information reproduces the Schrödinger equation, but that the Lorentz clock factor and the Schrödinger equation arise from sibling uniqueness structures on the same probability geometry: Bhattacharyya affinity and Fisher-Rao metric. The only undetermined dimensional constants are c and ℏ.