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Zloshchastiev investigates a logarithmic modification of the quantum wave equation within a generally covariant framework. For uncorrelated subsystems sharing the same nonlinear coefficient, the logarithmic term preserves energy additivity and separability. The proposed construction connects time evolution to a modular automorphism group, with a common energy-independent time recovered only at low energies. The paper develops energy-dependent propagation and dispersion relations, distinguishing a linear approximation from an extreme ultrarelativistic regime. A note added in proof acknowledges that Fermi observations ruled out the stated linear velocity dispersion, while arguing that the other regime should apply instead. Possible vacuum shock phenomena and astrophysical applications remain theoretical interpretations whose observational support and applicability the paper itself qualifies.

Logarithmic nonlinearity in theories of quantum gravity: origin of time and observational consequences