Fundamental Dynamical Equations For Two And Four-component Spinors In Galilean And Minkowski Space-time
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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 space-time. 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 two-component spinor wave functions thus the Pauli equation is not fundamental. It is formally proved that the Lévy-Leblond and Schrö̈dinger equations are the only Galilean invariant 4-component spinor equations for the Schrö̈dinger phase factor. New fundamental dynamical equations for four-component spinors are found using generalized phase factors. For the extended Galilei group a generalized Lé́vy-Leblond equation is found to be the only first order Galilean invariant four-component spinor equation. For the Poincaré́ group a generalized Dirac equation is found to be the only first order Poincaré́ invariant four-component spinor equation. In the non-relativistic limit the generalized Dirac equation is shown to reduce to the generalized Lé́vy-Leblond equation. A new momentum-energy relation is derived from the analysis of stationary states of the generalized Dirac equation. The new energy-momentum 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 well-known energy-momentum relationship of Special Theory of Relativity, it cannot describe ordinary matter. Hence, it is suggested that the new energy-momentum relationship represents a different form of matter that may be identifed as Dark Matter.