[2][3] Wolfgang Pauli wrote in 1933 against Weyl's equation because it violated parity.
[4] 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.
[5] Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless).
[4] The same year the Wu experiment showed that parity could be violated by the weak interaction, addressing Pauli's criticism.
[4] As experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced.
[2] In 2015, the first Weyl semimetal was demonstrated experimentally in crystalline tantalum arsenide (TaAs) by the collaboration of M.Z.
Hasan's (Princeton University) and H. Ding's (Chinese Academy of Sciences) teams.
[5] Independently, the same year, M. Soljačić team (Massachusetts Institute of Technology) also observed Weyl-like excitations in photonic crystals.
Both have the form where is a momentum-dependent two-component spinor which satisfies or By direct manipulation, one obtains that and concludes that the equations correspond to a particle that is massless.
is the Hermitian transpose, provided that the right-handed field transforms as The matrix
The Lorentz transform, in coordinates, is or, equivalently, This leads to In order to make use of the Weyl map a few indexes must be raised and lowered.
[a] The Weyl equation is conventionally interpreted as describing a massless particle.
The right handed field, as noted earlier, transforms as and so the complex conjugate field transforms as Applying the defining relationship, one concludes that which is exactly the same Lorentz covariance property noted earlier.
transforms in a covariant fashion; setting this to zero gives the complex two-component Majorana equation.
Similarly, the left-chiral Majorana equation (including an arbitrary phase factor
) is As noted earlier, the left and right chiral versions are related by a parity transformation.
Thus, the matched combinations of these are Lorentz covariant, and one may take as a pair of complex 2-spinor Majorana equations.
The equations are obtained from the Lagrangian densities By treating the spinor and its conjugate (denoted by
The term Weyl spinor is also frequently used in a more general setting, as an element of a Clifford module.
This general setting has multiple strengths: it clarifies their interpretation as fermions in physics, and it shows precisely how to define spin in General Relativity, or, indeed, for any Riemannian manifold or pseudo-Riemannian manifold.
This means that, as boosts and rotations are applied, the form of the equation itself does not change.
This is entirely analogous to how one might talk about a vector, and how it transforms under the rotation group, except that now, it has been adapted to the case of spinors.
, one may construct a pair of Weyl spinors as[11] and When properly examined in light of the Clifford algebra, these are naturally anti-commuting, that is, one has that
dimensional Minkowski space-time, there are only two such spinors possible, by convention labelled "left" and "right", as described above.
A more formal, general presentation of Weyl spinors can be found in the article on the spin group.
The abstract, general-relativistic form of the Weyl equation can be understood as follows: given a pseudo-Riemannian manifold
Selecting a single point on the fiber corresponds to selecting a local coordinate frame for spacetime; two different points on the fiber are related by a (Lorentz) boost/rotation, that is, by a local change of coordinates.
The electric charge arises because the Dirac field transforms under the action of the complexified spin group
This again renders it electrically neutral, but introduces a number of other quite surprising properties.