Distinguishability, Entropy, and the Emergence of Spacetime, Gauge Structure, and Renormalization

Physical observations access only coarse-grained information about underlying quantum systems. We investigate the consequences of the principle that only distinguishable information under finite, coarse-grained physical processes is meaningful. This requirement induces an information-geometric structure on the space of effective states, described by the Fisher metric and relative entropy. We show that conserved distinguishable quantities naturally give rise to gauge structure, while extremality of relative entropy under local perturbations yields effective dynamics structurally equivalent to Einstein’s equations. Scale-dependent coarse-graining induces flow of the information metric, leading to renormalization group dynamics and running couplings. In holographic regimes, this flow corresponds to an emergent radial direction, with spacetime geometry arising from the structure of distinguishability. Particle presence is reinterpreted as distinguishability from vacuum, formalized via POVMs and relative entropy. Gaussian states and field-theoretic mode decompositions provide explicit realizations. Black hole entropy appears as maximal distinguishability compatible with geometry, and a monotonic c-function is identified with relative entropy under coarse-graining.