| Weyl | |
|---|---|
| Name | Hermann Weyl |
| Birth date | 1885 |
| Death date | 1955 |
| Nationality | German |
| Fields | Mathematics, Theoretical physics |
| Institutions | ETH Zurich, Princeton University, Institute for Advanced Study |
| Alma mater | University of Göttingen |
| Known for | Weyl equation, Weyl quantization, Weyl tensor, group representation theory |
Weyl
Hermann Weyl's ideas pervade modern Quantum Physics through concepts that bear his name, including the Weyl equation, Weyl fermions, Weyl quantization, and notions of scale invariance and gauge symmetry. Weyl's contributions bridge mathematical physics, group theory, and quantum field theory, underpinning theoretical descriptions of chiral fermions, topological materials, and anomalies.
Weyl fermions are massless, chiral solutions of the Weyl equation that act as low-energy quasiparticles in certain crystalline solids. In condensed matter physics, Weyl semimetal phases host band crossings—Weyl nodes—that act as monopoles of Berry curvature in momentum space, characterized by an integer topological charge (chirality). Experimental realizations appeared in materials such as TaAs and related transition-metal pnictides, discovered using angle-resolved photoemission spectroscopy (ARPES) and transport probes showing the chiral anomaly in solids. Weyl semimetals connect concepts from high-energy particle physics (chiral fermions and anomalies) to topology and materials science, enabling studies of surface Fermi arc states, anomalous Hall effects, and magnetotransport phenomena tied to Berry phase physics.
The Weyl equation is a two-component, first-order relativistic wave equation for massless spin-1/2 particles. Derived as a chiral reduction of the Dirac equation, it employs Pauli matrices to encode spin and is invariant under proper orthochronous Lorentz transformations for representations of definite chirality. In quantum field theory, Weyl spinors furnish the building blocks of chiral fermion fields, appearing in the Standard Model's description of left-handed neutrinos prior to neutrino mass terms. The Weyl equation clarifies helicity versus chirality distinctions and underlies theoretical analyses of anomalies, CP violation, and the construction of gauge-invariant Lagrangians for massless fermions.
Weyl spinors transform in the fundamental, two-dimensional complex representations of the SL(2,C) double cover of the Lorentz group. These chiral spinor representations—often denoted (1/2,0) and (0,1/2)—are inequivalent and irreducible, in contrast to the four-component Dirac spinor which is their direct sum. Weyl's group-theoretic approach influenced the application of representation theory to particle classification and the construction of relativistic quantum fields. In gauge theories such as Quantum electrodynamics and the Electroweak interaction in the Standard Model, Weyl spinors are used to assign chiral quantum numbers, implement left–right asymmetry, and derive anomaly cancellation conditions (e.g., through assignments across fermion generations and gauge groups like SU(2), U(1)).
Weyl introduced the idea of a local rescaling symmetry of the metric—now called Weyl symmetry or local scale invariance—linking scale transformations to a gauge field. Although Weyl's original geometric program was superseded by General relativity, the concept evolved into modern notions of conformal symmetry and dilatation currents in conformal field theory (CFT) and beyond. In quantum contexts, Weyl symmetry constrains renormalization group flows and the structure of effective actions; models with exact or approximate Weyl invariance feature in approaches to quantum gravity, cosmology, and attempts at scale-invariant extensions of the Standard Model. Couplings of Weyl gauge fields to matter illustrate tensions between local scale symmetry and quantum anomalies.
The Weyl anomaly (trace anomaly) arises when classical scale or conformal invariance is broken by quantum effects after regularization and renormalization. In curved spacetime and renormalizable quantum field theory, the expectation value of the stress–energy tensor acquires a nonzero trace proportional to geometric invariants (e.g., the Ricci scalar and Weyl tensor squared) and beta functions of coupling constants. The Weyl anomaly is central to the study of anomalies in gauge theories, the derivation of central charges in two-dimensional CFTs (related to the Virasoro algebra), and holographic computations via the AdS/CFT correspondence. It affects the consistency conditions for quantized Weyl-invariant models and informs calculations of entanglement entropy and trace Ward identities.
Weyl quantization is a systematic map between classical phase-space functions and operators in Hilbert space, foundational to deformation quantization and phase-space quantum mechanics. Introduced by Weyl and later formalized by Wigner and Moyal, the Weyl transform yields the Wigner quasi-probability distribution and the Moyal bracket as a deformation of the Poisson bracket. These tools provide semiclassical expansions, operator ordering prescriptions, and computational techniques for quantum optics, quantum chaos, and quantization on manifolds. Weyl quantization connects to mathematical structures such as the Heisenberg group, symplectic geometry, and pseudodifferential operator theory, and underlies numerical phase-space methods used in quantum transport and many-body approximations.
Category:Quantum physics Category:Mathematical physics Category:Hermann Weyl