| Weyl equation | |
|---|---|
| Name | Weyl equation |
| Field | Quantum mechanics; Quantum field theory |
| Introduced | 1929 |
| Author | Hermann Weyl |
Weyl equation
The Weyl equation is a relativistic wave equation describing massless spin-1/2 particles, formulated by Hermann Weyl in 1929. It captures the dynamics of two-component Weyl spinor fields and underlies the concept of chirality in Quantum field theory, with direct relevance to the Standard Model of particle physics and to emergent quasiparticles in condensed matter physics such as Weyl semimetals. The equation is central to understanding helicity, anomalies, and the interplay between Lorentz symmetry and massless fermions.
The Weyl equation models elementary fermions in the massless limit and isolates chiral degrees of freedom that are otherwise combined in the Dirac equation. Its significance in particle physics stems from the chiral structure of the electroweak interaction in the Standard Model, where left-handed fermions and right-handed antifermions transform differently under SU(2) gauge symmetry. Historically, Weyl's formulation influenced the development of spinor representation theory for the Lorentz group and provided groundwork for later concepts such as chiral anomalys and neutrino physics. In condensed matter, low-energy excitations in certain crystals realize Weyl fermions as quasiparticles, connecting the equation to topological materials research pursued at institutions like Bell Labs and universities with active experimental programs.
The Weyl equation is a first-order linear partial differential equation for a two-component complex spinor ψ(x). In natural units (ħ = c = 1), the equation is commonly written as σ^μ ∂_μ ψ = 0 or explicitly (∂_t ± σ·∇)ψ = 0, where σ^μ = (I, σ^i) and σ^i are the Pauli matrices. The two choices of sign correspond to left-handed and right-handed Weyl equations (chirality eigenstates). Mathematically, Weyl spinors furnish the two inequivalent two-dimensional complex spinor representations (denoted (1/2,0) and (0,1/2)) of the SL(2,C) double cover of the Lorentz group. The absence of a mass term prevents coupling the two chiralities within this minimal representation.
The Weyl equation is a chiral reduction of the Dirac equation: a massive Dirac spinor can be decomposed into two Weyl spinors of opposite chirality, and the Dirac mass term couples these components. Conversely, combining two Weyl spinors with opposite chirality yields a Dirac fermion, as used in the description of charged leptons and quarks in the Standard Model. The Majorana equation describes neutral fermions that are their own antiparticles; a Majorana spinor can be constructed from a single Weyl spinor by imposing a real condition (Majorana condition) in appropriate dimensions. These relationships underpin model-building for neutrino masses (Dirac vs Majorana) and searches at experiments such as CERN and Super-Kamiokande.
Free solutions of the Weyl equation are plane waves ψ(x) = u(p) e^{-ip·x} with spinor amplitudes u(p) satisfying p_μ σ^μ u(p) = 0. For massless particles, chirality equals helicity: left-chiral solutions correspond to negative helicity states for positive-energy particles (and vice versa for antiparticles). Helicity eigenstates are labeled by eigenvalues ±1/2 under the spin operator aligned with momentum. These solutions are central to scattering amplitudes in quantum electrodynamics and perturbative quantum field theory calculations (e.g., spinor-helicity formalism used in N=4 supersymmetric Yang–Mills theory and collider physics at the Large Hadron Collider).
The Weyl equation is Lorentz invariant under the (1/2,0) or (0,1/2) representations of SL(2,C), preserving the causal structure of relativistic theories. It exhibits a U(1) phase symmetry yielding conserved currents when appropriately coupled, but pure Weyl fermions lack parity (P) symmetry because chirality is not invariant under spatial inversion. Charge conjugation (C) maps a Weyl spinor to the opposite chirality, so a single Weyl spinor does not possess an independent C symmetry; combined CP and time-reversal (T) considerations relate to CPT theorem constraints. The chiral character leads to quantum anomalies, notably the Adler–Bell–Jackiw anomaly (chiral anomaly), relevant to processes in quantum chromodynamics and early-universe baryogenesis scenarios studied in cosmology.
In particle physics, Weyl fermions model massless neutrinos in early theoretical work and appear effectively in high-energy limits of fermions at collider energies. The chiral structure is fundamental to electroweak theory and to mechanisms for generating fermion masses such as the Higgs mechanism. In condensed matter, Weyl equations describe low-energy quasiparticles in Weyl semimetals realized experimentally in materials like TaAs and studied using techniques such as angle-resolved photoemission spectroscopy (ARPES). These materials display topologically protected Fermi arc surface states and phenomena analogous to the chiral anomaly, measurable in magnetotransport experiments.
Quantization of Weyl fields proceeds by promoting classical spinor fields to operators satisfying anticommutation relations and constructing a Fock space of particle and antiparticle states. Lagrangian formulations use two-component notation with kinetic terms i ψ† σ^μ ∂_μ ψ; gauge couplings to groups like U(1), SU(2), and SU(3) follow the prescription of gauge theory. Regularization of chiral gauge theories requires care to preserve gauge invariance in the presence of anomalies, an issue central to the consistency of the Standard Model and addressed via anomaly cancellation conditions across fermion representations. Quantized Weyl fermions also feature in theoretical developments such as topological quantum field theory and in condensed matter field-theoretic descriptions of topological phases.
Category:Quantum mechanics Category:Quantum field theory Category:Spinors