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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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