Generalized equations and their solutions in the (S,0)×(0,S) representations of the Lorentz group

Abstract

In this talk I present three explicit examples of generalizations in relativistic quantum mechanics. First of all, I discuss the generalized spin-1/2 equations for neutrinos. They have been obtained by means of the Gersten-Sakurai method for derivations of arbitrary-spin relativistic equations. Possible physical consequences are discussed. Next, it is easy to check that both Dirac algebraic equation Det (\hat p - m) =0 and Det (\hat p + m) =0 for u- and v- 4-spinors have solutions with p_0= \pm E_p =\pm \sqrt{{\bf p}^2 +m^2}. The same is true for higher-spin equations. Meanwhile, every book considers the equality p_0=E_p for both $u-$ and $v-$ spinors of the (1/2,0)+(0,1/2)) representation only, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of the negative-energy solutions. The recent Ziino works (and, independently, the articles of several others) show that the Fock space can be doubled. We re-consider this possibility on the quantum field level for both S=1/2 and higher spin particles. The third example is: we postulate the non-commutativity of 4-momenta, and we derive the mass splitting in the Dirac equation. The applications are discussed.