A geometric extension of quantum mechanics based on an energy-time ambiguity in the Schrödinger picture

Abstract In this letter, we show that, within the Schrödinger picture, quantum mechanics exhibits a fundamental ambiguity in the roles of energy and time, opening the possibility that neither should be regarded as ontologically primitive. Motivated by this ambiguity, we propose a geometric reformulation in which the total curvature of a surface bulk generates quantum evolution. We show that all physically detectable quantum transformations are encoded in the boundary geometry, so that the effective structure acquires a holographic character. We further consider the model from a relational perspective, examine special cases, and discuss qualitative and quantitative connections with the role of geometry in general relativity.