WeylGleichungen

WeylGleichungen are a set of first‑order relativistic field equations introduced by Hermann Weyl in 1929 to describe massless spin‑one‑half particles. They are the simplest example of a two‑component spinor field theory and constitute the massless limit of the Dirac equation. The equations are written in terms of left‑handed and right‑handed two‑component Weyl spinors, ψL and ψR, which transform independently under the proper Lorentz group. Each component satisfies a linear equation of the form i∂t ψ = ±i σ·∇ ψ, where σ denotes the Pauli matrices. The ± sign distinguishes the two chiralities and leads to opposite helicities for the corresponding particle states.

Weyl equations are Lorentz invariant and invariant under global U(1) phase transformations, providing a basis for quantum electrodynamics in the massless fermion limit. When coupled to an electromagnetic gauge field, they form the Weyl–Dirac equation, which exhibits gauge symmetry and the properties necessary for chiral quantum field theories, such as the standard model’s treatment of neutrinos. Because the chirality projector (1±γ5)/2 annihilates a mass term, Weyl fermions cannot acquire a Dirac mass without mixing left‑ and right‑handed components, which is essential for understanding neutrino oscillations and possible Majorana masses. Mathematically, Weyl equations are closely related to the representation theory of the Lorentz group, and their solutions are helicity eigenstates with definite chirality, making them central to the study of topological phases and Weyl semimetals in condensed matter physics.