Standard Model of particle physics
Elementary particles of the Standard Model
Background Particle physics Standard Model Quantum field theory Gauge theory Spontaneous symmetry breaking Higgs mechanism
Constituents Electroweak interaction Quantum chromodynamics CKM matrix Standard Model mathematics
Limitations Strong CP problem Hierarchy problem Neutrino oscillations Physics beyond the Standard Model
Scientists Rutherford Thomson Chadwick Bose Sudarshan Davis Jr Anderson Fermi Dirac Feynman Rubbia Gell-Mann Kendall Taylor Friedman Powell Anderson Glashow Iliopoulos Lederman Maiani Meer Cowan Nambu Chamberlain Cabibbo Schwartz Perl Majorana Weinberg Lee Ward Salam Kobayashi Maskawa Mills Yang Yukawa 't Hooft Veltman Gross Pais Pauli Politzer Reines Schwinger Wilczek Cronin Fitch Vleck Higgs Englert Brout Hagen Guralnik Kibble de Mayolo Lattes Zweig
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A Majorana fermion (/maɪəˈrɑːnə/^[1]), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.

With the exception of neutrinos, all of the Standard Model fermions are known to behave as Dirac fermions at low energy (lower than the electroweak symmetry breaking temperature), and none are Majorana fermions. The nature of neutrinos is not settled – they may turn out to be either Dirac or Majorana fermions.

In condensed matter physics, quasiparticle excitations can appear like bound Majorana fermions. However, instead of a single fundamental particle, they are the collective movement of several individual particles (themselves composite) which are governed by non-Abelian statistics.

Theory

The concept goes back to Majorana's suggestion in 1937^[2] that electrically neutral spin-1/2 particles can be described by a real-valued wave equation (the Majorana equation), and would therefore be identical to their antiparticle, because the wave functions of particle and antiparticle are related by complex conjugation, which leaves the Majorana wave equation unchanged.

The difference between Majorana fermions and Dirac fermions can be expressed mathematically in terms of the creation and annihilation operators of second quantization: The creation operator ${\displaystyle \gamma _{j}^{\dagger }}$ creates a fermion in quantum state ${\displaystyle j}$ (described by a real wave function), whereas the annihilation operator ${\displaystyle \gamma _{j}}$ annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators ${\displaystyle \gamma _{j}^{\dagger }}$ and ${\displaystyle \gamma _{j}}$ are distinct, whereas for a Majorana fermion they are identical. The ordinary fermionic annihilation and creation operators ${\displaystyle f}$ and ${\displaystyle f^{\dagger }}$ can be written in terms of two Majorana operators ${\displaystyle \gamma _{1}}$ and ${\displaystyle \gamma _{2}}$ by

${\displaystyle f={\tfrac {1}{\sqrt {2}}}(\gamma _{1}+i\gamma _{2}),}$

${\displaystyle f^{\dagger }={\tfrac {1}{\sqrt {2}}}(\gamma _{1}-i\gamma _{2})~.}$

In supersymmetry models, neutralinos – superpartners of gauge bosons and Higgs bosons – are Majorana fermions.

Identities

Another common convention for the normalization of the Majorana fermion operator ${\displaystyle \gamma }$ is

${\displaystyle f={\tfrac {1}{2}}(\gamma _{1}+i\gamma _{2}),}$

${\displaystyle f^{\dagger }={\tfrac {1}{2}}(\gamma _{1}-i\gamma _{2}),}$

which can be rearranged to obtain the Majorana fermion operators as

${\displaystyle \gamma _{1}=f^{\dagger }+f,}$

${\displaystyle \gamma _{2}=i(f^{\dagger }-f).}$

It is easy to see that ${\displaystyle \gamma _{i}=\gamma _{i}^{\dagger }}$ is indeed fulfilled. This convention has the advantage that the Majorana operator squares to the identity, i.e. ${\displaystyle \gamma _{i}^{2}=(\gamma _{i}^{\dagger })^{2}=1}$. Using this convention, a collection of ${\displaystyle 2n}$ Majorana fermions (${\displaystyle n}$ ordinary fermions), ${\displaystyle \gamma _{i}}$ (${\displaystyle i=1,2,..,2n}$) obey the following anticommutation identities

${\displaystyle \{\gamma _{i},\gamma _{j}\}=2\delta _{ij}}$

and

${\displaystyle \sum _{ijkl}=4\sum _{ij}\gamma _{i}_{ij}\gamma _{j}}$

where ${\displaystyle A}$ and ${\displaystyle B}$ are antisymmetric matrices. These are identical to the commutation relations for the real Clifford algebra in ${\displaystyle n}$ dimensions (${\displaystyle \mathrm {Cl} (\mathbb {R} ^{n})}$).

Elementary particles

Because particles and antiparticles have opposite conserved charges, Majorana fermions have zero charge, hence among the fundamental particles, the only fermions that could be Majorana are sterile neutrinos, if they exist. All the other elementary fermions of the Standard Model have gauge charges, so they cannot have fundamental Majorana masses: Even the Standard Model's left-handed neutrinos and right-handed antineutrinos have non-zero weak isospin, ${\displaystyle T_{3}=\pm {\tfrac {1}{2}},}$ a charge-like quantum number. However, if they exist, the so-called "sterile neutrinos" (left-handed antineutrinos and right-handed neutrinos) would be truly neutral particles (assuming no other, unknown gauge charges exist).

The sterile neutrinos introduced to explain neutrino oscillation and anomalously small S.M. neutrino masses could have Majorana masses. If they do, then at low energy (after electroweak symmetry breaking), by the seesaw mechanism, the neutrino fields would naturally behave as six Majorana fields, with three of them expected to have very high masses (comparable to the GUT scale) and the other three expected to have very low masses (below 1 eV). If right-handed neutrinos exist but do not have a Majorana mass, the neutrinos would instead behave as three Dirac fermions and their antiparticles with masses coming directly from the Higgs interaction, like the other Standard Model fermions.

The seesaw mechanism is appealing because it would naturally explain why the observed neutrino masses are so small. However, if the neutrinos are Majorana then they violate the conservation of lepton number and even of B − L.

Neutrinoless double beta decay has not (yet) been observed,^[3] but if it does exist, it can be viewed as two ordinary beta decay events whose resultant antineutrinos immediately annihilate each other, and is only possible if neutrinos are their own antiparticles.^[4]

The high-energy analog of the neutrinoless double beta decay process is the production of same-sign charged lepton pairs in hadron colliders;^[5] it is being searched for by both the ATLAS and CMS experiments at the Large Hadron Collider. In theories based on left–right symmetry, there is a deep connection between these processes.^[6] In the currently most-favored explanation of the smallness of neutrino mass, the seesaw mechanism, the neutrino is “naturally” a Majorana fermion.

Majorana fermions cannot possess intrinsic electric or magnetic moments, only toroidal moments.^[7][8][9] Such minimal interaction with electromagnetic fields makes them potential candidates for cold dark matter.^[10][11]

Majorana bound states

In superconducting materials, a quasiparticle can emerge as a Majorana fermion (non-fundamental), more commonly referred to as a Bogoliubov quasiparticle in condensed matter physics. Its existence becomes possible because a quasiparticle in a superconductor is its own antiparticle.

Mathematically, the superconductor imposes electron hole "symmetry" on the quasiparticle excitations, relating the creation operator ${\displaystyle \gamma (E)}$ at energy ${\displaystyle E}$ to the annihilation operator ${\displaystyle {\mathrm{\gamma <!--\; \gamma \; -->}}^{†<!--\; †\; -->}(-<!--\; -\; -->E)}$Zdroj:https://en.wikipedia.org?pojem=Majorana_fermion
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