Other meanings of Spinor
Mathematics & physics
A spinor is a mathematical object that represents rotations and reflections in a way suited to quantum mechanics. Unlike an ordinary vector, a spinor can require a full 720-degree rotation to return to its original state, a property expressed by the double-cover relationship between the rotation group and the spin group.1 Spinors provide the natural language for describing fermions, including electrons and quarks, and arise from Clifford algebras and the geometry of spacetime.
A spinor is an element of a representation space for a spin group, rather than an ordinary representation space for rotations. The spin group Spin(n) is a double cover of the special orthogonal group SO(n), so each spatial rotation corresponds, in general, to two spin-group elements.1 This distinction explains why a spinor may change sign after a 360-degree rotation and return to its original value only after 720 degrees.
The construction is built from a Clifford algebra, whose multiplication encodes the metric through relations involving vectors and their squared lengths. A spinor is not simply a column of numbers independent of coordinates: its components depend on a chosen representation, while the geometric object is defined by how those components transform.
In quantum mechanics, spinors describe states with half-integer angular momentum, especially spin-1/2 particles. A two-component Pauli spinor represents the spin state of a nonrelativistic electron, with rotations acting through two-by-two unitary matrices rather than the three-by-three matrices used for ordinary spatial vectors.
The sign change under a 360-degree rotation does not directly alter a single measurement probability, because an overall phase is unobservable in isolation. It can nevertheless affect interference between alternatives, making the phase physically significant. The spinor formalism also separates intrinsic spin from orbital angular momentum; their combination produces the total angular momentum used in atomic and nuclear physics.
Relativistic quantum theory uses four-component Dirac spinors to combine particle states with special relativity. The Dirac equation couples these spinors to electromagnetic fields and predicts the existence of corresponding antiparticle solutions; its matrices furnish a representation of the spacetime Clifford algebra.2 In a chiral or Weyl basis, a Dirac spinor can be decomposed into left-handed and right-handed two-component spinors.
On curved spacetime, spinors require more structure than a metric alone: the manifold must admit a suitable spin structure, which lifts the orthonormal frame bundle to a principal spin bundle. Spin connections then define covariant differentiation, allowing equations for fermions to be formulated in general relativity and in quantum field theory on curved backgrounds.
Spinors have important uses beyond elementary-particle wavefunctions. In differential geometry, the Dirac operator acts on sections of a spinor bundle and links analysis to topology; its index can encode global information about a manifold. Spinors also occur in the geometry of minimal surfaces, special holonomy, supergravity, and the classification of possible fermion representations.
Not every manifold admits spinors globally. The obstruction is captured by the second Stiefel–Whitney class; when it does not vanish, one may sometimes replace a spin structure with a related spinc structure. Another subtlety is that the number and type of spinor components depend on dimension and signature: real, complex, and quaternionic forms can occur, and chirality is available only in appropriate even dimensions.
The precise dimension, reality condition, chirality, and number of components of a spinor depend on the underlying metric signature and representation convention.
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