Holonomy and Landau-type Quantization in Randers-Finsler Spacetimes

We develop a geometric formulation of nonrelativistic quantum mechanics in Randers-Finsler backgrounds based on the effective quantum connection induced by the Randers one-form. Starting from the classical Randers action, we derive the nonrelativistic Hamiltonian through the Legendre transformation and obtain the corresponding Schrödinger equation by canonical quantization. The resulting formalism naturally yields the kinetic momentum, probability current, effective force, and the curvature associated with the induced quantum connection. We show that the nontrivial quantum effects associated with the spatial effective connection are characterized by its local curvature and global holonomy, leading to three complementary geometric regimes. Globally trivial flat connections leave the local dynamics unchanged apart from uniform energy shifts, flat connections with nontrivial holonomy generate geometric phases, twisted boundary conditions, and persistent probability currents, whereas nonvanishing effective curvature gives rise to noncommuting kinetic momenta, Landau-type quantization, a characteristic geometric length, and curvature-controlled degeneracy. The formalism is further extended to nonuniform backgrounds through the first Born approximation, where the vector contribution to elastic scattering is shown to probe the transverse component of the effective connection and, consequently, the Fourier components of its effective curvature. These results provide a unified geometric framework for nonrelativistic quantum mechanics in Randers-Finsler backgrounds and identify holonomy and effective curvature as the fundamental global and local mechanisms through which the induced quantum connection modifies quantum behavior.