Solutions in the (1/2,0)+(0,1/2) Representation of the Lorentz Group

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 $Det(\hat{p}-m)=0$ and $Det(\hat{p}+m)=0$ for $u-$ and $v-$ 4-spinors have solutions with $p_0=\pm E_p=\pm \sqrt{{\mathbf{p}}^2+m^2}$. 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 $(1/2,0)\oplus (0,1/2))$ representation only, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of the negative-energy solutions. The recent work by Ziino (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 $S=1/2$ 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.

arXiv comment11 pp., Talk at the 11th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields (IARD 2018), June 4-7, Merida, Yuc., México. Accepted