Majorana fermion

A Majorana fermion (/maɪəˈrɒnə ˈfɛərmiːɒn/), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesized 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 the neutrino, all of the Standard Model fermions are known to behave as Dirac fermions at low energy (after electroweak symmetry breaking), and none are Majorana fermions. The nature of the neutrinos is not settled—they may be either Dirac or Majorana fermions. In condensed matter physics, bound Majorana fermions can appear as quasiparticle excitations—the collective movement of several individual particles, not a single one, and they are governed by non-abelian statistics. The concept goes back to Majorana's suggestion in 1937 that neutral spin-1⁄2 particles can be described by a real 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). 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 In supersymmetry models, neutralinos—superpartners of gauge bosons and Higgs bosons—are Majorana. Another common convention for the normalization of the Majorana fermion operator is This convention has the advantage that the Majorana operator squares to the identity.