In particle physics a Majorana fermion () or 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 Dirac fermion, which describes fermions that are not their own antiparticles. With the exception of neutrinos, all of the Standard Model elementary 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 be either Dirac or Majorana fermions. In condensed matter physics, quasiparticle excitations can appear like bound Majorana states. 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 that electrically neutral spin- 1 2 {\displaystyle {\tfrac {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 γ j † {\displaystyle \gamma _{j}^{\dagger }} creates a fermion in quantum state j {\displaystyle j} (described by a real wave function), whereas the annihilation operator γ j {\displaystyle \gamma _{j}} annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators γ j † {\displaystyle \gamma _{j}^{\dagger }} and γ j {\displaystyle \gamma _{j}} are distinct, whereas for a Majorana fermion they are identical. The ordinary fermionic annihilation and creation operators f {\displaystyle f} and f † {\displaystyle f^{\dagger }} can be written in terms of two Majorana operators γ 1 {\displaystyle \gamma _{1}} and γ 2 {\displaystyle \gamma _{2}} by
f = 1 2 ( γ 1 + i γ 2 ) , {\displaystyle f={\tfrac {1}{\sqrt {2}}}(\gamma _{1}+i\gamma _{2}),}
f † = 1 2 ( γ 1 − i γ 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 γ {\displaystyle \gamma } is
f = 1 2 ( γ 1 + i γ 2 ) , {\displaystyle f={\tfrac {1}{2}}(\gamma _{1}+i\gamma _{2}),}
f † = 1 2 ( γ 1 − i γ 2 ) , {\displaystyle f^{\dagger }={\tfrac {1}{2}}(\gamma _{1}-i\gamma _{2}),}
which can be rearranged to obtain the Majorana fermion operators as
γ 1 = f † + f , {\displaystyle \gamma _{1}=f^{\dagger }+f,}
γ 2 = i ( f † − f ) . {\displaystyle \gamma _{2}=i(f^{\dagger }-f).}
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