| Weyl equation | |
|---|---|
| Name | Weyl equation |
| Field | Quantum Physics |
| Introduced | 1929 |
| Introduced by | Hermann Weyl |
| Related | Dirac equation, Majorana fermion, Chirality |
Weyl equation
The Weyl equation is a relativistic wave equation describing massless spin-1/2 fermions with definite chirality. It occupies a central role in Quantum Field Theory and particle physics as the simplest relativistic fermionic equation, underpinning descriptions of neutrinos (in some approximations), chiral anomalies, and effective models in condensed matter physics such as Weyl semimetal. The equation's clarity about handedness makes it important for understanding symmetry and conservation in fundamental interactions.
The Weyl equation was formulated by Hermann Weyl in 1929 as a two-component reduction of the Dirac equation for massless particles. Weyl's work followed early developments by Paul Dirac (Dirac equation, 1928) and built on group-theoretic methods influenced by Élie Cartan and Emmy Noether. Historically, Weyl's proposal intersected with debates on parity and the nature of the neutrino; the discovery of parity violation in weak interactions by Chien-Shiung Wu and collaborators in 1956–1957 made chiral formulations especially relevant. The Weyl formalism later re-emerged in condensed-matter contexts with the theoretical and experimental identification of Weyl fermions in materials by groups such as those at Princeton University and Max Planck Institute for Chemical Physics of Solids.
The Weyl equation is expressed in terms of two-component spinors transforming under the fundamental representations of the Lorentz group SL(2,C). In natural units (ħ = c = 1), a left-handed Weyl spinor ψ_L satisfies ∂_μ σ^μ ψ_L = 0, and a right-handed spinor ψ_R satisfies ∂_μ \bar{σ}^μ ψ_R = 0, where σ^μ = (I, σ^i) and \bar{σ}^μ = (I, -σ^i) use the Pauli matrices σ^i. The Weyl equation can be derived from the Dirac equation by setting the mass to zero and projecting with the chiral projection operators (1 ± γ^5)/2, linking to the gamma matrices algebra of Clifford algebra. Representations are labeled by (1/2,0) and (0,1/2) of SL(2,C) and relate to the concept of helicity and chirality.
Solutions of the Weyl equation are plane waves with lightlike four-momentum p^μ satisfying p^2 = 0. For a given momentum, eigenstates are characterized by helicity ±1/2. The two-component formalism yields dispersion relation E = |p| and simple spinor structures described by helicity spinors used in scattering theory and the spinor-helicity formalism. The Weyl equation admits both positive- and negative-energy solutions interpreted within quantum field theory by second quantization, giving rise to particle and antiparticle states for massless fermions. Mathematically, solutions are sections of spinor bundles over Minkowski space and extend to curved spacetimes via the spin connection and vierbein formalism in general relativity.
The Weyl equation is a special case of the Dirac equation with vanishing mass, obtained by chiral projection. Conversely, combining a left- and right-handed Weyl spinor yields a four-component Dirac spinor describing a massive fermion. The Majorana equation describes self-conjugate fermions; a Majorana spinor can be constructed from two Weyl spinors related by charge conjugation, making Weyl spinors fundamental building blocks in constructing Dirac and Majorana representations. These relationships are central in model building for neutrino mass mechanisms such as the seesaw mechanism and in classifications of fermions by representations of the Poincaré group.
In particle physics, Weyl spinors are used in the Standard Model to represent chiral fermions that participate asymmetrically in the weak interaction mediated by W and Z bosons. The historically massless approximation for the neutrino motivated Weyl descriptions until discovery of neutrino oscillations implying mass. In condensed matter, emergent low-energy excitations in Weyl semimetals behave as Weyl fermions; materials such as TaAs and experiments at facilities including Argonne National Laboratory and Stanford University probed Fermi arcs and chiral anomaly signatures. Weyl equations also underpin techniques in high-energy physics computations (helicity amplitudes), in topological phases of matter, and in the description of chiral edge modes in quantum Hall effect contexts.
The Weyl equation manifests exact chiral symmetry for massless fermions: left- and right-handed components decouple. Classical currents associated with global phase transformations yield conserved vector currents, while separate chiral currents are classically conserved but may exhibit chiral anomaly in the quantum theory when coupled to gauge fields, a phenomenon elucidated by Adler–Bell–Jackiw anomaly analyses. Discrete symmetries—parity, charge conjugation, and time reversal—act differently on Weyl spinors leading to important implications for weak interaction phenomenology and tests of symmetry breaking in particle physics.
Direct detection of fundamental Weyl fermions as elementary particles is not established; however, the chiral behavior predicted by the Weyl equation was indirectly confirmed via parity-violating weak processes and later neutrino physics. In condensed matter physics, angle-resolved photoemission spectroscopy (ARPES) and transport experiments observed Weyl nodes and chiral anomaly-related negative magnetoresistance in materials such as TaAs, NbAs, and engineered heterostructures. These observations have implications for electronic device engineering, topological quantum materials research at institutions like MIT and University of Cambridge, and for precision tests of symmetry principles in both high-energy and solid-state systems.