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

In this paper 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$$EQUATION$$ (p–m) = 0 and Det$$EQUATION$$(p+m) = 0 for u– and v– 4-spinors have solutions with p0 = ±Ep = ± √p2+m2. The same is true for higher-spin equations. Meanwhile, every book considers the equality p0 = Ep for both u– and v– spinors of the (1/2,0)+(0,1/2)) representation only, thus applying the Dirac-Feynman- Stueckelberg procedure for eliminating 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 noncommutativity of 4-momenta, and we derive the mass splitting in the Dirac equation. The applications are discussed.