Abstract:

A search for new fundamental (Galilean and Poincaré invariant) dynamical equations for free elementary particles represented by spinor state functions is conducted in Galilean and Minkowski spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operations of the group of the spacetime 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 twocomponent spinor wave functions thus the Pauli equation is not fundamental. It is formally proved that the LévyLeblond and Schrö̈dinger equations are the only Galilean invariant 4component spinor equations for the Schrö̈dinger phase factor. New fundamental dynamical equations for fourcomponent spinors are found using generalized phase factors. For the extended Galilei group a generalized Lé́vyLeblond equation is found to be the only first order Galilean invariant fourcomponent spinor equation. For the Poincaré́ group a generalized Dirac equation is found to be the only first order Poincaré́ invariant fourcomponent spinor equation. In the nonrelativistic limit the generalized Dirac equation is shown to reduce to the generalized Lé́vyLeblond equation. A new momentumenergy relation is derived from the analysis of stationary states of the generalized Dirac equation. The new energymomentum relationship is used to show that the behavior of a particle obeying the generalized Dirac equation is different from that of a particle governed by the standard Dirac equation because of the existence of additional momentum and energy terms. Since this new energymomentum relationship differs from the wellknown energymomentum relationship of Special Theory of Relativity, it cannot describe ordinary matter. Hence, it is suggested that the new energymomentum relationship represents a different form of matter that may be identifed as Dark Matter. 