Majorana-Fermion

Named after Ettore Majorana spinors are Majorana in elementary particle physics for the mathematical description of fermions ( ie, particles with half-integer spin ) when they are equal to their own antiparticles: the so-called Majorana fermions. This property implies that the particles described may not carry electric charge.

Otherwise, it would be to oppositely charged and thus clearly distinguishable particles and antiparticles, such as Electrons and positrons. Such fermions that can carry a charge are called Dirac fermions.

To be distinguished from the Majorana fermions are also the hypothetical Major Onen, which, although also named after Ettore Majorana, but enter Goldstone bosons integer spin.

Occurrence

In the Standard Model: unresolved position of the neutrinos

In the standard model of particle physics (SM ) is none of the elementary particles, a Majorana fermion. Instead, here all fermions are described by Dirac spinors, the neutrinos, which would thus be distinguishable from antineutrinos. However the neutrinos in the standard model in contradiction to experimental results massless. A popular explanation for the observed neutrino masses, the see-saw mechanism, however, the description of neutrinos Majorana spinors, and thus the equality of neutrinos and antineutrinos requires. This in turn would imply a violation of the lepton number conservation, because particles and antiparticles have the same lepton number is assigned.

Whether one can distinguish between neutrinos and antineutrinos, is still open. One possibility for experimental clarification provides the neutrinoless double beta decay, which is only possible if neutrinos Majorana particles. After this decay mode as the Enriched Xenon Observatory is searched for in experiments.

In the MSSM

In supersymmetric extensions of the Standard Model, such as the minimal supersymmetric standard model ( MSSM ), both the gluinos and the neutralino by Majorana spinors are described. Neutralinos are candidates for WIMPs and dark matter.

Solid State Physics

In solid-state physics takes the place of the particle-antiparticle identity, the particle-hole identity.

Theoretical calculations predict that could be found in solids quasiparticles with the properties of Majorana fermions. In April 2012, researchers from the TU Delft have demonstrated to Leo Kouwenhoven this may experimentally, a review was in 2012 still pending. Then to Majorana fermions occur preferentially in one-dimensional systems, as "exactly half - occupied or semi - unoccupied " states - perfect particle-hole symmetry - exactly in the middle between two sequences of occupied and unoccupied states (1- s or 0's in the following example sequence: ... 00001111111111000000000111100000000 ... ).

The center of the (color -coded ) " Kink " excitations between two adjacent different sequences of digits (eg ... 01 .. - 10 .. or ... ) can be used as place each one Majorana - quasiparticle be understood (eg, left or right to the sequence 111111111 adjacent, with occupation number or Non -setting number 1/2). The length of the preceding or the following sequences can be interpreted as the respective distance between the quasiparticles.

This influences the coupling of adjacent superconductors at these one-dimensional insulating systems, because now the energy gap between the ground state and the lowest excited state of the superconductor as a s = 1 interaction exactly in the center can couple to the Majorana quasiparticles, while this center otherwise, for example, in the conventional spin- (± 1 /2) superconductivity, no state corresponds.

Generalizations and equivalences

The specified model, although seemingly very special, allows a number of equivalences and generalizations: eg on the ( renormierbare! ) Thermodynamics of Ising model in one-dimensional, the pseudo- euklische field theoretical thermodynamics of real (! ) model, again in the one-dimensional, and to the specified in the following classical field-theoretical description of non- Euclidean time in the special case d = 1 the

Mathematical Description

Similarly, the massless Weyl fermions, for which decouples the Dirac equation, are Majorana fermions 2- component particles, but with Majorana mass.

The Lagrangian of a Majorana particle is:

Where as usual in relativistic quantum mechanics.

The corresponding Dirac equation is:

If, as in the Weyl fermions, noting that applies under a Lorentz transformation,

So you can set

And it results in the Majorana equation for the 2-component field:

The Pauli spin matrices according contains.

The Majorana equation is Lorentz invariant and implies the Klein-Gordon equation, which determines the energy - momentum relation.