Question

solution of dirac equations giving dirac weyl and majorana fermions

Answer 1

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▪There are three different kinds of spin-1/2 particles: Dirac, Majorana, and Weyl.
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The first, Dirac fermions, have non-zero mass, and are represented as four component complex spinors. This is the mathematical description of fermions which emerges naturally from the massive Dirac equation, which Paul Dirac invented to describe the electron, a massive spin-1/2 particle, relativistically.
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The electron has charge as well as mass, and it also has two possible helicity states, which mix with each other under boosts. It is associated with an antiparticle which is distinguishable from itself by its opposite charge: the positron.
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When beta decays were first found, Wolfgang Pauli suggested that a third undetectable particle was being produced, since otherwise beta decays looked as if they violated energy conservation. Many people were even proposing that energy was only conserved statistically in beta decay, so Pauli's suggestion saved the day - the new particle was called the neutrino, because it had to be electrically neutral. But at that point the neutrino appeared to be massless. It could be argued that the neutrino spin was 1/2, just like that of the electron.
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The massless case is always a special case in the representation theory of the Lorentz group - it turns out that a massless spin-1/2 particle can be represented by a two component spinor, rather than a four component spinor.
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Hermann Weyl showed that the massless Dirac equation could be reduced to a two-component equation. The solutions of that equation are called Weyl spinors, or Weyl fermions.
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Weyl fermions are two-component spinors. They are irreducible representations of the Lorentz group, and since they are massless, their helicity is a good quantum number - it is boost invariant, since Weyl spinors move at the speed of light. So there is a well-defined notion of a left-handed or a right-handed Weyl spinor.
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Massive Dirac spinors can be written as combinations of left and right handed massless Weyl fermions.
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But there is another way of representing the massive Dirac equation found by Ettore Majorana, which makes its solutions into real four component spinors. Majorana fermions correspond to another possible type of four component spinor, which can have mass, called the Majorana mass. But the reality condition means that the particle is indistinguishable from the antiparticle for a Majorana fermion. The reality condition also means that Majorana spinors have a two component representation.
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Both Weyl and Majorana spinors can be regarded as constrained cases of Dirac spinors.
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Weyl fermions at first appeared to be the ones that were involved in the theory of the weak interactions. The particle is distinguishable from its anti-particle because the spinors are complex.
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Majorana fermions on the other hand are four component spinors for which the particle and antiparticle are indistinguishable. A Majorana fermion is its own antiparticle - much like the neutral spin-1 photon. Massless Majorana fermions are the natural objects to act as partners to neutral spin-1 or spin-0 particles, for people who like to build supersymmetric theories.
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At the current time, things are somewhat up in the air, since neutrinos have been found to be massive rather than massless - even if their masses are very small.
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It is not known whether Majorana fermions exist in nature, it is not known if neutrinos are actually Majorana fermions. They may be - if so then certain decays are allowed which wouldn't otherwise be, such as the very exotic neutrinoless double beta decay. Searches are under way for such decays.
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It was long thought that neutrinos were Weyl fermions - as it turns out, at least two of the three known types of neutrinos are not, since neutrino oscillations have been observed. Neutrinos may also be Dirac fermions.


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