Generalized Equations and Their Solutions in the (S, 0) ⊕ (0, S) Representations of the Lorentz Group

We present three explicit examples of generalizations in relativistic quantum mechanics. First of all, we 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 $p_0=\pm E_p=\pm \sqrt{p^2+m^2}$. The same is true for higher-spin equations. Meanwhile, every book considers the equality p₀ = 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. Some applications are discussed. (paper)