| spinor | |
|---|---|
| Name | Spinor |
| Field | Quantum mechanics; Mathematical physics |
| Introduced | 1929 |
| Notable people | Wolfgang Pauli; Paul Dirac; Élie Cartan |
| Related | Spin (physics); Clifford algebra; Dirac equation |
spinor
A spinor is a mathematical object that transforms under rotations in a way that generalizes vectors and tensors, characterized by projective representations of the rotation group and SO(n). In quantum mechanics and quantum field theory, spinors provide the minimal linear representation required to describe half-integer spin degrees of freedom such as electrons and neutrinos, and they underpin relativistic wave equations like the Dirac equation. Spinors are essential for linking algebraic structures like Clifford algebra and geometric notions such as spin structure on manifolds.
A spinor is defined algebraically as an element of a vector space carrying a representation of the spin group Spin(n), the double cover of SO(n), rather than a tensor representation of SO(n) itself. Physically, spinors encode intrinsic angular momentum (spin) in quantum systems; for example, a two-component complex spinor represents a spin-1/2 particle under nonrelativistic SU(2) rotations, while four-component Dirac spinors describe relativistic fermions under Lorentz transformations. The double-valued nature of spinor representations explains phenomena such as the need to rotate by 720° to return a fermionic state to its original phase. Key historical figures in interpreting spinors include Paul Dirac (relativistic electron theory) and Wolfgang Pauli (Pauli matrices).
Mathematically, spinors arise from the representation theory of Clifford algebra Cl(p,q) associated with a quadratic form of signature (p,q). Construction methods include using minimal left ideals of Clifford algebras, or inducing projective representations of SO(n) via the algebraic spin group Spin(n). Low-dimensional cases yield familiar objects: in three dimensions spinors correspond to two-dimensional complex representations of SU(2), generated by the Pauli matrices; in four-dimensional spacetime the relevant algebra is Cl(1,3) and spinors appear as four-component Dirac spinors that can be decomposed into two Weyl (chiral) spinors. Additional structures include Majorana spinors (real representations when allowed by signature and charge conjugation) and symplectic Majorana conditions in supersymmetric contexts. The classification of spinor modules is treated in works by Élie Cartan and in modern texts on representation theory and algebraic topology.
In nonrelativistic quantum mechanics, spinors provide the state space for intrinsic spin observables via the representation of SU(2); a spin-1/2 particle is described by a two-component complex spinor with expectation values of the Pauli matrices giving physical spin projections. Spinor wavefunctions combine with orbital degrees of freedom in the full Hilbert space used in atomic and molecular models (e.g., Pauli equation as a nonrelativistic limit including spin–orbit coupling). Quantum measurements and entanglement of spinor degrees of freedom are central in experiments of Stern–Gerlach experiment and in foundations of quantum information, such as spin qubits in quantum computing hardware (e.g., spintronics devices and NV center implementations). The transformation properties under rotations and time reversal are critical for selection rules and for classifying fermionic states in many-body systems.
In relativistic quantum field theory, spinor fields are sections of spinor bundles associated to a principal Spin(1,3) bundle over spacetime and are quantized to produce fermionic particles obeying the Fermi–Dirac statistics and canonical anticommutation relations. The prototypical equation is the Dirac equation, coupling Dirac spinors to gauge fields such as the electromagnetic field via minimal coupling; chiral symmetry and its breaking are analyzed using Weyl spinors in the context of the Standard Model with gauge groups like SU(3)×SU(2)×U(1). Anomalies related to fermion path integrals involve topological properties of spinor determinants and appear in computations by Edward Witten and others. Spinor helicity methods and spinor–helicity formalism are widely used in modern scattering amplitude calculations in perturbative quantum field theory and string theory.
Geometrically, spinors require the underlying manifold to admit a spin structure; not all manifolds do, with obstructions given by second Stiefel–Whitney classes. Spinor bundles arise in Riemannian geometry and index theory, where the Atiyah–Singer index theorem relates analytical indices of Dirac operators on spin manifolds to topological invariants. Twistor theory links spinors to complex geometry and was developed by Roger Penrose to recast aspects of field theory and gravity. In condensed matter and topological phases, spinor behavior and spin structure inform classifications of topological insulators and superconductors studied by groups like those at MIT and Microsoft Research.
Spinors model fundamental fermions (electrons, muons, quarks, neutrinos) in particle physics and are integral to calculations in quantum electrodynamics (QED) and quantum chromodynamics (QCD). In atomic physics, spinors explain fine and hyperfine structure via spin–orbit and magnetic interactions. Spinor representations underlie approaches to superconductivity (Bogoliubov–de Gennes formalism) and to relativistic electronic structure methods in computational chemistry, used in packages developed by research groups at institutions like Argonne National Laboratory. In gravitational contexts, spinor techniques are employed in supergravity and in attempts to quantize gravity, including applications in loop quantum gravity and supersymmetric model building. Experimental platforms exploiting spinor properties include spin-resolved spectroscopy, polarized beam experiments at facilities such as CERN, and spin qubit implementations in silicon quantum dot devices. Category:Mathematical physics