Emergent Quantum Dynamics of Vortex-Line under Linear Local Induction Approximation

Using the linear local induction approximation, we investigate the self-induced motion of a vortex-line that corresponds to the motion of a particle in quantum mechanics. Provided Kelvin waves, the effective Schrödinger equation, physical quantity operators, and the corresponding path-integral formula can be obtained. In particular, the effective Planck constant defined by parameters of vortex-line motion shows the mathematical relation between the two fields. The vortexline–particle mapping may help in understanding particle motion in quantum mechanics.