Representation Formalism and Quantum Mechanics in Curved Spacetime

We extend the representation frame formalism, previously introduced to account for key cosmological observations in the Einstein static universe, to non-relativistic quantum mechanics. In this framework, each inertial observer is associated with a flat representation referential $R_{obs}$, defined as the tangent space to the spatial manifold at the observer’s position, in which all measurements are represented. The Euclidean structure of $R_{obs}$ allows quantum systems to be described using the standard Schrödinger formalism, avoiding the technical ambiguities that arise when quantising directly on curved manifolds. We derive the relation between the Hamiltonian governing quantum dynamics in $R_{obs}$ and its counterpart defined on the physical manifold U, and show that curvature effects enter as observer-dependent modifications of effective potentials. Although the resulting quantum description depends on the observer’s representation frame, we show that this does not lead to contradictions between observers: consistency of measurement outcomes follows from the standard structure of quantum correlations established by physical interactions. We illustrate the formalism with explicit applications, including the hydrogen atom in an Einstein static universe and quantum systems in the vicinity of a black hole, highlighting how spacetime curvature manifests itself in the observer’s quantum description.