Extending geodesic motion to localized quantum states

A quantum state cannot be modeled as a point mass following a single geodesic when it has non-zero position fluctuations. Instead, a state initially in a spatial superposition may require an averaging over multiple classical geodesics. Here, we propose a model that addresses this feature of freely falling quantum objects. Our methods, based on a geometrical formulation of quantum mechanics, bridge the geometrical nature of gravity with quantum effects by treating both classical spacetime degrees of freedom and quantum properties on equal footing within a quantum phase space. In the relativistic case, the Riemannian structure of the theory becomes parametrically dependent on new degrees of freedom describing the quantum state. This quantum-deformed geometry produces semiclassical geodesic equations. Classical geodesic motion emerges when the state is sufficiently localized. However, we find that a spatially extended state experiences tidal forces more complex than those affecting a classical extended object, due to quantum features such as correlations, impurity, and entanglement. Detailed derivations offer potentially testable corrections to gravitational time dilation in Schwarzschild spacetime.