Weyl equation

Weyl equation
Joel Holdsworth (Joelholdsworth) · Public domain · source
Name	Weyl equation
Field	Theoretical physics
Introduced	1929
Introduced by	Hermann Weyl
Related	Dirac equation, Majorana equation, neutrino

Weyl equation The Weyl equation is a relativistic wave equation formulated for massless spin-1/2 particles, introduced by Hermann Weyl in 1929. It played a central role in the development of quantum field theory and particle physics, influencing work at institutions such as University of Göttingen, Princeton University, and Institute for Advanced Study. Weyl's formulation informed later developments by figures including Paul Dirac, Ettore Majorana, Wolfgang Pauli, Enrico Fermi, and Freeman Dyson.

Introduction

The Weyl equation arises in the context of attempts to reconcile Albert Einstein's theory of Special relativity with quantum mechanics as pursued by researchers at University of Leipzig, University of Zurich, and ETH Zürich. Hermann Weyl proposed a two-component spinor description distinct from Paul Dirac's four-component formalism; contemporaries such as Max Born, Werner Heisenberg, Erwin Schrödinger, and Wolfgang Pauli debated its implications. Experimental developments at laboratories including Cavendish Laboratory, Rutherford Laboratory, and CERN later connected Weyl's theoretical idea to observations attributed to particles like the neutrino and phenomena investigated by researchers including C. N. Yang and Robert Mills.

Mathematical formulation

The Weyl equation is written for two-component spinor fields transforming under the Lorentz group representations associated with Weyl spinors studied by mathematicians and physicists in the tradition of Élie Cartan, Hermann Weyl, and Évariste Galois-era group theory. In modern notation the equation uses Pauli matrices introduced by Wolfgang Pauli and the four-momentum operators familiar from work by P. A. M. Dirac, Paul Langevin, and Hendrik Lorentz. The equation can be expressed using objects from representation theory related to SL(2,C), Spin group, and constructions appearing in texts by Roger Penrose, Wolfgang Rindler, and Michael Peskin. The Weyl operator is linear and first-order, reflecting symmetry principles emphasized by Emmy Noether and structural analysis developed in Hilbert space methods used by John von Neumann.

Solutions and properties

Solutions of the Weyl equation describe plane waves and localized wavepackets whose dispersion relations match the lightlike characteristics central to Special relativity and analyses by Hermann Minkowski and Arthur Eddington. The spinor solutions decompose into helicity eigenstates connected to experimental asymmetries studied by Chien-Shiung Wu and implications for weak interactions explored by Tsung-Dao Lee, Chen-Ning Yang, and Sheldon Glashow. Mathematical properties such as normalization and current conservation reflect structures analyzed by Paul Dirac, Julian Schwinger, and Gerard 't Hooft; singular solutions and Green's functions link to techniques developed by Siméon Denis Poisson and George Green.

Representation theory and chirality

Weyl spinors furnish the two inequivalent chiral representations of SL(2,C) that correspond to left-handed and right-handed chirality, a distinction emphasized in the work of Steven Weinberg, Murray Gell-Mann, and Yoichiro Nambu. Chirality in Weyl theory underpins parity-violation results recorded in experiments at facilities such as Brookhaven National Laboratory and Los Alamos National Laboratory and theorized by Lev Landau and Richard Feynman. The representation-theoretic structure connects to branchings studied by Élie Cartan and to classification schemes used by Roger Penrose and Abraham Pais in particle taxonomies catalogued at CERN and in the Particle Data Group compilations.

Physical applications

Weyl's formalism has been applied to neutrino physics traced through contributions of Wolfgang Pauli, Enrico Fermi, and Bruno Pontecorvo; it guided early models of neutrino behavior in beta decay experiments associated with C. S. Wu and theoretical frameworks by Hans Bethe and Robert Oppenheimer. In condensed matter physics Weyl quasiparticles appear in topological semimetals investigated by groups at Stanford University, MIT, and University of Cambridge and experiments performed at facilities like Los Alamos National Laboratory and National Institute of Standards and Technology. Cosmological and astrophysical contexts invoking Weyl-type massless fermions were explored in work by Subrahmanyan Chandrasekhar, George Gamow, and Andrei Sakharov.

Relation to Dirac and Majorana equations

The Weyl equation is a massless limit of the Dirac equation introduced by Paul Dirac; combining left- and right-chiral Weyl spinors yields Dirac spinors central to quantum electrodynamics developed by Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman. The Majorana equation formulated by Ettore Majorana provides an alternative real representation where particles are their own antiparticles, a concept furthered in neutrino model building by Bruno Pontecorvo and Rudolf Peierls. Interrelations among Weyl, Dirac, and Majorana descriptions inform searches at Fermilab, CERN, and Super-Kamiokande and theoretical programs by Steven Weinberg and Edward Witten concerning mass generation, symmetry breaking, and beyond-Standard-Model scenarios.

Category:Quantum field theory