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Concept# Weyl equation

Summary

In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.
None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion). In condensed matter physics, some materials can display quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals.
Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term.
The Dirac equation was published in 1928 by Paul Dirac, and was first used to model spin-1⁄2 particles in the framework of relativistic quantum mechanics. Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation. Wolfgang Pauli wrote in 1933 against Weyl’s equation because it violated parity. However, three years before, Pauli had predicted the existence of a new elementary fermion, the neutrino, to explain the beta decay, which eventually was described using the Weyl equation.
In 1937, Conyers Herring proposed that Weyl fermions may exist as quasiparticles in condensed matter.
Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless). The same year the Wu experiment showed that parity could be violated by the weak interaction, addressing Pauli's criticism. This was followed by the measurement of the neutrino's helicity in 1958. As experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced. Thus, the Standard Model was built under the assumption that neutrinos were Weyl fermions.

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In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields. Elements of a spin representation are called spinors.

Dirac fermion

In physics, a Dirac fermion is a spin-1⁄2 particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. In particle physics, all fermions in the standard model have distinct antiparticles (perhaps excepting neutrinos) and hence are Dirac fermions. They are named after Paul Dirac, and can be modeled with the Dirac equation. A Dirac fermion is equivalent to two Weyl fermions. The counterpart to a Dirac fermion is a Majorana fermion, a particle that must be its own antiparticle.

Higher-dimensional gamma matrices

In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors.

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