Article,

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

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(2016)cite arxiv:1607.03011Comment: 20 pp., Talk at the 31st International Colloquium on Group Theoretical Methods in Physics, Rio de Janeiro, Brasil, JUne 19-25, 2016.

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 $Det (p - m) =0$ and $Det (p + m) =0$ for $u-$ and $v-$ 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 $p_0=E_p$ 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.

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