Fundamental dynamical equations for spinor wavefunctions: I. Lévy-Leblond and Schr?dinger equations

摘要

A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wavefunctions. A method to derive the Lévy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Lévy-Leblond and Schr?dinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schr?dinger phase factor. Physical implications of the obtained results and their relationships to the PauliSchr?dinger equation are discussed.