Quantization of geometric-like coupling in gravitational field based on characterization and transformation

Connections between the formulations of physics and geometry have been evident throughout history, from classical mechanics to general relativity. Independently, quantum mechanics has been established in flat space. In this study, we investigate the geometric-like coupling of a test particle and its quantized form in a gravitational field. The main text consists of three parts: the characterization of the particle and its operator form is pointwise established. Minimal coupling with the electromagnetic potential is analyzed as a reference via an infinitesimal transform. Subsequently, the geometric coupling from the geodesic strain and its associated phase transform of the initial parallel test particle is formally studied. Within the gravitational field equation, the phase transform is specified via metric tensors in general. The linearized field condition is analyzed explicitly to show the local analogy between the four-potential in gauge-like and geometric-like couplings. From the isomorphism and function maps, the amplitude and phase in the operator form are translated to the quantum form and the Schrödinger-like equation is obtained. The representation of local equivalence and the phase transform condition is formulated. Examples of the phase shift and potential well, as well as the geometric Aharonov-Bohm effect, are studied for potential applications. Finally, the geometric-like and gauge-like couplings are summarized based on the concept of pointwise characterization and associated matrix transformation.