Weyl

Weyl is a surname and a term used in mathematics and physics, most often in reference to Hermann Weyl (1885–1955), a German mathematician and theoretical physicist. His work spanned differential geometry, representation theory, and quantum mechanics, and many concepts bear his name as a result.

Hermann Weyl contributed to the development of gauge invariance in physics, as well as foundational ideas in differential geometry and the relationship between geometry and physical theories. His proposal of a scale gauge theory, now known as Weyl geometry, sought to unify gravity and electromagnetism and influenced later developments in gauge theory and geometric methods in physics.

Several mathematical objects are named Weyl: the Weyl tensor, the trace-free part of the Riemann curvature tensor that encodes conformal curvature in a space-time; the Weyl group, a finite reflection group associated with root systems in Lie algebras and central to representation theory; the Weyl character formula, an explicit expression for the characters of irreducible representations of semisimple Lie algebras; the Weyl law, the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold; Weyl quantization, a procedure that maps classical observables to quantum operators; Weyl spinors, two-component spinor representations of the Lorentz group; and Weyl geometry, a generalization of Riemannian geometry with a length-scale connection.

In physics, Weyl fermions are massless, chiral fermions described by Weyl equations; they appear as fundamental particles in high-energy theories and as emergent quasiparticles in condensed matter systems, notably in Weyl semimetals, where low-energy excitations behave like Weyl fermions.