| Weyl spinors | |
|---|---|
| Name | Weyl spinor |
| Caption | Conceptual depiction of a two-component spinor and chirality |
| Type | Mathematical object / Quantum field |
| Introduced | 1929 |
| Introduced by | Hermann Weyl |
| Related | Dirac spinor, Majorana spinor, SL(2,C), Lorentz group |
Weyl spinors
Weyl spinors are two-component spinor fields that furnish the fundamental, irreducible spin-½ representations of the proper, orthochronous Lorentz group via its double cover SL(2,C). They are central in relativistic quantum mechanics and quantum field theory as the simplest building blocks for chiral fermions, underlying the description of massless particles and the chiral structure of the Standard Model. Weyl spinors illuminate concepts such as chirality, helicity, and anomalies in gauge theories.
A Weyl spinor is a complex two-component object transforming under one of the two inequivalent fundamental representations of SL(2,C), commonly denoted (½,0) or (0,½). The two types correspond to left-handed and right-handed chiralities. In the massless limit, Weyl spinors describe particles with definite chirality, which for free particles coincides with helicity up to sign. Weyl spinors are significant because the weak interactions in the Standard Model couple only to left-chiral fermions, a discovery that led to understanding parity violation in weak interactions and informed the formulation of electroweak theory by Sheldon Glashow, Abdus Salam, and Steven Weinberg.
Mathematically, a Weyl spinor ψα (α = 1,2) is an element of the fundamental representation space of SL(2,C). Its transformation law under a matrix S ∈ SL(2,C) is ψ → Sψ for the (½,0) representation and ψ̄̇ → S̄ψ̄̇ for the conjugate (0,½). One uses the Pauli matrices σα to map between four-vectors and 2×2 Hermitian matrices via xμσμ, linking the spinor formalism to the Minkowski space vector representation of the Poincaré group. Index notation commonly employs undotted and dotted indices (α, ˙α) following conventions from texts such as those by Steven Weinberg and J. J. Sakurai.
Under the proper, orthochronous Lorentz group L↑+, the double cover SL(2,C) provides two inequivalent fundamental representations. Weyl spinors in the (½,0) representation transform as left-chiral objects, while (0,½) transform as right-chiral. Chirality is a Lorentz-invariant property for massless fields and is encoded algebraically using projection operators constructed from the gamma matrices in four-component notation or directly by the choice of representation in two-component formalism. The discovery of chirality's role in parity violation by experiments such as those by Chien-Shiung Wu highlighted the physical importance of Weyl spinors.
The Weyl equation is the relativistic first-order differential equation satisfied by a free massless two-component spinor. In natural units it reads iσμ∂μψ = 0 for a left-handed spinor and iσ̄μ∂μχ = 0 for a right-handed spinor, where σμ = (I, σi) and σ̄μ = (I, -σi). The Weyl equation is obtained as the massless limit of the Dirac equation and yields propagation at the speed of light with definite chirality and dispersion relation pμpμ = 0. Solutions are plane-wave spinors labeled by four-momentum and helicity; quantization of these solutions leads to Fock-space states used in scattering computations in quantum electrodynamics and quantum chromodynamics.
A Dirac spinor can be decomposed into two Weyl spinors of opposite chirality, providing a manifestly Lorentz-covariant way to include mass terms that couple left- and right-handed components. A Majorana spinor can be constructed from a Weyl spinor by imposing a reality (charge-conjugation) condition that identifies the spinor with its charge conjugate; this is possible in four dimensions with appropriate representations. The existence or absence of independent Weyl components determines whether mass terms are Dirac, Majorana, or forbidden by gauge symmetries; for example, the Standard Model initially contained only left-handed Weyl neutrinos before neutrino mass measurements motivated mechanisms such as the seesaw mechanism.
Weyl spinors are extensively used in model building and computations in particle physics. The left-handed lepton and quark doublets of the Standard Model are Weyl fields transforming under SU(2)L gauge symmetry of the electroweak sector developed by Glashow, Weinberg, Salam. Neutrino phenomenology—oscillations observed by experiments such as Super-Kamiokande and Sudbury Neutrino Observatory—has driven the inclusion of right-handed components or Majorana masses for Weyl neutrinos. In perturbative calculations, two-component spinor techniques are prevalent in modern amplitude methods such as spinor-helicity formalism used at collaborations like CERN and in software tools like MadGraph.
In quantum field theory, Weyl fields are quantized by expanding in mode solutions of the Weyl equation and promoting coefficients to annihilation and creation operators satisfying canonical anticommutation relations. Gauge interactions are introduced by covariant derivatives acting on Weyl multiplets charged under groups like SU(2), U(1), or SU(3). Anomalies—quantum violations of classical symmetries—are computed using Weyl fermions and were crucial in determining anomaly cancellation conditions for the Standard Model (notably by analyses by Adler and Bell and Jackiw). Modern treatments employ path integrals, regularization schemes, and renormalization group methods developed by theorists at Princeton University, Harvard University, and institutions worldwide.
Category:Spinors Category:Quantum field theory Category:Particle physics