A Direct Road to Majorana Fields

Abstract

A concise discussion of spin-1/2 field equations with a special focus on Majorana spinors is presented. The Majorana formalism which describes massive neutral fermions by the help of two-component or four-component spinors is of fundamental importance for the understanding of mathematical aspects of supersymmetric and other extensions of the Standard Model of particle physics, which may play an increasingly important role at the beginning of the LHC era. The interplay between the two-component and the four-component formalism is highlighted in an introductory way. Majorana particles are predicted both by grand unified theories, in which these particles are neutrinos, and by supersymmetric theories, in which they are photinos, gluinos and other states

A New Route to the Majorana Equation

“…The present paper corroborates and expands the earlier work [8,9] on this subject and the largely tutorial papers by Aste [10] and Pal [11], who also discussed Weyl neutrinos [12] and defined the operation of Lorentz-invariant complex conjugation, which is important for the Majorana spinor field. At the outset, we here make use of only the complex conjugation operator and the Pauli spin matrices, corresponding to the irreducible representation of the Lorentz group.…”

Weyl, Majorana and Dirac Fields from a Unified Perspective

“…In analogy to the construction given by Equation (25), one defines the matrix-valued differential operator [9]:…”

Majorana spinor from the point of view of geometric algebra

“…Thus, the respective Majorana fields represent the particles that are their own antiparticles. Since the particles and antiparticles coincide, from this it follows that the Majorana particle is a neutral spin- 2 1 particle (fermion) carrying no charge. The concept of Majorana spinor, especially in the condensed matter physics, is frequently generalized assuming that the spinor is real but not necessarily related to the Dirac equation.…”

“…where m is the mass, µ γî are real 4×4 Majorana matrices [7] and ψ M is a real spinor, as a rule, is analyzed in terms of the complex Hilbert space [2,3]. In the present paper the DM equation is formulated and solved in terms of real geometric algebra (GA) [8][9][10][11], by mathematicians also called the Clifford algebra, where the quantum mechanics is formulated in graded vector spaces that accommodate non-commuting multivectors.…”

“…The concept of Majorana spinor, especially in the condensed matter physics, is frequently generalized assuming that the spinor is real but not necessarily related to the Dirac equation. In terms of the Hilbert space the properties of Majorana spinors were discussed most thoroughly in articles [2,3], where the earlier literature is cited.…”