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

Abstract

Abstract I present 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( p ^ – m) = 0 and Det( p ^ + m) = 0 for u – and υ – 4-spinors have solutions with p 0 = ± E p = ± p 2 + m 2 . The same is true for higher-spin equations. Meanwhile, every book considers the equality p0 = Ep for both u – and υ – spinors of the (1/2, 0) ⊕ (0, 1/2) representation, 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.