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

TL;DR

This paper explores generalized relativistic equations for spin-1/2 particles, including neutrinos, discusses negative-energy solutions, and investigates mass splitting through non-commutative momenta, offering new insights into relativistic quantum mechanics.

Contribution

It introduces generalized spin-1/2 equations, examines negative-energy solutions in a broader context, and proposes non-commutative momenta leading to mass splitting, expanding relativistic quantum theory.

Findings

Generalized spin-1/2 equations for neutrinos derived.

Negative-energy solutions are present in higher-spin equations.

Mass splitting arises from non-commutative 4-momenta.

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 $p_0= \pm E_p =\pm \sqrt{{\bf p}^2 +m^2}$. 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.